Fundamentals of Acoustics

FUNDAMENTALS OF ACOUSTICS
AND NOISE CONTROL
Finn Jacobsen, Torben Poulsen, Jens Holger Rindel,

Anders Christian Gade and Mogens Ohlrich
Printed at Ørsted • DTU, Technical University of Denmark

August 2006 Note no 31200
ii
CONTENTS
Page
1 An elementary introduction to acoustics....................................................................1

Finn Jacobsen
1.1 Introduction ....................................................................................................................1
1.2 Fundamental acoustic concepts......................................................................................1
1.2.1 Plane sound waves .............................................................................................4

1.2.2 Spherical sound waves .....................................................................................12
1.3 Acoustic measurements................................................................................................15
1.3.1 Frequency analysis ...........................................................................................15

1.3.2 Levels and decibels .........................................................................................18
1.3.3 Noise measurement techniques and instrumentation .......................................21
1.4 The concept of impedance............................................................................................27
1.5 Sound energy, sound intensity, sound power and sound absorption............................31
1.5.1 The energy in a sound field ..............................................................................32

1.5.2 Sound absorption..............................................................................................35
1.6 Radiation of sound .......................................................................................................36
1.6.1 Point sources ....................................................................................................36

1.6.2 Sound radiation from a circular piston in an infinite baffle .............................41
1.7 References ....................................................................................................................49
1.8 Bibliography.................................................................................................................50
1.9 Appendix: Complex notation .......................................................................................51
2 Ear, Hearing and Speech ...........................................................................................55

Torben Poulsen
2.1 Introduction ..................................................................................................................55
2.2 The ear..........................................................................................................................55
2.2.1 The outer ear.....................................................................................................56

2.2.2 The middle ear..................................................................................................56
iii
2.2.3 The inner ear.....................................................................................................57
2.2.4 The frequency analyzer at the Basilar membrane ............................................59
2.3 Human hearing .............................................................................................................61
2.3.1 The hearing threshold.......................................................................................61

2.3.2 Audiogram........................................................................................................62
2.3.3 Loudness level..................................................................................................63
2.4 Masking........................................................................................................................64
2.4.1 Complete masking............................................................................................65

2.4.2 Partial masking.................................................................................................67
2.4.3 Forward masking..............................................................................................67
2.4.4 Backward masking ...........................................................................................67
2.5 Loudness.......................................................................................................................68
2.5.1 The loudness curve...........................................................................................68

2.5.2 Temporal integration ........................................................................................69
2.5.3 Measurement of loudness.................................................................................69
2.6 The auditory filters .......................................................................................................71
2.6.1 Critical bands....................................................................................................71

2.6.2 Equivalent rectangular bands ...........................................................................72
2.7 Speech ..........................................................................................................................73
2.7.1 Speech production ............................................................................................73

2.7.2 Speech spectrum, speech level .........................................................................74
2.7.3 Speech intelligibility ........................................................................................75
2.8 References ....................................................................................................................78
3 An introduction to room acoustics............................................................................81

Jens Holger Rindel
3.1 Sound waves in rooms..................................................................................................81
3.1.1 Standing waves in a rectangular room .............................................................81

3.1.2 Transfer function in a room..............................................................................83
3.1.3 Density of natural frequencies..........................................................................83
3.2 Statistical room acoustics .............................................................................................85
3.2.1 The diffuse sound field.....................................................................................85
iv
3.2.2 Incident sound power on a surface...................................................................86
3.2.3 Equivalent absorption area ...............................................................................87
3.2.4 Energy balance in a room.................................................................................87
3.2.5 Reverberation time. Sabine’s formula..............................................................88
3.2.6 Stationary sound field in a room. Reverberation distance ...............................89
3.3 Geometrical room acoustics .........................................................................................92
3.3.1 Sound rays and a general reverberation formula..............................................92

3.3.2 Sound absorption in the air...............................................................................93
3.3.3 Sound reflections and image sources ...............................................................94
3.3.4 Reflection density in a room ............................................................................95
3.4 Room acoustical design................................................................................................96
3.4.1 Choice of room dimensions..............................................................................96

3.4.2 Reflection control.............................................................................................96
3.4.3 Calculation of reverberation time.....................................................................97
3.4.4 Reverberation time in non-diffuse rooms.........................................................98
3.4.5 Optimum reverberation time and acoustic regulation of rooms.......................99
3.4.6 Measurement of reverberation time ...............................................................100
3.5 References ..................................................................................................................101
4 Sound absorbers and their application in room design ........................................113

Anders Christian Gade
4.1 Introduction ................................................................................................................103
4.2 The room method for measurement of sound absorption ..........................................103
4.3 Different types of sound absorbers.............................................................................104
4.3.1 Porous absorbers ............................................................................................105

4.3.2 Membrane absorbers ......................................................................................106
4.3.3 Resonator absorbers .......................................................................................108
4.4 Application of sound absorbers in room acoustic design...........................................109
4.5 References ..................................................................................................................112
5 An introduction to sound insulation .......................................................................113

Jens Holger Rindel
5.1 The sound transmission loss.......................................................................................113
5.1.1 Definition .......................................................................................................113
v
5.1.2 Sound insulation between two rooms.............................................................113
5.1.3 Measurement of sound insulation ..................................................................114
5.1.4 Multi-element partitions and apertures ..........................................................114
5.2 Single leaf constructions ............................................................................................117
5.2.1 Sound transmission through a solid material .................................................117

5.2.2 The mass law..................................................................................................119
5.2.3 Sound insulation at random incidence ...........................................................120
5.2.4 The critical frequency.....................................................................................121
5.2.5 A general model of sound insulation of single constructions ........................122
5.3 Double leaf constructions...........................................................................................123
5.3.1 Sound transmission through a double construction........................................123

5.3.2 The mass-air-mass resonance frequency........................................................124
5.3.3 A general model of sound insulation of double constructions .......................124
5.4 Flanking transmission ................................................................................................126
5.5 Enclosures ..................................................................................................................127
5.6 Impact sound insulation .............................................................................................127
5.7 Single-number rating of sound insulation ..................................................................128
5.7.1 The weighted sound reduction index .............................................................128

5.7.2 The weighted impact sound pressure level ....................................................129
5.8 Requirements for sound insulation.............................................................................131
5.9 References ..................................................................................................................131
6 Mechanical vibration and structureborne sound..................................................133

Mogens Ohlrich
6.1 Introduction ................................................................................................................133
6.1.1 Sources of vibration .......................................................................................134

6.1.2 Measurement quantities..................................................................................134
6.1.3 Linear mechanical systems............................................................................ 135
6.2 Simple mechanical resonators....................................................................................136
6.2.1 Equation of motion for simple resonator........................................................137

6.2.2 Forced harmonic response of simple resonator..............................................138
6.2.3 Frequency response functions ........................................................................145
vi
6.2.4 Forced vibration caused by motion excitation ...............................................147
6.3 Vibration and waves in continuous systems ..............................................................148
6.3.1 Longitudinal waves ........................................................................................149

6.3.2 Shear waves....................................................................................................150
6.3.3 Bending waves ...............................................................................................151
6.3.4 Input mobilities of infinite systems................................................................152
6.4 Vibration isolation and power transmission...............................................................153
6.4.1 Estimation of spring stiffness and natural frequency .....................................154

6.4.2 Transmission of power in rigidly coupled systems........................................155
6.4.3 Vibration isolated source................................................................................156
6.4.4 Design considerations for resilient elements..................................................159
6.5 References ..................................................................................................................162
List of symbols .......................................................................................................................163
Index…...................................................................................................................................167
vii
viii
1. AN ELEMENTARY INTRODUCTION TO ACOUSTICS
Finn Jacobsen
1.1 INTRODUCTION
Acoustics is the science of sound, that is, wave motion in gases, liquids and solids,
and the effects of such wave motion. Thus the scope of acoustics ranges from fundamental
physical acoustics to, say, bioacoustics, psychoacoustics and music, and includes technical
fields such as transducer technology, sound recording and reproduction, design of theatres
and concert halls, and noise control.
The purpose of this chapter is to give an introduction to fundamental acoustic con-
cepts, to the physical principles of acoustic wave motion, and to acoustic measurements.
1.2 FUNDAMENTAL ACOUSTIC CONCEPTS
One of the characteristics of fluids, that is, gases and liquids, is the lack of constraints
to deformation. Fluids are unable to transmit shearing forces, and therefore they react against
a change of shape only because of inertia. On the other hand a fluid reacts against a change in
the volume with a change of the pressure. Sound waves are compressional oscillatory distur-
bances that propagate in a fluid. The waves involve molecules of the fluid moving back and
forth in the direction of propagation (with no net flow), accompanied by changes in the pres-
sure, density and temperature; see figure 1.2.1. The sound pressure, that is, the difference be-
tween the instantaneous value of the total pressure and the static pressure, is the quantity we
hear. It is also much easier to measure the sound pressure than the other quantities. Note that
sound waves are longitudinal waves, unlike bending waves on a beam or waves on a
stretched string, which are transversal waves in which the particles move back and forth in a
direction perpendicular to the direction of propagation.
Figure 1.2.1 Fluid particles in the sound field generated by a pulsating sphere. (From ref. [1].)
In most cases the oscillatory changes undergone by the fluid are extremely small. One
can get an idea about the orders of magnitude of these changes by considering the variations
1
in air corresponding to a sound pressure level1 of 120 dB, which is a very high sound pressure
level, close to the threshold of pain. At this level the fractional pressure variations are about
2 × 10 −4 , the fractional changes of the density are about 1.4 × 10 −4 , the oscillatory changes of
the temperature are less than 0.02 °C, and the particle velocity2 is about 50 mm/s, which at
1000 Hz corresponds to a displacement of less than 8 µm . In fact at 1000 Hz the particle dis-
placement at the threshold of hearing is less than the diameter of a hydrogen atom!3
Sound waves exhibit a number of phenomena that are characteristics of waves; see
figure 1.2.2. Waves propagating in different directions interfere; waves will be reflected by a
rigid surface and more or less absorbed by a soft one; they will be scattered by small obsta-
cles; because of diffraction there will only partly be shadow behind a screen; and if the me-
dium is inhomogeneous for instance because of temperature gradients the waves will be re-
fracted, which means that they change direction as they propagate. The speed with which
sound waves propagate in fluids is independent of the frequency, but other waves of interest
in acoustics, bending waves on plates and beams, for example, are dispersive, which means
that the speed of such waves depends on the frequency content of the waveform.
Figure 1.2.2 Various wave phenomena.
A mathematical description of the wave motion in a fluid can be obtained by combin-

ing equations that express the facts that i) mass is conserved, ii) the local longitudinal force
caused by a difference in the local pressure is balanced by the enertia of the medium, and iii)
sound is very nearly an adiabatic phenomenon, that is, there is no flow of heat. The observa-
tion that most acoustic phenomena involve perturbations that are several orders of magnitude
smaller than the equilibrium values of the medium makes it possible to simplify the mathe-
matical description by neglecting higher-order terms. The result is the linearised wave equa-
tion. This is a second-order partial differential equation that, expressed in terms of the sound
pressure p, takes the form
∂2 p ∂2 p ∂2 p 1 ∂2 p
+ + = (1.2.1)
∂x 2 ∂y 2 ∂z 2 c 2 ∂t 2
1
See section 1.3.2 for a definition of the sound pressure level.
2
The concept of fluid particles refers to a macroscopic average, not to individual molecules; therefore
the particle velocity can be much less than the velocity of the molecules.
3
At these conditions the fractional pressure variations amount to about 2 × 10−10 . By comparison, a
change in altitude of one metre gives rise to a fractional change in the static pressure that is far and away larger,
about 10-4.
2
in a Cartesian coordinate system.4 The physical unit of the sound pressure is pascal (1 Pa = 1
Nm-2). As we shall see later the quantity
c = KS ρ (1.2.2a)
is the speed of sound. The quantity Ks is the adiabatic bulk modulus and ρ is the (equilibrium)
density of the medium. For gases, Ks = γp0, where ( is the ratio of the specific heat at con-
stant pressure to that at constant volume ( 1.401 for air) and p0 is the static pressure (
101.3 kPa for air under normal ambient conditions). The adiabatic bulk modulus can also be
expressed in terms of the gas constant R ( 287 J·kg-1K-1 for air), the absolute temperature T,
and the equilibrium density of the medium,
c = γ p0 ρ = γ RT , (1.2.2b)
which shows that the equilibrium density of a gas can be written as

ρ = p0 RT . (1.2.3)
At 293.15 K = 20°C the speed of sound in air is 343 m/s. Under normal ambient conditions
(20°C, 101.3 kPa) the density of air is 1.204 kgm-3. Note that the speed of sound of a gas does
not depend on the static pressure.
Adiabatic compression
Because the process is adiabatic, the fractional pressure variations in a small cavity driven by a vibrat-
ing piston, say, a pistonphone for calibrating microphones, equal the fractional density variations multiplied by
the ratio of specific heats γ . The physical explanation for the ‘additional’ pressure is that the pressure in-
crease/decrease caused by the reduced/expanded volume of the cavity is accompanied by an increase/decrease
of the temperature, which increases/reduces the pressure even further. The fractional variations in the density are
of course identical with the fractional change of the volume (except for the sign); therefore,
p ∆ρ ∆V
=γ = −γ .
p0 ρ V
In section 1.4 we shall derive a relation between the volume velocity (= the volume displacement ∆V per unit
of time) and the resulting sound pressure.
Figure 1.2.3 A small cavity driven by a vibrating piston.
The linearity of eq. (1.2.1) is due to the absence of higher-order terms in p in combi-
nation with the fact that ∂ 2 ∂x 2 and ∂ 2 ∂t 2 are linear operators.5 This is an extremely impor-
4
The left-hand side of eq. (1.2.1) is the Laplacian of the sound pressure, that is, the divergence of the
gradient. A negative value of this quantity at a certain point implies that the gradient converges towards the
point, indicating a high local value. The wave equation states that this high local pressure tends to decrease.
5
This follows from the fact that ∂ 2 ( p1 + p2 ) ∂t 2 = ∂ 2 p1 ∂t 2 + ∂ 2 p2 ∂t 2 .
3
tant property. It implies that a sinusoidal source will generate a sound field in which the pres-
sure at all positions varies sinusoidally. It also implies linear superposition: sound waves do
not interact, they simply pass through each other (see figure 1.2.5).6
The diversity of possible sound fields is of course enormous, which leads to the con-
clusion that we must supplement eq. (1.2.1) with some additional information about the
sources that generate the sound field, surfaces that reflect or absorb sound, objects that scatter
sound, etc. This information is known as the boundary conditions. The boundary conditions
are often expressed in terms of the particle velocity. For example, the normal component of
the particle velocity u is zero on a rigid surface. Therefore we need an additional equation
that relates the particle velocity to the sound pressure. This relation is known as Euler’s equa-
tion of motion,
∂u
ρ + ∇p = 0, (1.2.4)
∂t
which is simply Newton’s second law of motion for a fluid. The operator ∇ is the gradient
(the spatial derivative ( ∂ ∂x , ∂ ∂y , ∂ ∂z )). Note that the particle velocity is a vector, unlike
the sound pressure, which is a scalar.
Sound in liquids
The speed of sound is much higher in liquids than in gases. For example, the speed of sound in water is
about 1500 ms-1. The density of liquids is also much higher; the density of water is about 1000 kgm-3. Both the
density and the speed of sound depend on the static pressure and the temperature, and there are no simple gen-
eral relations corresponding to eqs. (1.2.2b) and (1.2.3).
1.2.1 Plane sound waves

The plane wave is a central concept in acoustics. Plane waves are waves in which any
acoustic variable at a given time is a constant on any plane perpendicular to the direction of
propagation. Such waves can propagate in a duct. In a limited area at a distance far from a
source of sound in free space the curvature of the spherical wavefronts is negligible and the
waves can be regarded as locally plane.
Figure 1.2.4 The sound pressure in a plane wave of arbitrary waveform at two different instants of time.
The plane wave is a solution to the one-dimensional wave equation,

∂2 p 1 ∂2 p
= , (1.2.5)
∂x 2 c 2 ∂t 2
cf. equation (1.2.1). It is easy to show that the expression
6
At very high sound pressure levels, say at levels in excess of 140 dB, the linear approximation is no
longer adequate. This complicates the analysis enormously. Fortunately, we can safely assume linearity under
practically all circumstances encountered in daily life.
4
p = f1 (ct − x) + f 2 (ct + x), (1.2.6)
where f1 and f2 are arbitrary functions, is a solution to eq. (1.2.5), and it can be shown this is
the general solution. Since the argument of f1 is constant if x increases as ct it follows that the
first term of this expression represents a wave that propagates undistorted and unattenuated in
the x-direction with constant speed, c, whereas the second term represents a similar wave
travelling in the opposite direction. See figures 1.2.4 and 1.2.5.
Figure 1.2.5 Two plane waves travelling in opposite directions are passing through each other.
The special case of a harmonic plane progressive wave is of great importance. Har-
monic waves are generated by sinusoidal sources, for example a loudspeaker driven with a
pure tone. A harmonic plane wave propagating in the x-direction can be written
⎛ω ⎞
p = p1 sin ⎜ (ct − x) + ϕ ⎟ = p1 sin(ωt − kx + ϕ ), (1.2.7)
⎝c ⎠
where ω = 2πf is the angular (or radian) frequency and k = ω c is the (angular) wavenum-
ber. The quantity p1 is known as the amplitude of the wave and φ is a phase angle. At any po-
sition in this sound field the sound pressure varies sinusoidally with the angular frequency ω,
and at any fixed time the sound pressure varies sinusoidally with x with the spatial period
c 2πc 2π
λ= = = . (1.2.8)
f ω k
The quantity λ is the wavelength, which is defined as the distance travelled by the wave in
one cycle. Note that the wavelength is inversely proportional to the frequency. At 1000 Hz
the wavelength in air is about 34 cm. In rough numbers the audible frequency range goes
from 20 Hz to 20 kHz (see section 2.3), which leads to the conclusion that acousticians are
faced with wavelengths in the range from 17 m at the lowest audible frequency to 17 mm at
the highest audible frequency (in air). Since the efficiency of a radiator of sound or the effect
of an obstacle on the sound field depends very much on its size expressed in terms of the
acoustic wavelength, it can be realised that the wide frequency range is one of the challenges
5
in acoustics. It simplifies the analysis enormously if the wavelength is very long or very short
compared with typical dimensions.
Figure 1.2.6 The sound pressure in a plane harmonic wave at two different instants of time.
Sound fields are often studied frequency by frequency. As already mentioned, linear-
ity implies that a sinusoidal source with the frequency ω will generate a sound field that var-
ies harmonically with this frequency at all positions.7 Since the frequency is given, all that
remains to be determined is the amplitude and phase at all positions. This leads to the intro-
duction of the complex exponential representation, where the sound pressure is written as a
complex function of the position multiplied with a complex exponential. The former function
takes account of the amplitude and phase, and the latter describes the time dependence. Thus
at any given position the sound pressure can be written as a complex function of the form8
pˆ = A e jωt = A e jϕ e jωt = A e j(ωt +ϕ ) (1.2.9)
(where φ is the phase of the complex amplitude A), and the real, physical sound pressure is
the real part of the complex pressure,
{ }
p = Re { pˆ } = Re A e j(ωt +ϕ ) = A cos(ωt + ϕ ). (1.2.10)
Since the entire sound field varies as ejωt, the operator ∂ ∂t can be replaced by jω
(because the derivative of ejωt with respect to time is jωejωt),9 and the operator ∂ 2 ∂t 2 can be
replaced by -ω2. It follows that Euler’s equation of motion can now be written
jωρuˆ + ∇pˆ = 0, (1.2.11)
and the wave equation can be simplified to
∂ 2 pˆ ∂ 2 pˆ ∂ 2 pˆ
+ + + k 2 pˆ = 0, (1.2.12)
∂x 2 ∂y 2 ∂z 2
which is known as the Helmholtz equation. See the Appendix for further details about com-
plex representation of harmonic signals. We note that the use of complex notation is mathe-
matically very convenient, which will become apparent later.
Written with complex notation the equation for a plane wave that propagates in the x-
7
If the source emitted any other signal than a sinusoidal the waveform would in the general case
change with the position in the sound field, because the various frequency components would change amplitude
and phase relative to each other. This explains the usefulness of harmonic analysis.
8
Throughout this note complex variables representing harmonic signals are indicated by carets.
9
The sign of the argument of the exponential is just a convention. The ejωt convention is common in
electronic engineering, in audio and in most areas of acoustics. The alternative convention e-jωt is favoured by
mathematicians, physicists and acousticians concerned with sound propagation. With the alternative sign con-
vention ∂ ∂t should obviously be replaced by -jω. Mathematicians and physicists also tend to prefer ‘i’ to ‘j’.
6
direction becomes
pˆ = pi e j(ωt − kx ) . (1.2.13)
Equation (1.2.11) shows that the particle velocity is proportional to the gradient of the pres-
sure. It follows that the particle velocity in the plane propagating wave given by eq. (1.2.13)
is
1 ∂pˆ k p pˆ
uˆ x = − = pi e j(ωt − kx ) = i e j(ωt − kx ) = . (1.2.14)
jωρ ∂x ωρ ρc ρc
Thus the sound pressure and the particle velocity are in phase in a plane propagating wave
(see also figure 1.2.10), and the ratio of the sound pressure to the particle velocity is ρc, the
characteristic impedance of the medium. As the name implies, this quantity describes an im-
portant acoustic property of the fluid, as will become apparent later. The characteristic im-
pedance of air at 20°C and 101.3 kPa is about 413 kg·m-2s-1.
Figure 1.2.7 A semi-infinite tube driven by a piston.
Example 1.2.1
An semi-infinite tube is driven by a piston with the vibrational velocity U as shown in figure 1.2.7.
Because the tube is infinite there is no reflected wave, so the sound field can be written
pi j(ωt − kx )
pˆ ( x) = pi e j(ωt − kx ) , uˆ x ( x) = e .
ρc
The boundary condition at the piston implies that the particle velocity equals the vibrational velocity of the pis-
ton:
p
uˆ x (0) = i e jωt = Ue jωt .
ρc
It follows that the sound pressure generated by the piston is
pˆ ( x) = U ρ ce j(ωt − kx ) .
The general solution to the one-dimensional Helmholtz equation is

pˆ = pi e j(ωt − kx ) + pr e j(ωt + kx ) , (1.2.15)
which can be identified as the sum of a wave that travels in the positive x-direction and a
wave that travels in the opposite direction (cf. eq. (1.2.6)). The corresponding expression for
the particle velocity becomes, from eq. (1.2.11),
7
1 ∂pˆ k k
uˆ x = − = pi e j(ωt − kx ) − pr e j(ωt + kx )
jωρ ∂x ωρ ωρ
(1.2.16)
p p
= i e j(ωt − kx ) − r e j(ωt + kx ) .
ρc ρc
It can be seen that whereas pˆ = uˆ x ρc in a plane wave that propagates in the positive x-
direction, the sign is the opposite, that is, pˆ = −uˆ x ρc , in a plane wave that propagates in the
negative x-direction. The reason for the change in the sign is that the particle velocity is a
vector, unlike the sound pressure, so û x is a vector component. It is also worth noting that the
general relation between the sound pressure and the particle velocity in this interference field
is far more complicated than in a plane propagating wave.
Figure 1.2.8 Sound pressure in a wave that is reflected from a rigid surface. (Adapted from ref. [2].)
A plane wave that strikes a plane rigid surface perpendicular to the direction of
propagation will be reflected. This phenomenon is illustrated in figure 1.2.8, which shows
how an incident transient disturbance is reflected. Note that the normal component of the
gradient of the pressure is identically zero on the surface for all values of t. This is a conse-
quence of the fact that the boundary condition at the surface implies that the particle velocity
must equal zero here, cf. eq. (1.2.4).
However, it is easier to analyse the phenomenon assuming harmonic waves. In this
case the sound field is given by the general expressions (1.2.15) and (1.2.16), and our task is
to determine the relation between pi and pr from the boundary condition at the surface, say at
x = 0. As mentioned, the rigid surface implies that the particle velocity must be zero here,
which with eq. (1.2.16) leads to the conclusion that pi = pr , so the reflected wave has the
same amplitude as the incident wave. Equation (1.2.15) now becomes
pˆ = pi (e j(ωt − kx ) + e j(ωt + kx ) ) = pi (e − j kx + e j kx )e jωt = 2 pi cos kx e jωt , (1.2.17)
and eq. (1.2.16) becomes
2 pi
uˆ x = − j sin kx e jωt . (1.2.18)
ρc
8
Note that the amplitude of the sound pressure is doubled on the surface (cf. figure
1.2.8). Note also the nodal planes10 where the sound pressure is zero at x = - λ/4, x = - 3λ/4,
etc., and the planes where the particle velocity is zero at x = - λ/2, x = - λ, etc. The interfer-
ence of the two plane waves travelling in opposite directions has produced a standing wave
pattern, shown in figure 1.2.9. The physical explanation of the fact that the sound pressure is
identically zero at a distance of a quarter of a wavelength from the reflecting plane is that the
incident wave must travel a distance of half a wavelength before it returns to the same point;
accordingly the incident and reflected waves are in antiphase (that is, 180° out of phase), and
since they have the same amplitude they cancel each other. This phenomenon is called de-
structive interference. At a distance of half a wavelength from the reflecting plane the inci-
dent wave must travel one wavelength before it returns to the same point. Accordingly, the
sound pressure is doubled here (constructive interference). Another interesting observation
from eqs. (1.2.17) and (1.2.18) is that the resulting sound pressure and particle velocity at any
position are temporarily 90° out of phase (if the sound pressure as a function of time is a co-
sine then the particle velocity is a sine). As we shall see later this indicates that there is no net
flow of sound energy towards the rigid surface. See also figure 1.2.10.
Figure 1.2.9 Standing wave pattern caused by reflection from a rigid surface at x = 0.
Example 1.2.2
The standing wave phenomenon can be observed in a tube terminated by a rigid cap. When the length
of the tube, l, equals an odd-numbered multiple of a quarter of a wavelength the sound pressure is zero at the
input, which means that it would take very little force to drive a piston here. This is an example of an acoustic
resonance. In this case it occurs at the frequency
c
f0 = ,
4l
and at odd-numbered multiples of this frequency, 3f0, 5f0, 7f0, etc. Note that the resonances are harmonically
related. This means that if some mechanism excites the tube the result will be a musical sound with the funda-
mental frequency f and overtones corresponding to odd-numbered harmonics.11
10
A node on, say, a vibrating string is a point that does not move, and an antinode is a point with
maximum displacement. By analogy, points in a standing wave at which the sound pressure is identically zero
are called pressure nodes. In this case the pressure nodes coincide with velocity antinodes.
11
A musical note is not a pure (sinusoidal) tone but a periodic signal consisting of the fundamental and
a number of its harmonics, also called partials. The n’th harmonic (or partial) is also called the (n-1)’th over-
tone, and the fundamental is the first harmonic. The pitch of the musical tone is determined by the fundamental
frequency, which is also the distance between the harmonic components.
9
Standing waves in tubes are of interest with many musical instruments. For example, closed organ
pipes are tubes closed at one end and driven at the other, open end, and such pipes have only odd-numbered
harmonics. See also example 1.4.4.
The ratio of pr to pi is the (complex) reflection factor R. The amplitude of this quantity
describes how well the reflecting surface reflects sound. In the case of a rigid plane R = 1, as
we have seen, which implies perfect reflection with no phase shift, but in the general case of
a more or less absorbing surface R will be complex and less than unity (|R| ≤ 1), indicating
partial reflection with a phase shift at the reflection plane.
If we introduce the reflection factor in eq. (1.2.15) it becomes
pˆ = pi ( e j(ωt − kx ) + R e j(ωt + kx ) ) , (1.2.19)
from which it can be seen that the amplitude of the sound pressure varies with the position in
the sound field. When the two terms in the parenthesis are in phase the sound pressure as-
sumes its maximum value,
pmax = pi (1 + R ) , (1.2.20a)
and when they are in antiphase the sound pressure assumes the minimum value
pmin = pi (1 − R ) . (1.2.20b)
The ratio of pmax to pmin is called the standing wave ratio,

pmax 1 + R
s= = . (1.2.21)
pmin 1 − R
From eq. (1.2.21) it follows that
s −1
R= , (1.2.22)
s +1
which leads to the conclusion that it is possible to determine the acoustic properties of a ma-
terial by exposing it to normal sound incidence and measuring the standing wave ratio in the
resulting interference field. See also section 1.5.
Figure 1.2.10 Spatial distributions of instantaneous sound pressure and particle velocity at two different in-
stants of time. (a) Case with no reflection (R = 0).
10
Figure 1.2.10 – contd. (b) Case with partial reflection from a soft surface; (c) case with perfect reflection from a
rigid surface (R = 1). (From ref. [3].)
Sound transmission between two fluids
Figure 1.2.11 Reflection and transmission of a plane wave incident on the interface between two fluids.
When a sound wave in one fluid is incident on the boundary of another fluid, say, a sound wave in air
is incident on the surface of water, it will be partly reflected and partly transmitted. For simplicity let us assume
that a plane wave in fluid 1 strikes the surface of fluid 2 at normal incidence as shown in figure 1.2.11. Antici-
pating a reflected wave we can write
pˆ1 = pi e j(ωt − kx ) + pr e j(ωt + kx )
for fluid 1, and

pˆ 2 = pt e j(ωt − kx )
for fluid 2. There are two boundary conditions at the interface: the sound pressure must be the same in fluid 1
and in fluid 2 (otherwise there would be a net force), and the particle velocity must be the same in fluid 1 and in
fluid 2 (otherwise the fluids would not remain in contact). It follows that
11
pi − pr p
pi + pr = pt and = t .
ρ1c1 ρ 2 c2
Combining these equations gives
pr ρ c − ρ1c1
=R= 2 2 ,
pi ρ 2 c2 + ρ1c1
which shows that the wave is almost fully reflected in phase ( R 1 ) if ρ2c2 >> ρ1c1, almost fully reflected in
antiphase ( R −1 ) if ρ2c2 << ρ1c1, and not reflected at all if ρ2c2 = ρ1c1, irrespective of the individual properties
of c1, c2, ρ1 and ρ2.
Because of the significant difference between the characteristic impedances of air and water (the ratio
is about 1: 3600) a sound wave in air that strikes a surface of water at normal incidence is almost completely
reflected, and so is a sound wave that strikes the air-water interface from the water, but in the latter case the
phase of the reflected wave is reversed, as shown in figure 1.2.12. Compare figures 1.2.8 and 1.2.12, and figures
1.2.9 and 1.2.13.
Figure 1.2.12 Reflection with phase reversal from a medium of low characteristic impedance. (Adapted from
ref. [2].)
Figure 1.2.13 Standing wave pattern caused by reflection from a medium of low characteristic impedance.
1.2.2 Spherical sound waves

The wave equation can be expressed in other coordinate systems than the Cartesian. If
sound is generated by a source in an environment without reflections (which is usually re-
ferred to as a free field) it will generally be more useful to express the wave equation in a
spherical coordinate system (r, θ, φ). The resulting equation is more complicated than eq.
(1.2.1). However, if the source under study is spherically symmetric there can be no angular
12
dependence, and the equation becomes quite simple,12
∂ 2 p 2 ∂p 1 ∂ 2 p
+ = . (1.2.23a)
∂r 2 r ∂r c 2 ∂t 2
If we rewrite in the form
∂ 2 (rp ) 1 ∂ 2 (rp )
= 2 , (1.2.23b)
∂r 2 c ∂t 2
it becomes apparent that this equation is identical in form with the one-dimensional wave
equation, eq. (1.2.5), although p has been replaced by rp. (It is easy to get from eq. (1.2.23b)
to eq. (1.2.23a); it is more difficult the other way.) It follows that the general solution to eq.
(1.2.23) can be written
rp = f1 (ct − r ) + f 2 (ct + r ), (1.2.24a)
that is
1
p= ( f1 (ct − r ) + f 2 (ct + r ) ), (1.2.24b)
r
where f1 and f2 are arbitrary functions. The first term is wave that travels outwards, away
from the source (cf. the first term of eq. (1.2.6)). Note that the shape of the wave is preserved.
However, the sound pressure is seen to decrease in inverse proportion to the distance. This is
the inverse distance law.13 The second term represents a converging wave, that is, a spherical
wave travelling inwards. In principle such a wave could be generated by a reflecting spherical
12
This can be seen as follows. Since the sound pressure depends only on r we have
∂p ∂p ∂r
= ,
∂x ∂r ∂x
which, with
r = x2 + y2 + z 2 ,
becomes
∂p x ∂p
= .
∂x r ∂r
Similar considerations leads to the following expression for the second-order derivative,
∂ 2 p 1 ∂p ∂ ⎛ 1 ∂p ⎞ 1 ∂p x 2 ∂ ⎛ 1 ∂p ⎞ 1 ∂p x 2 ∂ 2 p x 2 ∂p
= +x ⎜ = + = + − .
∂x 2
r ∂r ∂x ⎝ r ∂r ⎟⎠ r ∂r r ∂r ⎜⎝ r ∂r ⎟⎠ r ∂r r 2 ∂r 2 r 3 ∂r
Combining eq. (1.2.1) with this expression and the corresponding relations for y and z finally yields eq.
(1.2.23a):
∂ 2 p ∂ 2 p ∂ 2 p 3 ∂p x 2 + y 2 + z 2 ∂ 2 p x 2 + y 2 + z 2 ∂p 2 ∂p ∂ 2 p 1 ∂ 2 p
+ + = + − = + = .
∂x 2 ∂y 2 ∂z 2 r ∂r r2 ∂r 2 r3 ∂r r ∂r ∂r 2 c 2 ∂t 2
13
The inverse distance law is also known as the inverse square law because the sound intensity is in-
versely proportional to the square of the distance to the source. See sections 1.5 and 1.6.
13
surface centred at the source, but that is a rare phenomenon indeed. Accordingly we will ig-
nore the second term when we study sound radiation in section 1.6.
A harmonic spherical wave is a solution to the Helmholtz equation
∂ 2 (rpˆ )
+ k 2 rpˆ = 0. (1.2.25)
∂r 2
Expressed in the complex notation the diverging wave can be written

e j(ωt − kr )
pˆ = A . (1.2.26)
r
The particle velocity component in the radial direction can be calculated from eq. (1.2.11),
1 ∂pˆ A e j(ωt − kr ) ⎛ 1 ⎞ pˆ ⎛ 1 ⎞
uˆr = − = ⎜⎜1 + ⎟⎟ = ⎜⎜1 + ⎟. (1.2.27)
jωρ ∂r ρc r ⎝ jkr ⎠ ρc ⎝ jkr ⎟⎠
Because of the spherical symmetry there are no components in the other directions. Note that
far14 from the source the sound pressure and the particle velocity are in phase and their ratio
equals the characteristic impedance of the medium, just as in a plane wave. On the other
hand, when kr << 1 the particle velocity is larger than pˆ ρc and the sound pressure and the
particle velocity are almost in quadrature, that is, 90° out of phase. These are nearfield char-
acteristics, and such a sound field is also known as a reactive field. See figure 1.2.14.
Figure 1.2.14 (a) Measurement far from a spherical source in free space; (b) measurement close to a spherical
source. ––, Sound pressure; - - -, particle velocity multiplied by Dc. (From ref. [4].)
14
In acoustics, dimensions are measured in terms of the wavelength, so ‘far from’ means that r >> λ
(or kr >> 1), just as ‘near’ means that r << λ (or kr << 1).
14
1.3 ACOUSTIC MEASUREMENTS
The most important measure of sound is the rms sound pressure,15 defined as
½
⎛ 1 T ⎞
prms = p 2 (t ) = ⎜ lim ∫ p 2 (t )dt ⎟ . (1.3.1)
T →∞ T 0
⎝ ⎠
However, as we shall see, a frequency weighting filter16 is usually applied to the signal before
the rms value is determined. Quite often such a single value does not give sufficient informa-
tion about the nature of the sound, and therefore the rms sound pressure is determined in fre-
quency bands. The resulting sound pressures are practically always compressed logarithmi-
cally and presented in decibels.
Example 1.3.1
The fact that sin2ωt = ½ (1 - cos2ωt) leads to the conclusion that the rms value of a sinusoidal signal
with the amplitude A is A / 2 .
1.3.1 Frequency analysis

Single frequency sound is useful for analysing acoustic phenomena, but most sounds
encountered in practice have ‘broadband’ characteristics, which means that they cover a wide
frequency range. If the sound is more or less steady, it will practically always be more useful
to analyse it in the frequency domain than to look at the sound pressure as a function of time.
Figure 1.3.1 The keyboard of a small piano. The white keys from C to B correspond to the seven notes of the C
major scale. (Adapted from ref. [7].)
Frequency (or spectral) analysis of a signal involves decomposing the signal into its
spectral components. This analysis can be carried out by means of digital analysers that em-
ploy the discrete Fourier transform (‘FFT analysers’). This topic is outside the scope of this
note, but see eg refs. [5, 6]. Alternatively, the signal can be passed through a number of ana-
logue or digital bandpass filters17 with different centre frequencies, a ‘filter bank.’ The filters
can have the same bandwidth or they can have constant relative bandwidth, which means that
the bandwidth is a certain percentage of the centre frequency. Constant relative bandwidth
corresponds to uniform resolution on a logarithmic frequency scale. Such a scale is in much
15
Root mean square value, usually abbreviated rms. This is the square root of the mean square value.
16
A filter is a device that modifies a signal by attenuating some of its frequency components.
17
The ideal bandpass filter would allow frequency components in the passband to pass unattenuated,
but would completely remove frequency components outside the passband. Real filters have, of course, a certain
passband ripple and a finite stopband attenuation.
15
better agreement with the subjective pitch of musical notes than a linear scale, and therefore
frequencies are often represented on a logarithmic scale in acoustics, and frequency analysis
is often carried out with constant percentage filters. The most common filters in acoustics are
octave band filters and one-third octave band filters.
An octave18 is a frequency ratio of 2:1, known from musical scales. Accordingly, the
lower limiting frequency of an octave band is half the upper frequency limit, and the centre
frequency is the geometric mean, that is,
f l = f c 2½ , f u = 2½ f c , fc = fl fu , (1.3.2a, 1.3.2b, 1.3.2c)
where fc is the centre frequency. In a similar manner a one-third octave19 band is a band for
which fu = 2⅓ fl , and
fl = fc 2 6 , fu = 2 6 fc , fc =
1 1
fl fu , (1.3.3a, 1.3.3b, 1.3.3c)
Since 210 = 1024 103 it follows that 210 3 10 and 21 3 101 10 , that is, ten one-third
octaves very nearly make a decade, and a one-third octave is almost identical with one tenth
of a decade. Table 3.1 gives the nominal centre frequencies of standardised octave and one-
third octave band filters.20 As mentioned earlier, the human ear may respond to frequencies in
the range from 20 Hz to 20 kHz, that is, a range of three decades, ten octaves or thirty one-
third octaves.
20 25 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000
1250 1600 2000 2500 3150 4000 5000 6300 8000 10000 12500 16000 20000
Table 1.3.1 Standardised one-third octave and octave (bold characters) band centre frequencies (in hertz).
An important property of the mean square value of a signal is that it can be partitioned
into frequency bands. This means that if we analyse a signal in, say, one-third octave bands,
the sum of the mean square values of the filtered signals equals the mean square value of the
unfiltered signal. The reason is that products of different frequency components average to
zero, so that all the cross terms vanish; the different frequency components are uncorrelated
signals. This can be illustrated by analysing a sum of two pure tones with different frequen-
cies,
( A sin ω1t + B sin ω 2t ) 2 = A2 sin 2 ω1t + B 2 sin 2 ω 2t + 2 ABsin ω1t sin ω 2t
(1.3.4)
= ( A 2 + B 2 ) / 2.
18
Musical tones an octave apart sound very similar. The diatonic scale contains seven notes per octave
corresponding to the white keys on a piano keyboard; see figure 1.3.1. Thus an octave spans eight notes, say
from C to C'; hence the name octave (from Latin octo: eight).
19
A semitone is one twelfth of an octave on the equally tempered scale (a frequency ratio of 21/12:1).
Since 2 = 24/12 it can be seen that a one-third octave is identical with four semitones or a major third (eg from C
⅓
to E, cf. figure 1.3.1). Accordingly, one-third octave band filters are called Terzfilters in German.
20
Round numbers are convenient. The standardised nominal centre frequencies are based on the fact
that the series 1.25, 1.6, 2, 2.5, 3.15, 4, 5, 6.3, 8, 10 is in reasonable agreement with 10n/10, with n = 1, 2,....., 10.
16
Note that the mean square values of the two signals are added unless ω1 = ω2. The validity of
this rule is not restricted to pure tones of different frequency; the mean square value of any
stationary signal equals the sum of mean square values of its frequency components, which
can be determined with a parallel bank of contiguous filters. Thus
2
prms = ∑ prms,
2
i, (1.3.5)
i
where prms,i is the rms value of the output of the i’th filter. Equation (1.3.5) is known as
Parseval’s formula.
Random noise
Many generators of sound produce noise rather than pure tones. Whereas pure tones and other periodic
signals are deterministic, noise is a stochastic or random phenomenon. Stationary noise is a stochastic signal
whose statistical properties do not change with time.
White noise is stationary noise with a flat spectral density, that is, constant mean square value per hertz.
The term white noise is an analogy to white light. When white noise is passed through a bandpass filter, the
mean square of the output is directly proportional to the bandwidth of the filter. It follows that when white noise
is analysed with constant percentage filters, the mean square of the output is proportional to the centre frequency
of the filter. For example, if white noise is analysed with a bank of octave band filters, the output mean square
values of two adjacent filters differ by a factor of two.
Pink noise is stationary noise that has the same mean square value in bands with constant relative
width, eg octave bands. Thus compared with white noise low frequencies are emphasised by pink noise; hence
the name, which is an analogy to an optical phenomenon. It follows that the mean square value of pink noise in
octave bands is three times larger than the mean square value of the noise in one-third octave bands.
Example 1.3.2
The fact that noise, unlike periodic signals, has a finite mean square value per hertz implies that one
can detect a pure tone in noise irrespective of the signal-to-noise ratio by analysing with sufficiently fine spec-
tral resolution: As the bandwidth is reduced, less and less noise passes through the filter, and the tone will
emerge. Compared with filter bank analysers FFT analysers have the advantage that the spectral resolution can
be varied over a wide range [6]; therefore FFT analysers are particular suitable for detecting tones in noise.
When several independent sources of noise are present at the same time the mean
square sound pressures generated by the individual sources are additive. This is due to the
fact that independent sources generate uncorrelated signals, that is, signals whose product av-
erage to zero; therefore the cross terms vanish:
( p1 (t ) + p2 (t )) 2 = p12 (t ) + p22 (t ) + 2 p1 (t ) p2 (t ) = p12 (t ) + p22 (t ). (1.3.6)
It follows that
tot = ∑ prms,i .
2 2
prms, (1.3.7)
i
Note the similarity between eqs. (1.3.5) and (1.3.7). It is of enormous practical importance
that the mean square values of uncorrelated signals are additive, because signals generated by
different mechanisms are invariably uncorrelated. Almost all signals that occur in real life are
mutually uncorrelated.
17
Example 1.3.3
Equation (1.3.7) leads to the conclusion that the mean square pressure generated by a crowd of noisy
people in a room is proportional to the number of people. Thus the rms value of the sound pressure in the room
is proportional to the square root of the number of people.
Example 1.3.4
Consider the case where the rms sound pressure generated by a source of noise is to be measured in the
presence of background noise that cannot be turned off. It follows from eq. (1.3.7) that it is possible in principle
to correct the measurement for the influence of the stationary background noise; one simply subtracts the mean
square value of the background noise from the total mean square pressure. For this to work in practice the back-
ground noise must not be too strong, though, and it is absolutely necessary that it is completely stationary.
1.3.2 Levels and decibels
The human auditory system can cope with sound pressure variations over a range of
more than a million times (see section 2.3). Because of this wide range, the sound pressure
and other acoustic quantities are usually measured on a logarithmic scale. An additional rea-
son is that the subjective impression of how loud noise sounds correlates much better with a
logarithmic measure of the sound pressure than with the sound pressure itself. The unit is the
decibel,21 abbreviated dB, which is a relative measure, requiring a reference quantity. The
results are called levels. The sound pressure level (sometimes abbreviated SPL) is defined as
2
prms p
L p = 10 log10 2
= 20 log10 rms , (1.3.8)
pref pref
where pref is the reference sound pressure, and log10 is the logarithm to the base of 10, hence-
forth written log. The reference sound pressure is 20 µPa for sound waves in air,22 corre-
sponding roughly to the lowest audible sound at 1000 Hz. Some typical sound pressure levels
are given in figure 1.3.2.
The fact that the mean square sound pressures of independent sources are additive (cf.
eq. (1.3.7)) leads to the conclusion that the levels of such sources are combined as follows:
⎛ 0.1L ⎞
L p ,tot = 10 log ⎜ ∑10 p ,i ⎟ . (1.3.9)
⎝ i ⎠
Another consequence of eq. (1.3.7) is that one can correct for the influence of steady
background noise on a measurement of the sound pressure level generated by a source as fol-
lows:
(
L p ,source = 10 log 10
0.1L p ,tot
− 10
0.1L p ,background
). (1.3.10)
21
As the name implies, the decibel is one tenth of a bel. However, the bel is rarely used today. The use
of decibels rather than bels is probably due to the fact that most sound pressure levels encountered in practice
take values between 0 and 120 when measured in decibels, as can be seen in figure 1.3.2. Another reason might
be that to be audible, the change of the level of a given (broadband) sound must be of the order of one decibel.
22
For sound in other fluids than atmospheric air (water, for example) the reference sound pressure is 1
:Pa. To avoid possible confusion it is sometimes advisable to state the reference sound pressure explicitly, eg
‘the sound pressure level is 77 dB re 20 µPa.’
18
This corresponds to subtracting the mean square sound pressure of the background noise
from the total mean square sound pressure as described in example 1.3.5. However, since all
measurements are subject to random errors, the result of the correction will be reliable only if
the background level is at least, say, 3 dB below the total sound pressure level. If the back-
ground noise is more than 10 dB below the total level the correction is less than 0.5 dB.
Figure 1.3.2 Typical sound pressure levels. (Source: Brüel & Kjær.)
19
Example 1.3.5
Expressed in terms of sound pressure levels the inverse distance law states that the level decreases by 6
dB when the distance to the source is doubled.
Example 1.3.6
When each of two independent sources in the absence of the other generates a sound pressure level of
70 dB at a certain point, the resulting sound pressure level is 73 dB (not 140 dB!), because 10 log 2 3 . If one
source creates a sound pressure level of 65 dB and the other a sound pressure level of 59 dB, the total level is
10 log(106.5 + 105.9 ) 66 dB .
Example 1.3.7
Say the task is to determine the sound pressure level generated by a source in background noise with a
level of 59 dB. If the total sound pressure level is 66 dB, it follows from eq. (1.3.10) that the source would have
produced a sound pressure level of 10 log(106.6 − 105.9 ) 6.5 dB in the absence of the background noise.
Example 1.3.8
When two sinusoidal sources emit pure tones of the same frequency they create an interference field,
and depending on the phase difference the total sound pressure amplitude at a given position will assume a value
between the sum of the two amplitudes and the difference:
A − B ≤ Ae jωt + Be jωt = A + B = A e jϕA + B e jϕB ≤ A + B .
For example, if two pure tone sources of the same frequency each generates a sound pressure level of 70 dB in
the absence of the other source then the total sound pressure level can be anywhere between 76 dB (constructive
interference) and - ∞ dB (destructive interference). Note that eqs. (1.3.7) and (1.3.9) do not apply in this case
because the signals are not uncorrelated. See also figure 1.9.2 in the Appendix.
Other first-order acoustic quantities, for example the particle velocity, are also often
measured on a logarithmic scale. The reference velocity is 1 nm/s = 10-9 m/s.23 This reference
is also used in measurements of the vibratory velocities of vibrating structures.
The acoustic second-order quantities sound intensity and sound power, defined in sec-
tion 1.5, are also measured on a logarithmic scale. The sound intensity level is
I
LI = 10 log , (1.3.11)
I ref
where I is the intensity and Iref = 1 pWm-2 = 10-12 Wm-2,24 and the sound power level is
Pa
LW = 10 log , (1.3.12)
Pref
where Pa is the sound power and Pref = 1 pW. Note than levels of linear quantities are defined
as twenty times the logarithm of the ratio of the rms value to a reference value, whereas lev-
els of second-order quantities are defined as ten times the logarithm, in agreement with the
fact that if the linear quantities are doubled then quantities of second order are quadrupled.
23
The prefix n (for ‘nano’) represents a factor of 10-9.
24
The prefix p (for ‘pico’) represents a factor of 10-12.
20
Example 1.3.9
It follows from the constant spectral density of white noise that when such a signal is analysed in one-
third octave bands, the level increases 1 dB from one band to the next (10 log(21 3 ) 1dB) .
1.3.3 Noise measurement techniques and instrumentation

A sound level meter is an instrument designed to measure sound pressure levels. To-
day such instruments can be anything from fairly simple devices with analogue filters and
detectors and a moving coil meter to advanced digital analysers. Figure 1.3.3 shows a block
diagram of a simple sound level meter. The microphone converts the sound pressure to an
electrical signal, which is amplified and passes through various filters. After this the signal is
squared and averaged with a detector, and the result is finally converted to decibels and
shown on a display. In the following a very brief description of such an instrument will be
given; see eg refs. [8, 9] for further details.
The most commonly used microphones for this purpose are condenser microphones,
which are more stable and accurate than other types. The diaphragm of a condenser micro-
phone is a very thin, highly tensioned foil. Inside the housing of the microphone cartridge is
the other part of the capacitor, the back plate, placed very close to the diaphragm (see figure
1.3.4). The capacitor is electrically charged, either by an external voltage on the back plate or
(in case of prepolarised electret microphones) by properties of the diaphragm or the back
plate. When the diaphragm moves in response to the sound pressure, the capacitance changes,
and this produces an electrical voltage proportional to the instantaneous sound pressure.
Figure 1.3.3 A sound level meter. (From ref. [10].)
Figure 1.3.4 A condenser microphone. (Source: Brüel & Kjær.)
21
Figure 1.3.5 The ‘free-field correction’ of a typical measurement microphone for sound coming from various
directions. The free-field correction is the fractional increase of the sound pressure (usually expressed in dB)
caused by the presence of the microphone in the sound field. (From ref. [11].)
Figure 1.3.6 Free-field response of a microphone of the ‘free-field’ type at axial incidence. (From ref. [11].)
The microphone should be as small as possible so as not to disturb the sound field.
However, this is in conflict with the requirement of a high sensitivity and a low inherent
noise level, and typical measurement microphones are ‘½-inch’ microphones with a diameter
of about 13 mm. At low frequencies, say below 1 kHz, such a microphone is much smaller
than the wavelength and does not disturb the sound field appreciably. In this frequency range
the microphone is omnidirectional as of course it should be since the sound pressure is a sca-
lar and has no direction. However, from a few kilohertz and upwards the size of the micro-
phone is no longer negligible compared with the wavelength, and therefore it is no longer
omnidirectional, which means that its response varies with the nature of the sound field; see
figure 1.3.5.
One can design condenser microphones to have a flat response in as wide a frequency
range as possible under specified sound field conditions. For example, ‘free-field’ micro-
phones are designed to have a flat response for axial incidence (see figure 1.3.6), and such
microphones should therefore be pointed towards the source. ‘Random-incidence’ micro-
phones are designed for measurements in a diffuse sound field where sound is arriving from
all directions, and ‘pressure’ microphones are intended for measurements in small cavities.
22
Figure 1.3.7 Standardised frequency weighting curves. (From ref. [8].)
Centre frequency (Hz) A-weighting (dB) C-weighting (dB)

20 -50.5 -6.2
25 -44.7 -4.4
31.5 -39.4 -3.0
40 -34.6 -2.0
50 -30.2 -1.3
63 -26.2 -0.8
80 -22.5 -0.5
100 -19.1 -0.3
125 -16.1 -0.2
160 -13.4 -0.1
200 -10.9 0.0
250 -8.6 0.0
315 -6.6 0.0
400 -4.8 0.0
500 -3.2 0.0
630 -1.9 0.0
800 -0.8 0.0
1000 0.0 0.0
1250 0.6 0.0
1600 1.0 -0.1
2000 1.2 -0.2
2500 1.3 -0.3
3150 1.2 -0.5
4000 1.0 -0.8
5000 0.5 -1.3
6300 -0.1 -2.0
8000 -1.1 -3.0
10000 -2.5 -4.4
12500 -4.3 -6.2
16000 -6.6 -8.5
20000 -9.3 -11.2
Table 1.3.2 The response of standard A- and C-weighting filters in one-third octave bands.
23
The sensitivity of the human auditory system varies significantly with the frequency
in a way that changes with the level (see section 2.3). In particular the human ear is, at low
levels, much less sensitive to low frequencies than to medium frequencies. This is the back-
ground for the standardised frequency weighting filters shown in figure 1.3.7. The original
intention was to simulate a human ear at various levels, but it has long ago been realised that
the human auditory system is far more complicated than implied by such simple weighting
curves, and B- and D-weighting filters are little used today. On the other hand the A-
weighted sound pressure level is the most widely used single-value measure of sound, be-
cause the A-weighted sound pressure level correlates in general much better with the subjec-
tive effect of noise than measurements of the sound pressure level with a flat frequency re-
sponse. C-weighting, which is essentially flat in the audible frequency range, is sometimes
used in combination with A-weighting, because a large difference between the A-weighted
level and the C-weighted level is a clear indication of a prominent content of low frequency
noise. The results of measurements of the A- and C-weighted sound pressure level are de-
noted LA and LC respectively, and the unit is dB.25 If no weighting filter is applied, the level is
sometimes denoted LLin.
In the measurement instrument the frequency weighting filter is followed by a squar-
ing device, a lowpass filter that smooths out the instantaneous fluctuations, and a logarithmic
converter. The lowpass filter corresponds to applying a time weighting function. The most
common time weighting in sound level meters is exponential, which implies that the squared
signal is smoothed with a decaying exponential so that recent data are given more weight
than older data:
⎛⎛ 1 t ⎞ 2 ⎞
L p (t ) = 10 log ⎜ ⎜ ∫ p 2 (u ) e − (t −u ) /τ du ⎟ pref ⎟. (1.3.13)
⎝ ⎝ τ −∞ ⎠ ⎠
Two values of the time constant τ are standardised: S (for ‘slow’) is exponential averaging
with a time constant of 1 s, and F (for ‘fast’) is exponential averaging with a time constant of
125 ms.
The alternative to exponential averaging is linear (or integrating) averaging, in which
all the sound is weighted uniformly during the integration. The equivalent sound pressure
level is defined as
⎛⎛ 1 t2 ⎞ 2 ⎞
Leq = 10 log ⎜⎜ ⎜ ∫t
2
p (t )dt ⎟ pref ⎟⎟ . (1.3.14)
⎝ ⎝ t2 − t1 1 ⎠ ⎠
Measurements of random noise with a finite integration time are subject to random
errors that depend on the bandwidth of the signal and on the integration time. It can be shown
that the variance of the measurement result is inversely proportional to the product of the
bandwidth and the integration time [6].26
25
In practice the unit is often written dB (A) and dB (C), respectively.
26
In the literature reference is sometimes made to the equivalent integration time of exponential de-
tectors. This is two times the time constant (eg 250 ms for ‘F’), because a measurement with an exponential
detector with a time constant of τ has the same statistical uncertainty as a measurement with linear averaging
over a period of 2τ if the signal is random noise [9].
24
As can be seen by comparing with eqs. (1.3.1) and (1.3.8), the equivalent sound pres-
sure level is just the sound pressure level corresponding to the rms sound pressure determined
with a specified integration period. The A-weighted equivalent sound pressure level LAeq is
the level corresponding to a similar time integral of the A-weighted instantaneous sound
pressure. Sometimes the quantity is written LAeq,T where T is the integration time.
Whereas exponential averaging corresponds to a running average and thus gives a
(smoothed) measure of the sound at any instant of time, the equivalent sound pressure level
(with or without A-weighting) can be used for characterising the total effect of fluctuating
noise, for example noise from road traffic. Typical values of T are 30 s for measurement of
noise from technical installations, 8 h for noise in a working environment and 24 h for traffic
noise.
Sometimes it is useful to analyse noise signals in one-third octave bands, cf. section
1.3.1. From eq. (1.3.5) it can be seen that the total sound pressure level can be calculated
from the levels in the individual one-third octave bands as follows:
⎛ ⎞
LLin = 10 log⎜ ∑100.1Li ⎟. (1.3.15)
⎝ i ⎠
In a similar manner one can calculate the A-weighted sound pressure level from the one-third
octave band values and the attenuation data given in table 1.3.2,
⎛ ⎞
LA = 10 log⎜ ∑ 100.1( Li + K i ) ⎟, (1.3.16)
⎝ i ⎠
where Ki is the relative response of the A-weighting filter (in dB) in the i’th band, given in
table 1.3.2.
Example 1.3.10
A source gives rise to the following one-third octave band values of the sound pressure level at a cer-
tain point,
Centre frequency (Hz) Sound pressure level (dB)
315 52
400 68
500 76
630 71
800 54
and less than 50 dB in all the other bands. It follows that
( )
LLin = 10 log 105.2 + 106.8 + 107.6 + 107.1 + 105.4 77.7 dB,
and
( )
LA = 10 log 10(5.2 − 0.66) + 10(6.8− 0.48) + 10(7.6 − 0.32) + 10(7.1− 0.19) + 10(5.4 − 0.08) 74.7 dB,
Noise that changes its level in a regular manner is called intermittent noise. Such
noise is for example generated by machinery that operates in cycles. If the noise occurs at
several steady levels, the equivalent sound pressure level can be calculated from the formula
25
⎛ t ⎞
Leq,T = 10log ⎜ ∑ i 100.1Li ⎟ . (1.3.17)
⎝ i T ⎠
This corresponds to adding the mean square values with a weighting that reflects the duration
of each level.
Example 1.3.11
The A-weighted sound pressure level at a given position in an industrial hall changes periodically be-
tween 84 dB in intervals of 15 minutes, 95 dB in intervals of 5 minutes and 71 dB in intervals of 20 minutes.
From eq. (1.3.17) it follows that the equivalent sound pressure level over a working day is
⎛ 15 5 20 ⎞
LAeq = 10 log ⎜ 108.4 + 109.5 + 107.1 ⎟ 87.0 dB.
⎝ 40 40 40 ⎠
Most sound level meters have also a peak detector for determining the highest abso-
lute value of the instantaneous sound pressure (without filters and without time weighting),
ppeak. The peak level is calculated from this value and eq. (1.3.8) in the usual manner, that is,
ppeak
L p = 20 log . (1.3.18)
pref
Example 1.3.12
The crest factor of a signal is the ratio of its peak value to the rms value (sometimes expressed in dB).
From example 1.3.1 it follows that the crest factor of a pure tone signal is 2 or 3 dB.
The sound exposure level (sometimes abbreviated SEL) is closely related to LAeq, but
instead of dividing the time integral of the squared A-weighted instantaneous sound pressure
by the actual integration time one divides by t0 = 1 s. Thus the sound exposure level is a
measure of the total energy27 of the noise, normalised to 1 s:
⎛⎛ 1 t2 ⎞ 2 ⎞
LAE = 10 log ⎜⎜ ⎜ ∫ pA2 (t )dt ⎟ pref ⎟⎟ (1.3.19)
⎝ ⎝ t0
t1
⎠ ⎠
This quantity is used for measuring the total energy of a ‘noise event’ (say, a hammer blow or
the take off of an aircraft), independently of its duration. Evidently the measurement interval
should encompass the entire event.
Example 1.3.13
It is clear from eqs. (1.3.14) and (1.3.19) that LAeq,T of a noise event of finite duration decreases with
the logarithm of T if the T exceeds its duration:
⎛⎛ 1 ∞ ⎞ 2 ⎞ T
LAeq,T = 10 log ⎜ ⎜
⎝⎝ T
∫ −∞
pA2 (t )dt ⎟ pref
⎠
⎟ = LAE − 10 log .
⎠ t0
27
In signal analysis it is customary to use the term ‘energy’ in the sense of the integral of the square of
a signal, without regard to its units. This should not be confused with the potential energy density of the sound
field introduced in section 1.5.
26
Example 1.3.14
If n identical noise events each with a sound exposure level of LAE occur within a period of T then the
A-weighted equivalent sound level is
T
LAeq,T = LAE + 10 log n − 10 log ,
t0
because the integrals of the squared signals are additive; cf. eq. (1.3.7).28
1.4 THE CONCEPT OF IMPEDANCE
By definition an impedance is the ratio of the complex amplitudes of two signals rep-
resenting cause and effect, for example the ratio of an AC voltage across a part of an electric
circuit to the corresponding current, the ratio of a mechanical force to the resulting vibra-
tional velocity, or the ratio of the sound pressure to the particle velocity. The term has been
coined from the verb ‘impede’ (obstruct, hinder), indicating that it is a measure of the opposi-
tion to the flow of current etc. The reciprocal of the impedance is the admittance, coined from
the verb ‘admit’ and indicating lack of such opposition. Note that these concepts require
complex representation of harmonic signals; it makes no sense to divide, say, the instantane-
ous sound pressure with the instantaneous particle velocity. There is no simple way of de-
scribing properties corresponding to a complex value of the impedance without the use of
complex notation.
The mechanical impedance is perhaps simpler to understand than the other impedance
concepts, since it is intuitively clear that it takes a certain vibratory force to generate me-
chanical vibrations. The mechanical impedance of a structure at a given point is the ratio of
the complex amplitude of a harmonic point force acting on the structure to the complex am-
plitude of the resulting vibratory velocity at the same point,29
Fˆ
Zm = . (1.4.1a)
vˆ
The unit is kg/s. The mechanical admittance is the reciprocal of the mechanical impedance,
vˆ
Ym = . (1.4.1b)
Fˆ
This quantity is also known as the mobility. The unit is s/kg.
Example 1.4.1
It takes a force of F=a·M to set a mass M into the acceleration a (Newton’s second law of motion);
therefore the mechanical impedance of the mass is
Fˆ Fˆ
Zm = = = jω M .
vˆ aˆ jω
28
Strictly speaking this requires that the instantaneous product of the ‘event’ and any of its time shif-
ted versions time average to zero. In practice this will always be the case.
29
Note that the sign of the imaginary part of the impedance changes if the e-iωt convention is used in-
stead of the ejωt convention. Cf. footnote no 9 on p. 6.
27
Example 1.4.2
It takes a force of F=ξK to stretch a spring with the stiffness K a length of ξ (Hooke’s law); therefore
the mechanical impedance of the spring is
Fˆ Fˆ K
Zm = = = .
ˆ
vˆ jωξ jω
Figure 1. 4.1 A mass hanging from a spring.
Example 1.4.3
A simple mechanical oscillator consists of a mass M suspended on a spring with a stiffness constant of
K, as sketched in figure 1.4.1. In order to set the mass into vibrations one will have to move the mass and dis-
place the spring from its equilibrium value. It follows that the mechanical impedance of this system is the sum
of the impedance of the mass and the impedance of the spring,
Z m = jω M +
K
jω
⎛
⎝
K⎞
ω ⎠
( )
= j ⎜ ω M − ⎟ = jω M 1 − ( ω0 ω ) ,
2
where
ω0 = K M
is the angular resonance frequency. Note that the impedance is zero at the resonance, indicating that even a very
small harmonic force at this frequency will generate an infinite velocity. In practice there will always be some
losses, of course, so the impedance is very small but not zero at the resonance frequency.
The acoustic impedance is associated with average properties on a surface. This quan-
tity is mainly used under conditions where the sound pressure is more or less constant on the
surface. It is defined as the complex ratio of the average sound pressure to the volume veloc-
ity, which is the surface integral of the normal component of the particle velocity,
qˆ = ∫ uˆ ⋅ dS, (1.4.2)
S
where S is the surface area. Thus the acoustic impedance is

pˆ av
Za = . (1.4.3)
qˆ
The unit is kgm-4s-1. Since the total force acting on the surface equals the product of the aver-
age sound pressure and the area, and since qˆ = Suˆn if the velocity is uniform, it can be seen
that there is a simple relation between the two impedance concepts under such conditions:
Zm = Za S 2. (1.4.4)
This equation makes it possible to calculate the force it would take to drive a massless piston
with the velocity ûn . In other words, the acoustic impedance describes the load on a fictive
28
piston caused by the medium. If the piston is real, the impedance is called the radiation im-
pedance. This quantity is used for describing the load caused by the motion of the medium
on, for example, a loudspeaker membrane.30
The concept of acoustic impedance is essentially associated with approximate low-
frequency models. For example, it is a very good approximation to assume that the sound
field in a tube is one-dimensional when the wavelength is large compared with the cross-
sectional dimensions of the tube. Under such conditions the sound field can be described by
eqs. (1.2.15) and (1.2.16), and a tube of a given length behaves as an acoustic two-port.31 It is
possible to calculate the transmission of sound through complicated systems of pipes using
fairly simple considerations based the assumption of continuity of the sound pressure and the
volume velocity at each junction [12].32
The acoustic impedance is also useful in studying the properties of acoustic transduc-
ers. Such transducers are usually much smaller than the wavelength in a significant part of the
frequency range. This makes it possible to employ so-called lumped parameter models where
the system is described by an analogous electrical circuit composed of simple lumped ele-
ment, inductors, resistors and capacitors, representing masses, losses and springs [13, 14].
Finally it should be mentioned that the acoustic impedance can be used for describing the
acoustic properties of materials exposed to normal sound incidence.33
Example 1.4.4
The acoustic input impedance of a tube terminated by a rigid cap can be deduced from eqs. (1.2.17) and
(1.2.18) (with x = -l),
ρc
Za = − j cot kl ,
S
where l is the length of the tube and S is its cross-sectional area. Note that the impedance goes to infinity when l
equals a multiple of half a wavelength, indicating that it would take an infinitely large force to drive a piston at
the inlet of the tube at these frequencies (see figure 1.4.2). Conversely, the impedance is zero when l equals an
odd-numbered multiple of a quarter of a wavelength; at these frequencies the sound pressure on a vibrating pis-
ton at the inlet of the tube would vanish. Cf. example 1.2.3.
30
The load of the medium on a vibrating piston can be described either in terms of the acoustic radia-
tion impedance (the ratio of the sound pressure to the volume velocity) or the mechanical radiation impedance
(the ratio of the force to the velocity).
31
‘Two-port’ is a term from electric circuit theory denoting a network with two terminals. Such a net-
work is completely described by the relations between four quantities, the voltage and current at the input ter-
minal and the voltage and current at the output terminal. By analogy, an acoustic two-port is completely de-
scribed by the relations between the sound pressures and the volume velocities at the two terminals. In case of a
cylindrical tube such relations can easily be derived from eqs. (1.2.15) and (1.2.16) [12].
32
Such systems act as acoustic filters. Silencers (or mufflers) are composed of coupled tubes.
33
In the general case we need to describe the properties of acoustic materials with the local ratio of the
sound pressure on the surface to the resulting vibrational velocity. In most literature this quantity, which is used
mainly in theoretical work, is called the specific acoustic impedance. In practical applications the properties of
acoustic materials are described in terms of absorption coefficients (or absorption factors), assuming either nor-
mal or diffuse sound incidence (see section 1.5). It is possible to calculate the absorption coefficient of a mate-
rial from its specific acoustic impedance, but not the impedance from the absorption coefficient.
29
Figure 1.4.2 The acoustic input impedance of a tube terminated rigidly.
Example 1.4.5
At low frequencies the acoustic impedance of the rigidly terminated tube of example 1.4.4 can be sim-
plified. The factor cotkl approaches 1/kl, and the acoustic impedance becomes
ρc ρ c2
Za − j = ,
Slk jωV
where V = Sl is the volume of the tube, indicating that the tube acts as a spring, independently of its shape. This
is the acoustic impedance of a cavity much smaller than the wavelength. Since, from eq. (1.2.2b),
ρ c 2 = γ p0 ,
it can be seen that the acoustic impedance of a cavity at low frequencies can also be written
γ p0
Za = ,
jωV
in agreement with the considerations on p. 3.
Example 1.4.6
A Helmholtz resonator is the acoustic analogue to the simple mechanical oscillator described in exam-
ple 1.4.3; see figure 1.4.3. The dimensions of the cavity are much smaller that the wavelength; therefore it be-
haves as a spring with the acoustic impedance
ρc2
Za = ,
jωV
where V is the volume; cf. example 1.4.5. The air in the neck moves back and forth uniformly as if it were in-
compressible; therefore the air in the neck behaves as a lumped mass with the mechanical impedance
Z m = jωρ Sleff ,
where leff is the effective length and S is the cross-sectional area of the neck. (The effective length of the neck is
somewhat longer than the physical length, because some of the air just outside the neck is moving along with
the air in the neck.) The corresponding acoustic impedance follows from eq. (1.4.4):
jωρ leff
Za = ,
S
By analogy with example 1.4.3 we conclude that the angular resonance frequency is
S
ω0 = c .
Vleff
30
Note that the resonance frequency is independent of the density of the medium. See also section 4.3.
It is intuitively clear that a larger volume or a longer neck would correspond to a lower frequency, but
it is perhaps less obvious that a smaller neck area gives a lower frequency.
Figure 1.4.3 A Helmholtz resonator.
Yet another impedance concept, the characteristic impedance, has already been intro-
duced. As we have seen in section 1.2.1, the complex ratio of the sound pressure to the parti-
cle velocity in a plane propagating wave equals the characteristic impedance of the medium
(cf. eq. (1.2.14)), and it approximates this value in a free field far from the source (cf. eq.
(1.2.27)). Thus, the characteristic impedance describes a property of the medium, as we have
seen in example 1.2.4. The unit is kgm-2s-1.
1.5 SOUND ENERGY, SOUND INTENSITY, SOUND POWER AND SOUND AB-
SORPTION
The most important quantity for describing a sound field is certainly the sound pres-
sure. However, sources of sound emit sound power, and sound fields are also energy fields in
which potential and kinetic energies are generated, transmitted and dissipated. Some typical
sound power levels are given in table 1.5.1.
Aircraft turbojet engine 10 kW 160 dB

Gas turbine (1 MW) 32 W 135 dB
Small airplane 5W 127 dB
Tractor (150 hp) 100 mW 110 dB
Large electric motor (0.5 MW) 10 mW 100 dB
Vacuum cleaner 100 µW 80 dB
Office machine 32 µW 75 dB
Speech 10 µW 70 dB
Whisper 10 nW 40 dB
Table 1.5.1 Typical sound power levels.
It is apparent that the radiated sound power is a negligible part of the energy conver-
sion of almost any source. However, energy considerations are nevertheless of great practical
importance in acoustics. The usefulness is due to the fact that a statistical approach where the
energy of the sound field is considered turns out to give very useful approximations in room
31
acoustics and in noise control (see section 3.2). In fact determining the sound power of
sources is a central point in noise control engineering. The value and relevance of knowing
the sound power radiated by a source is due to the fact that this quantity is largely independ-
ent of the surroundings of the source in the audible frequency range.
1.5.1 The energy in a sound field

It can be shown that the instantaneous potential energy density in a sound field (the
potential sound energy per unit volume) is given by the expression
p 2 (t )
wpot (t ) = . (1.5.1)
2 ρc 2
This quantity describes the energy stored in a unit volume of the medium because of the
compression or rarefaction; the phenomenon is analogous to the potential energy in a com-
pressed or elongated spring, and the derivation is similar.
The instantaneous kinetic energy density in a sound field (the kinetic energy per unit
volume) is
1 2
wkin ( t ) = ρu (t ) . (1.5.2)
2
This quantity describes the energy per unit volume represented by the mass of the particles of
the medium moving with the velocity u. This corresponds to the kinetic energy of a moving
mass, and the derivation is similar.
The instantaneous sound intensity is the product of the instantaneous sound pressure
and the instantaneous particle velocity,
I (t ) = p (t ) u (t ) . (1.5.3)
This quantity, which is a vector, expresses the magnitude and direction of the instantaneous
flow of sound energy per unit area, or the work done by the sound wave per unit area of an
imaginary surface perpendicular to the vector.
Energy conservation
By combining the fundamental equations that govern a sound field (the conservation of mass, the rela-
tion between the sound pressure and density changes, and Euler’s equation of motion), one can derive the equa-
tion
∂w(t )
∇ ⋅ I(t ) = − ,
∂t
where ∇ ⋅ I (t ) is the divergence of the instantaneous sound intensity and w(t) is the sum of the potential and
kinetic energy densities. This is the equation of conservation of sound energy, which expresses the simple fact
that the rate of change of the total sound energy at a given point in a sound field is equal to the flow of converg-
ing sound energy; if the sound energy density at the point increases there must be a net flow of energy towards
the point, and if it decreases there must be net flow of energy diverging away from the point.
The global version of this equation is obtained using Gauss’s theorem,34
∫ V
∇ ⋅ I (t )dV = ∫ I (t ) ⋅ dS = −
S
∂
∂t ( ∫ w(t)dV ) = − ∂E∂(tt) ,
V
34
According to Gauss’s theorem the volume integral of the divergence of a vector equals the cor-
responding surface integral of the (outward pointing) normal component of the vector.
32
where S is the area of an arbitrary, closed surface, V is the volume inside the surface, and E(t) is the total instan-
taneous sound energy within the surface. This equation shows that the rate of change of the total sound energy
within a closed surface is identical with the surface integral of the normal component of the instantaneous sound
intensity.
In practice the time-averaged energy densities,

2
prms 1 2
wpot = , wkin = ρurms , (1.5.4a, 1.5.4b)
2 ρc 2 2
are more important than the instantaneous quantities, and the time-averaged sound intensity
(which is usually referred to just as the ‘sound intensity’),
I = I(t ) = p(t )u(t ), (1.5.5)
is more important than the instantaneous intensity I(t). It can be shown that the integral of the
normal component of the sound intensity over a closed surface is zero,
∫ I⋅dS = 0
S
(1.5.6)
in any sound field unless there is generation or dissipation of sound power within the surface
S. If the surface encloses a source the integral equals the radiated sound power of the source,
irrespective of the presence of other sources of noise outside the surface:
∫ I⋅dS = P
S a (1.5.7)
Often we will be concerned with harmonic signals and make use of complex notation,
as in sections 1.2 and 1.4. Expressed in the complex notation eqs. (1.5.4) and (1.5.5) become
2
pˆ 1 2
wpot = , wkin = ρ uˆ , (1.5.8a, 1.5.8b)
4ρ c 2
4
1
I= Re { pˆ uˆ *} . (1.5.9)
2
(Note that the two complex exponentials describing the time dependence of the sound pres-
sure and the particle velocity cancel each other because one of them is conjugated; see the
Appendix.) The component of the sound intensity in a certain direction is
1
Ir = Re { pu
ˆ ˆr*} . (1.5.10)
2
Inserting the expressions for the sound pressure (eq. (1.2.13)) and the particle velocity (eqs.
(1.2.14)) in a plane propagating wave into eq. (1.5.10) shows that
2
pˆ 2
prms
Ix = = (1.5.11)
2 ρc ρc
in this particular sound field. Moreover, inserting the corresponding expressions for a simple
spherical wave, eqs. (1.2.26) and (1.2.27), into eq. (1.5.10) gives the same relation:
33
⎧ A e j(ωt − kr ) A* e − j(ωt − kr ) ⎛
2 2
1 ⎞⎫
1
2
{ }
I r = Re pˆ uˆr = Re⎨
*
r
⎜⎜1 −
ρcr ⎝ jkr ⎠⎭ 2 ρcr
⎟⎟⎬ =
A
2
=
pˆ
2 ρc
. (1.5.12)
⎩
It is apparent that there is a simple relation between the sound intensity and the mean square
sound pressure in these two extremely important cases.35 However, it should be emphasised
that in the general case eq. (1.5.11) is not valid, and one will have to measure both the sound
pressure and the particle velocity simultaneously and time integrate the instantaneous product
in order to measure the sound intensity. Equipment for such measurements has been commer-
cially available since the early 1980s [3].
Example 1.5.1
It follows from eq. (1.5.11) that the sound intensity in a plane propagating wave with an rms sound
pressure of 1 Pa is (1 Pa) 2 (1.2 Kgm −3 ⋅ 343ms −1 ) 2.4 mW m 2 .
Example 1.5.2
The sound intensity in the interference field generated by a plane sound wave reflected from a rigid
surface at normal incidence can be determined by inserting eqs. (1.2.17) and (1.2.18) into eq. (1.5.10):
1 ⎧ j2 pi* ⎫ ⎧⎪ 2 j pi 2 ⎫⎪
Ix = Re ⎨2 pi cos kx sin kx ⎬ = Re ⎨ sin 2kx ⎬ = 0.
2 ⎩ ρc ⎭ ρc
⎩⎪ ⎭⎪
This result shows that there is no net flow of sound energy towards the rigid surface.
In an environment without reflecting surfaces the sound field far from any source of
finite extent is locally plane, as mentioned in section 1.2.1, and therefore the local sound in-
tensity is to a good approximation given by eq. (1.5.11). With eq. (1.5.7) we now conclude
that one can estimate the radiated sound power of a source by measuring the mean square
pressure generated by the source on a spherical surface centred at the source:
Pa = ∫ ( prms
2
( ρ c) ) d S . (1.5.13)
S
However, whereas eq. (1.5.7) is valid even in the presence of sources outside the measure-
ment surface eq. (1.5.11) is not, of course; therefore only the source under test must be pre-
sent. In practice one measures the sound pressure at a finite number of discrete points. This is
the free-field method of measuring sound power. Note that an anechoic room (a room without
any reflecting surfaces) is required.
Under conditions where the sound pressure and the particle velocity are constant over
a surface in phase as well as in amplitude we can write
pˆ = qˆZ a (1.5.14)
(cf. eq. (1.4.4)), and the sound power passing through the surface can be expressed in terms
of the acoustic impedance:
35
Eq. (1.5.11) implies that the sound intensity level is almost identical with the sound pressure level in
(
air at 20°C and 101.3 kPa: it is easy to show that LI = L p − 10 log ρ cI ref pref
2
)
Lp − 0.14 dB Lp .
34
2
1
Pa = Re { pq
2
1
ˆ ˆ *} = Re qˆ Z a =
2
2 qˆ
2
{
Re {Z a }. } (1.5.15)
This expression demonstrates that the radiated sound power of a vibrating surface is closely
related to the volume velocity and to the real part of the radiation impedance.
1.5.2 Sound absorption

Most materials absorb sound. As we have seen in section 1.2 we need a precise de-
scription of the boundary conditions for solving the wave equation, which leads to a descrip-
tion of material properties in terms of the specific acoustic impedance, as mentioned in sec-
tion 1.4. However, in practical applications, for example in architectural acoustics, a simpler
measure of the acoustic properties of materials, the absorption coefficient (or absorption fac-
tor), is more useful. By definition the absorption coefficient of a given material is the ab-
sorbed fraction of the incident sound power. From this definition it follows that the absorp-
tion coefficient takes values between naught and unity. A value of unity implies that all the
incident sound power is absorbed.
In general the absorption coefficient of a given material depends on the structure of
the sound field (plane wave incidence of a given angle of incidence, for example, or random
or diffuse incidence in a room). Here we will study only the absorption for plane waves of
normal incidence.
Figure 1.5.1 A standing wave tube for measuring the normal incidence absorption coefficient. (From ref. [15].)
Consider the sound field in a tube driven by a loudspeaker at one end and terminated
by the material under test at the other end, as sketched in figure 1.5.1. This is a one-
dimensional field, which means that it has the general form given by eqs. (1.2.15) and
(1.2.16). The amplitudes pi and pr depend on the boundary conditions, that is, the vibrational
velocity of the loudspeaker and the properties of the material at the end of the tube. The
sound intensity is obtained by inserting eqs. (1.2.15) and (1.2.16) into eq. (1.5.10):
2 2
p − pr
Ix =
1
2 ρc
{(
Re pi e − j kx
+ pr e j kx
)( p e
*
i
j kx
−p e *
r
− j kx
)} = i
2 ρc
. (1.5.16)
The incident sound intensity is the value associated with the incident wave, that is,
2
p
I inc = i . (1.5.17)
2 ρc
The absorption coefficient is the ratio of Ix to Iinc,
35
2 2 2
I p − p ⎛ s −1⎞
2 4s
α= x = i 2 r =1− R =1− ⎜ ⎟ = , (1.5.18)
I inc pi ⎝ s + 1⎠ (1 + s )2
where we have introduced the reflection factor and the standing wave ratio (cf. eqs. (1.2.19)
and (1.2.22)). Note that the absorption coefficient is independent of the phase angle of R,
which shows that there is more information in the complex reflection factor than in the ab-
sorption coefficient. Equation (1.5.18) demonstrates that one can determine the normal inci-
dence absorption of a material by exposing it to normal sound incidence in a tube and meas-
uring the standing wave ratio of the resulting interference field.
Figure 1.5.2 Standing wave patterns for various values of the absorption coefficient: 90 % (–––); 60 % (– – –);
30 % (···).
Example 1.5.3
If the material under test is completely reflecting then |R| = 1, corresponding to an absorption coeffi-
cient of zero. In this case the standing wave ratio is infinitely large. If the material is completely absorbing, R =
0, corresponding to an absorption coefficient of unity. In this case there is no reflected wave, so the sound pres-
sure amplitude is constant in the tube, corresponding to a standing wave ratio of one.
1.6 RADIATION OF SOUND
Sound can be generated by many different mechanisms. In this note we will study
only the simplest one, which is also the most important: that of a solid vibrating surface. As
we shall see, the most efficient mechanism for radiation of sound involves a net volume dis-
placement.
1.6.1 Point sources

The simplest source to describe mathematically is a pulsating sphere, that is, a sphere
that expands and contracts harmonically with spherical symmetry. In free space such a source
generates the simple spherical sound field we studied in section 1.2.2. Say the source has a
radius of a. From eq. (1.2.27) we know that the particle velocity on the surface of the source
is
A e j(ωt − ka ) ⎛ 1 ⎞
uˆr (a) = ⎜⎜1 + ⎟. (1.6.1)
ρc a ⎝ jka ⎟⎠
The boundary condition on the surface implies that the vibrational velocity U e jωt must equal
the normal component of the particle velocity; therefore
36
jρ cka 2U e j ka jρωQ e j ka
A= = , (1.6.2)
1 + jka 4π (1 + jka )
where we have introduced the volume velocity of the pulsating sphere,
Q = 4πa 2U , (1.6.3)
by multiplying with the surface area of the sphere. Inserting into eq. (1.2.26) gives an expres-
sion for the sound pressure generated by the source,
jρωQ e j(ωt − k ( r − a ))
pˆ = . (1.6.4)
4πr (1 + jka )
We can now calculate the radiation impedance of the pulsating sphere. This is the ratio of the
sound pressure on the surface of the sphere to the volume velocity (cf. eq. (1.4.3)):
pˆ (a) jρω ρ ck 2 jωρ
Z a,r = = + , (1.6.5)
Q e jωt 4πa (1 + jka ) 4π 4πa
where the approximation to the right is based on the assumption that ka << 1. Note that the
imaginary part of the radiation impedance is much larger than the real part at low frequencies,
indicating that most of the force it takes to expand and contract the sphere goes to moving the
mass of the air in a region near the sphere (cf. example 1.4.1). This air moves back and forth
almost as if it were incompressible.
In the limit of a vanishingly small sphere with a finite volume velocity, the source be-
comes a monopole, a point source or a simple source. With ka << 1, the expression for the
sound pressure generated by a point source with the volume velocity Q e jωt becomes
jρωQ e j(ωt − kr )
pˆ = . (1.6.6)
4πr
A vanishingly small sphere with a finite volume velocity36 may seem to be a rather academic
source. However, the monopole is a central concept in theoretical acoustics. At low frequen-
cies it is a good approximation to any source that produces a net displacement of volume, that
is, any source that is small compared with the wavelength and changes its volume as a func-
tion of time, irrespective of its shape and the way it vibrates. An enclosed loudspeaker is to a
good approximation a monopole at low frequencies. A source that injects fluid, the outlet of
an engine exhaust system, for example, is also in effect a monopole.
The sound intensity generated by the monopole can be determined from eq. (1.5.10):
1 ⎞ ⎫ ( ρω Q )
2
1 1 ⎧ jρωQ − jρωQ* ⎛
I r = Re { pu
ˆ ˆr*} = Re ⎨ ⎜1 − ⎟⎬ = . (1.6.7)
2 2 ⎩ 4πr 4πr ρ c ⎝ jkr ⎠ ⎭ 32π 2 r 2 ρ c
By multiplying with the surface of the area of a sphere with the radius r we get the sound
power radiated by the monopole,
36
The volume velocity of the monopole is sometimes referred to as the source strength. However,
some authors use other definitions of the source strength. The term ‘volume velocity’ is unambiguous.
37
( ρω Q )
2 2
ρ ck 2 Q
Pa = 4π r =
2
. (1.6.8)
32π 2 r 2 ρ c 8π
We could also obtain this result from eqs. (1.5.14) and (1.6.5), of course. Note that the sound
power is proportional to the square of the frequency, indicating that a small pulsating sphere
is not a very efficient radiator of sound at low frequencies.
Reciprocity
The reciprocity principle states that if a monopole source at a given point generates a certain sound
pressure at a another point then the monopole would generate the same sound pressure if we interchange listener
and source position, irrespective of the presence of reflecting or absorbing surfaces. This is a strong statement
with many practical implications.
The sound field generated by a monopole in front of a rigid plane surface of infinite
extent, say at x = 0, can easily be calculated if one makes use of the concept of image
sources. Because the surface is rigid we have the boundary condition ux = 0 at x = 0, and sim-
ple symmetry considerations show that this is satisfied if we replace the rigid plane with an
image source; see figure 1.6.1. It follows that the resulting sound pressure is the sum of the
sound pressures generated by the two monopoles:
jρωQ e j(ωt − kr1 ) jρωQ e j(ωt − kr2 )
pˆ = + . (1.6.9)
4πr1 4πr2
Figure 1.6.1 A monopole near a rigid plane surface.
Since r1 = r2 at all points on the plane it follows that the sound pressure is doubled
here. It is also apparent that the sound pressure at all positions in the field is doubled if the
monopole is placed on the plane. However, in the general case of the monopole being at dis-
tance of h from the plane we have an interference field.
If r >> h we can approximate r1 and r2 by
r1 r − h cos θ , r2 r + h cos θ , (1.6.10a, 1.6.10b)
in the complex exponentials. In the denominators the approximation r1 r2 r will do. The
result is
jρωQ j(ωt − k ( r − h cosθ )) j(ωt − k ( r + h cosθ )) jρωQ
pˆ
4πr
( e +e ) =
2πr
cos(kh cos θ ) e j(ωt − kr ) . (1.6.11)
Inspection of eq. (1.6.11) leads to the conclusion that the sound pressure in the far field de-
pends on kh and on θ unless kh << 1, in which case the sound pressure is simply doubled, as
mentioned above.
38
The sound power of the monopole is affected by the presence of the reflecting surface
unless it is far away, kh >> 1. We can calculate the sound power by integrating the sound in-
tensity over a hemisphere, cf. eq. (1.5.7). (Since the normal component of the particle veloc-
ity is zero at all points on the plane between the source and the image source, the normal
component of the intensity is also zero, so this surface does not contribute to the integral.)
Moreover, the considerations that lead to eq. (1.5.13) are also valid for combinations of
sources. It follows that
2 2
π 2 2π pˆ ρ ck 2 Q π 2
Pa = ∫ ∫ r sin θ dϕ dθ =
2
∫ cos 2 (kh cos θ ) sin θ dθ
0 0 2ρ c 4π 0
2 2
(1.6.12)
ρ ck 2 Q kh ρ ck 2 Q ⎛ sin(2kh) ⎞
=
4πkh ∫ 0
cos 2 xdx =
8π
⎜1 +
⎝ 2kh ⎠
⎟.
Figure 1.6.2 shows the factor in parentheses. It is apparent that the rigid surface has an insig-
nificant influence on the sound power output of the source when h exceeds a quarter of a
wavelength.
Figure 1.6.2 The influence of a rigid surface on the sound power of a monopole.
Example 1.6.1
The approximations of eqs. (1.6.10) and (1.6.11) are only valid in the far field. In the general case we
can rewrite eq. (1.6.9) in form
jρωQ j(ωt − kr1 ) ⎛ r1 jk (r − r ) ⎞

pˆ = e ⎜1 + e 1 2 ⎟ ,
4πr1 ⎝ r2 ⎠
where the parenthesis shows the effect of the reflecting plane, that is, it represents the sound pressure normalised
by the free field value. The normalised equation can be used for studying outdoor sound propagation over a hard
surface, and it is common practice to present the ‘ground effect’, that is, the effect of reflections from the ground
on outdoor sound propagation, in this form.
At very low frequencies k(r1 - r2) << 1, and the rigid surface has the effect of increasing the sound pres-
sure by a factor of 1+ r1/r2. Destructive interference occurs when the second term in the parenthesis is real and
negative, and the first interference dip occurs when k(r1 - r2) = π, corresponding to (r1 - r2) being half a wave-
length. Figure 1.6.3 shows the sound pressure relative to free field for sound propagation over a rigid plane sur-
face.
39
Figure 1.6.3 The sound pressure in one-third octave bands generated by a monopole above a rigid plane and
shown relative to free field for five different source-receiver distances.(From ref. [16].)
Example 1.6.2
It can be deduced from eq. (1.6.12) that two identical monopoles in close proximity (two enclosed
loudspeakers driven with the same signal, for example) will radiate twice as much sound power as they do when
they are far from each other at very low frequencies. The physical explanation is that the radiation load on each
source is doubled; the sound pressure on each source is not only generated by the source itself but also by the
neighbouring source.
Two monopoles of the same volume velocity but vibrating in antiphase constitute a
point dipole if the distance between the two monopoles is much less than the wavelength. It is
clear that the combined source has no net volume velocity. A point dipole is a good approxi-
mation to a small vibrating body that does not change its fixed volume as a function of time.
Such a source exerts a force on the fluid. The oscillating sphere shown in figure 1.6.5, for ex-
ample, is in effect a dipole, and so is an unenclosed loudspeaker unit. Other examples include
vibrating beams and wires.
Figure 1.6.4 A point dipole.
The sound pressure generated by the two monopoles is

jρωQ e j(ωt − kr1 ) jρωQ e j(ωt − kr2 )
pˆ = − . (1.6.13)
4πr1 4πr2
The nearfield of this combination of sources is fairly complicated. However, the far field is
40
relatively simple. We can calculate the sound pressure in the far field in the same way we
used in deriving eq. (1.6.11),
jρωQ j(ωt − k ( r + h cosθ )) j(ωt − k ( r − h cosθ )) ρωQ

pˆ
4πr
( e −e ) =
2πr
sin(kh cos θ ) e j(ωt − kr )
(1.6.14)
ρ chk Q
2
cos θ e j(ωt − kr ) .
2πr
Note that the sound pressure is proportional to h|Q|, varies as cosθ and is identically zero in
the plane between the two monopoles.37
Figure 1.6.5 Fluid particles in the sound field generated by an oscillating sphere. (From ref. [1].)
The sound power of the dipole is calculated by integrating the mean square sound
pressure over a spherical surface centred midway between the two monopoles:
2 2
π 2π pˆ ρ ch 2 k 4 Q π
Pa = ∫ ∫ r sin θ dϕ dθ =
2
∫ cos 2 θ sin θ dθ
0 0 2ρ c 4π 0
(1.6.15)
2 2
ρ ch k Q 2 4
1 ρ ch k Q
2 4
=
4π ∫ −1
x 2 dx =
6π
.
Note that the sound power of the dipole is proportional to the fourth power of the frequency,
indicating very poor sound radiation at low frequencies. The physical explanation of the poor
radiation efficiency of the dipole is of course that the two monopoles almost cancel each
other.
1.6.2 Sound radiation from a circular piston in an infinite baffle

Apart from the pulsating sphere, a vibrating circular piston in an infinite, rigid baffle
is one of the simplest cases of a spatially extended sound source that can be dealt with ana-
37
The quantity 2hQ is referred to by some authors as the dipole strength. However, other authors use
other definitions.
41
lytically. It is often used in connection with loudspeaker modelling. Here, it exemplifies two
important characteristics of any sound source: i) its directivity and ii) its radiation impedance.
These two characteristics are the most fundamental properties of a loudspeaker.
The basic approach to extended sound sources is to consider them as composed of
many simple sources, just as a dipole is made up of two monopoles. Thus, the piston is the
sum of many monopoles that all radiate in phase. Because of the presence of the infinite baf-
fle, just as many monopoles, vibrating in phase with the first ones, must be considered just on
the other side of the baffle (images sources, cf. eq. (1.6.9)). Let the complex volume velocity
of each elementary monopole be UdS, which implies that the complex velocity of the piston
is U. By linear superposition we conclude that the sound pressure radiated by the piston can
be evaluated at any position in front of the baffle simply by integrating over the surface of the
piston,
e j(ωt − kh )
pˆ = jωρ ∫ UdS . (1.6.16)
S 2πh
This is a special case of what is known as Rayleigh’s integral, which can be used for comput-
ing the sound radiation into half space of an infinite surface with a given vibrational velocity
[17]. The quantity h is the distance between the position of evaluation and the running posi-
tion on the piston, and S is the surface of the piston of radius a (see figure 1.6.6). Note the
factor of two in the denominator instead of four for the monopole, which is due to the contri-
bution of the image sources.
Figure 1.6.6 Definition of the variables. (From ref. [18].)
The far field sound pressure, that is, the sound pressure at long distances from the
centre of the piston compared with the radius and the wavelength, can be evaluated by ex-
panding h in the complex exponential,
y
h = r 2 + y 2 − 2ry sin θ cos ϕ r 1 − 2 sin θ cos ϕ
r (1.6.17)
r − y sin θ cos ϕ ,
42
while retaining only the first term of eq. (1.6.17) in the denominator. Thus the expression for
the sound pressure becomes
jωρU j(ωt − kr ) 2π a jkysinθ cosϕ

pˆ (r ,θ )
2πr
e ∫0 ∫0 e ydydϕ . (1.6.18)
The calculation makes use of the Bessel functions J0(z) and J1(z), defined by
1 2π jz cos β
2π ∫ 0
J0 ( z) = e dβ (1.6.19)
and
1 z
z ∫0
J1 ( z ) = β J 0 ( β )dβ (1.6.20)
(see figure 1.6.7), and leads to the following expression for the far field sound pressure,
Figure 1.6.7 Bessel functions of order 0, 1 and 2.
jωρU a J1 (ka sin θ ) j(ωt − kr ) jωρ Q ⎡ 2 J1 (ka sin θ ) ⎤ j(ωt − kr )

pˆ (r ,θ ) = e = e (1.6.21)
r k sin θ 2πr ⎢⎣ ka sin θ ⎥⎦
where we have introduced the volume velocity of the piston, Q = π a2U. The factor in brack-
ets is called the directivity of the piston, which is a frequency dependent function that de-
scribes the directional characteristics of the source in the far field,
⎡ 2 J (ka sin θ ) ⎤
D( f ,θ ) = ⎢ 1 . (1.6.22)
⎣ ka sin θ ⎥⎦
This function has its maximum value, unity, when θ = 0, indicating maximum radiation in the
axial direction all frequencies. Figure 1.6.8 shows the directivity for different values of the
normalised frequency ka. Note that the piston is an omnidirectional source (a monopole
placed on a rigid surface) at low frequencies, just as one would expect.
43
Figure 1.6.8 Directivity of the piston as a function of the normalised frequency ka. (From ref. [18].)
The sound pressure on the axis can also be evaluated fairly easily. Since sin θ = 0 on
the axis, the expression for the distance h reduces to
h = r 2 + y2 , (1.6.23)
from which,
ydy y
dh = = dy . (1.6.24)
r +y
2 2 h
Thus the sound pressure on the axis is given by
jωρU jωt 2π r 2 + a2
pˆ = e ∫ ∫ e- jkh dhdϕ = ρ cUe jωt ⎡⎢e- jkr − e- jk r 2 + a2 ⎤.
⎥⎦ (1.6.25)
2π 0 r ⎣
If we introduce the quantity
∆= (r 2
)
+ a2 − r 2 , (1.6.26)
the sound pressure can be written
pˆ = 2 jρcU e j(ωt −k [r + ∆ ]) sin(k∆). (1.6.27)
It can be seen that the sound pressure is zero when k) is a multiple of π, that is, when ) is a
multiple of half a wavelength, corresponding to the positions
44
⎡ 1 a n λ⎤
rp =0 = a ⎢ − (1.6.28)
⎣ 2n λ 2 a ⎥⎦
on the axis, where n is a positive integer. In a similar way, the sound pressure assumes a
maximum value for
2∆ = r 2 + a 2 − r = (2m + 1)λ 2 (1.6.29)
(where m is a positive integer), that is, for
⎡ 1 a 2m + 1 λ ⎤
rp =max = a ⎢ − . (1.6.30)
⎣ 2m + 1 λ 4 a ⎥⎦
Figure 1.6.9 shows the normalised sound pressure on the axis of the piston as a function of the
distance, which for a given frequency is defined by the corresponding ka-factor.
Figure 1.6.9 Sound pressure on the axis of a baffled piston for ka/2π = 5.5. (From Pierce 1989.)
It may seem surprising that the sound pressure is zero at some positions right in front of the
vibrating piston. The explanation is destructive interference, caused by the fact that the dis-
tance from such a position to the various parts of the piston varies in such a manner that the
contributions cancel out.
Example 1.6.3
In the far field, when r >> a and r >> a2/λ, one obtains
1⎡ ⎛ a2 ⎞ ⎤ a2
∆ ⎢ r ⎜1 + 2 ⎟ − r⎥ = ,
2 ⎣⎢ ⎝ 2r ⎠ ⎦⎥ 4r
and the sound pressure reduces to
⎛ ka 2 ⎞ j(ωt − kr ) jρ ckQ j(ωt − kr )

pˆ = jρ cU ⎜ ⎟e = e .
⎝ 2r ⎠ 2πr
45
This expression agrees with eq. (1.6.21) for θ = 0 (D(f) = 1), as of course it should. This asymptotic expression is
plotted as a dashed line in figure 1.6.9.
Example 1.6.4
The distances at which the minima occur, normalised by the radius of the piston, are given in terms of
normalised frequencies by
r = ⎡ ka − πn ⎤ .
a ⎢⎣ 4πn ka ⎥⎦
Minima of order n only occur for ka ≥ 2πn > 6. Thus for a loudspeaker with a radius of 50 mm, minima only
occur at frequencies higher than 6900 Hz, that is, far above the frequencies at which the piston approximation is
valid. It follows that the minima are never observed in real life!
In the nearfield there is no possible approximation. However, by developing the

spherical monopole field in cylindrical coordinates, the force exerted on the piston can be cal-
culated analytically. The calculations are rather complicated (see the book by Pierce or the
book by Morse and Ingard for a complete treatment), and lead to an expression in terms of
special functions such as Bessel and Struve functions. The result is,
⎡ J (2ka) H (2ka) ⎤
Fˆ = ∫ pˆ dS = ρ cπa 2U e jωt ⎢1 − 1 +j 1 , (1.6.31)
S
⎣ ka ka ⎥⎦
where H1 is the first Struve function.

The radiation impedance is the impedance seen by the piston, that is, the ratio of the
average sound pressure to the volume velocity,
< pˆ > Fˆ
Z a,r = = . (1.6.32)
Q e jωt SQ e jωt
Inserting eq. (1.6.31) gives
ρc ⎡ J1 (2ka) H (2ka) ⎤
Z a,r = 1− +j 1 . (1.6.33)
πa ⎢⎣
2
ka ka ⎥⎦
Figure 1.6.10 shows the normalised, dimensionless radiation impedance (the bracket in eq.
(1.6.33)),
Z a,r πa 2
= R1 + jX 1. (1.6.34)
ρc
At low frequencies and at high frequencies the radiation impedance takes simple expressions:
46
1 8
ka << 1 Z a,r = ra,r + jω ma,r = ρ ck 2 + jωρ , (1.6.35a)
2π 3aπ 2
ρc 2 ρc ⎛ 4 π⎞
ka >> 1 Z a,r = 2
+ jωρ 22 3
= 2 ⎜1 + j ⎟. (1.6.35b)
πa πk a πa ⎝ 2ka ⎠
Figure 1.6.10 Normalised radiation impedance of a piston as a function of the normalised frequency. (From
Pierce 1989.)
The first expression is fundamental for designing loudspeakers. Note that the real part of the
radiation impedance equals that of a small pulsating sphere, eq. (1.6.5), multiplied by a factor
of two because of the rigid plane. The quantity ma,r can be interpreted as the acoustic mass of
the air driven along by the piston in its movement. Interference effects in the nearfield make it
different from the imaginary part of impedance of the pulsating sphere. However, as in the
case of the pulsating sphere, the imaginary part of the acoustic radiation impedance diverges
when the radius a goes to zero.
Example 1.6.5
The mechanical radiation impedance is given by eqs. (1.4.4) and (1.6.34) as Zm,r = ρcπa2(R1+jX1). Its
low frequency approximation is therefore:
πa 4 ρ ck 2 8a 3
Z m,r = + jωρ .
2 3
The imaginary part of this impedance is the impedance of the mass of a layer of air in front of the piston. This
layer of air is moving back and forth as if it were incompressible.
47
The radiated sound power is defined in section 1.5 as the integral of the normal com-
ponent of the sound intensity over a surface than encloses the source. This method can also be
used for computing the sound power of a piston in an infinite baffle. However, by far the sim-
plest approach is to use eq. (1.5.15), which expresses the sound power in terms of the mean
square volume velocity and the real part of the acoustic radiation impedance:
ρc ρc ⎡ J1 (2ka ) ⎤
Pa = 12 Q Re {Z a,r } = 12 Q
2 2 2
R1 = 12 Q 1− . (1.6.36)
πa 2
πa ⎢⎣
2
ka ⎥⎦
At low frequencies this becomes, with eq. (1.6.35a),
2
ρ ck 2 Q
Pa = , (1.6.37)
4π
which is just what we would expect since the piston acts as a monopole on a rigid plane in
this frequency range (cf. eq. (1.6.12)).
Example 1.6.6
Instead of using the volume velocity and the acoustic impedance we could equally well compute the
sound power from the mean square velocity and the real part of the mechanical radiation impedance, since, with
eq. (1.4.4),
Q Re {Z a,r } = 12 U Re {Z m,r }.
2 2
Pa = 1
2
Example 1.6.7
Equation (1.6.37) shows that the sound power of the piston is proportional to |ωQ|2 at low frequencies,
that is, the sound power is independent of the frequency if the volume acceleration is independent of the fre-
quency. This implies that the displacement of the piston should be inversely proportional to the square of the
frequency if we want the sound power to be independent of the frequency. In other words, it implies very large
displacements at low frequencies. Since mechanical systems such as loudspeakers only allow a limited excur-
sion, the low frequency sound power output of a loudspeaker is limited: the only way left to increase the sound
power is to increase the size of the membrane. This explains why very large loudspeakers are found in subwoof-
ers.
The directivity factor of a source is the sound intensity on the axis in the far field nor-
malised by the sound intensity of an omnidirectional source with the same sound power. From
eq. (1.6.21) the sound intensity on the axis is
2
1 ⎛ Q ⎞
I r = ρck 2 ⎜⎜ ⎟
⎟ (1.6.38)
2 ⎝ 2πr ⎠
(see also example 1.6.3). Normalising with Pa/4πr2 (eq. (1.6.36)) gives the directivity factor
Q(f),
48
Q( f ) =
(ka )2 =
(ka )2 . (1.6.39)
R1 J (2ka)
1− 1
ka
The directivity factor of the piston is plotted in figure 1.6.11 as a function of the normalised
frequency ka. Note that the directivity factor approaches two at low frequencies rather than
one, reflecting the fact that all the sound power is radiated in only half a sphere.
In practice, one often uses the directivity index, defined by
DI ( f ) = 10 log Q( f ). (1.6.40)
Q(f)
15
10
ka
0
0 1 2 3 4
Figure 1.6.11 Directivity factor of a piston in a baffle. (From ref. [18].)
1.7 REFERENCES
1. L. Cremer and M. Hubert: Vorlesungen über Technische Akustik (3rd edition). Sprin-
ger-Verlag, Berlin, 1985.
2. T.D. Rossing, F.R. Moore and P.A. Wheeler: The Science of Sound (3nd edition). Ad-
dison Wesley, San Francisco, CA, 2002.
3. M.J. Crocker and F. Jacobsen: Sound intensity. Chapter 156 in Encyclopedia of
Acoustics, ed. M.J. Crocker. John Wiley & Sons, New York, 1997.
4. F. Jacobsen: A note on instantaneous and time-averaged active and reactive sound
intensity. Journal of Sound and Vibration 147, 1991, 489-496.
5. F. Jacobsen: An elementary introduction to applied signal analysis. Acoustic Tech-
nology, Ørsted•DTU, Technical University of Denmark, Note no 7001, 2004.
6. R.B. Randall: Frequency analysis (3rd edition). Brüel & Kjær, Nærum, 1987.
7. D.W. Martin and W.D. Ward: Subjective evaluation of musical scale temperament in
pianos. Journal of the Acoustical Society of America 33, 1961, 582-585.
49
8. R.W. Krug: Sound level meters. Chapter 155 in Encyclopedia of Acoustics, ed. M.J.
Crocker, John Wiley & Sons, New York, 1997.
9. J. Pope: Analyzers. Chapter 107 in Handbook of Acoustics, ed. M.J. Crocker. John
Wiley & Sons, New York, 1998.
10. P.V. Brüel, J. Pope and H.K. Zaveri: Introduction to acoustical measurement and in-
strumentation. Chapter 154 in Encyclopedia of Acoustics, ed. M.J. Crocker. John
Wiley & Sons, New York, 1997.
11. Anon.: Microphone Handbook. Brüel & Kjær, Nærum, 1996.
12. F. Jacobsen: Propagation of sound waves in ducts. Acoustic Technology, Ør-
sted•DTU, Technical University of Denmark, Note no 31260, 2005.
13. K. Rasmussen: Analogier mellem mekaniske, akustiske og elektriske systemer (4th
edition). Polyteknisk Forlag, Lyngby, 1994.
14. W. Marshall Leach, Jr.: Introduction to Electroacoustics and Audio Amplifier Design
(2nd edition). Kendall/Hunt Publishing Company, Dubuque, IA, 1999.
15. Z. Maekawa and P. Lord: Environmental and Architectural Acoustics. E & FN Spon,
London, 1994.
16. E.M. Salomons: Computational Atmospheric Acoustics. Kluwer Academic Publish-
ers, Dordrecht, 2001.
17. W.S. Rayleigh: On the passage of waves through apertures in plane screens, and al-
lied theorems. Philosophical Magazine 43, 1897, 259-272.
18. K. Rasmussen: Lydfelter. Acoustic Technology, Ørsted•DTU, Technical University
of Denmark, Note no 2107, 1996.
1.8 BIBLIOGRAPHY
Although there is an enormous amount of literature available on different aspects of

acoustics only relatively few textbooks are devoted to the general mathematical theory of
acoustic wave motion. Recommended reading includes the following. The books by Fahy and
by Kinsler et al. are introductory, and so is the text by Nelson. More advanced treatments can
be found in the books by Pierce (1989), Morse and Ingard (1968) and Filippi et al.
1. A.D. Pierce: Mathematical theory of wave propagation. Chapter 2 in Encyclopedia of

Acoustics, ed. M.J. Crocker, John Wiley & Sons, New York, 1997.
2. F. Fahy: Foundations of Engineering Acoustics. Academic Press, San Diego, 2000.
3. L.E. Kinsler, A.R. Frey, A.B. Coppens and J.V. Sanders: Fundamentals of Acoustics
(4th edition). John Wiley & Sons, New York, 2000.
4. P.A. Nelson: An introduction to acoustics. Chapter 1 in Fundamentals of Noise and
Vibration, ed. F.J. Fahy and J. Walker. E & FN Spon, London, 1998.
5. P.M. Morse: Vibration and Sound (2nd edition). The American Institute of Physics,
New York, 1983.
6. A.D. Pierce: Acoustics. An Introduction to Its Physical Principles and Applications.
The Acoustical Society of America, New York, 1989.
7. P.M. Morse and K.U. Ingard: Theoretical Acoustics. McGraw-Hill, New York, 1968.
8. P. Filippi, D. Habault, J.P. Lefebre and A. Bergassoli: Acoustics: Basic Physics,
Theory and Methods. Academic Press, London, 1999.
50
1.9 APPENDIX: COMPLEX NOTATION
In a harmonic sound field the sound pressure at any point is a function of the type
cos(Tt + n). It is common practice to use complex notation in such cases. This is a symbolic
method that makes use of the fact that complex exponentials give a more condensed notation
that trigonometric functions because of the complicated multiplication theorems of sines and
cosines.
We recall that a complex number A can be written either in terms of its real and
imaginary part or in terms of its magnitude (also called absolute value or modulus) and phase
angle,
A = Ar + jAi = A e jϕ A , (1.9.1)
where
j = −1 (1.9.2)
is the imaginary unit, and
Ar = Re{A} = A cos ϕ A , Ai = Im{A} = A sin ϕ A , (1.9.3, 1.9.4)

A= Ar2 + Ai2 (1.9.5)
(see figure 1.9.1). The complex conjugate of A is

A* = Ar − jAi = A e − jϕ A ; (1.9.6)
therefore the magnitude can also be written

A= A ⋅ A* . (1.9.7)
Multiplication and division of two complex numbers are most conveniently carried out if they
are given in terms of magnitudes and phase angles,
A j(ϕ A −ϕ B )
AB = A B e j(ϕ A +ϕ B ) , A B= e . (1.9.8, 1.9.9)
B
Figure 1.9.1. Complex representation of a harmonic signal.
51
Complex representation of harmonic signals makes use of the fact that
e jx = cos x + jsin x (1.9.10)
(Euler’s equation) or, conversely,
cos x =
2
(
1 jx
e + e − jx ,) sin x = − j
2
(
1 jx
)
e − e − jx . (1.911a, 1.9.11b)
In a harmonic sound field the sound pressure at a given position can be written
pˆ = Ae jωt , (1.9.12)
where A is the complex amplitude of the sound pressure. The real, physical sound pressure is
of course a real function of the time,
{ }
p = Re{pˆ } = Re A e j(ωt +ϕ A ) = A cos(ωt + ϕ A ), (1.9.13)
which is seen to be an expression of the form cos(ωt + φ). The magnitude of the complex
quantity |A| is called the amplitude of the pressure, and φA is its phase.
The mathematical convenience of the complex representation of harmonic signals can
be illustrated by an example. A sum of two harmonic signals of the same frequency, A1ejωt
and A2ejωt, is yet another harmonic signal with an amplitude of |A1 + A2| (see figure 1.9.2).
Evidently, this can also be derived without complex notation,
p = A1 cos(ωt + ϕ1 ) + A2 cos(ωt + ϕ 2 )
= ( A1 cos ϕ1 + A2 cos ϕ 2 )cos ωt − ( A1 sin ϕ1 + A2 sin ϕ 2 )sin ωt (1.9.14)
(
= ( A1 cos ϕ1 + A2 cos ϕ 2 ) + ( A1 sin ϕ1 + A2 sin ϕ 2 )
2
) cos(ωt + ϕ ),
2 ½
where
A1 sin ϕ1 + A2 sin ϕ 2
ϕ = arctan , (1.9.15)
A1 cos ϕ1 + A2 cos ϕ 2
but the expedience and convenience of the complex method seems indisputable.
Figure 1.9.2. Two simple harmonic signals with identical frequencies. (From Kinsler et al. 2000.)
52
Since
d jωt
e = jωe jωt , (1.9.16)
dt
it follows that differentiation with respect to time corresponds to multiplication by a factor of

jω. Conversely, integration with respect to time corresponds to division with jω. If, for exam-
ple, the vibrational velocity of a surface is, in complex representation,
vˆ = Be jωt = B e j(ωt +ϕB ) , (1.9.17)
which means that the real, physical velocity is
v = Re{vˆ} = B cos(ωt + ϕ B ), (1.9.18)
then the acceleration is written
aˆ = jωvˆ, (1.9.19)
which means that the physical acceleration is
a = Re{aˆ} = Re{jωBe jωt } = −ω B sin(ωt + ϕ B ). (1.9.20)
and this is seen to agree with the fact that
d
cos(ωt + ϕ B ) = −ω sin(ωt + ϕ B ). (1.9.21)
dt
In a similar manner we find the displacement,
vˆ
ξˆ = , (1.9.22)
jω
which means that
⎧ 1 ⎫ 1
ξ = Re{ξˆ} = Re⎨ Be jωt ⎬ = B sin(ωt + ϕ B ), (1.9.23)
⎩ jω ⎭ ω
in agreement with the fact that
d⎛1 ⎞
⎜ sin(ωt + ϕ B ) ⎟ = cos(ωt + ϕ B ). (1.9.24)
dt ⎝ ω ⎠
53
Acoustic second-order quantities involve time averages of squared harmonic signals
and, more generally, products of harmonic signals. Such quantities are dealt with in a special
way, as follows. Expressed in terms of the complex pressure amplitude p̂ , the mean square
pressure becomes
2
p 2 = prms
2
= pˆ 2, (1.9.25)
in agreement with the fact that the average value of a squared cosine is ½. Note that it is the
squared magnitude of p̂ that enters into the expression, not the square of p̂ , which in general
would be a complex number proportional to e2jωt.
The time average of a product is calculated as follows,
xy =
1
2
{ } 1
{ }
Re xˆyˆ ∗ = Re xˆ ∗ yˆ
2 (1.9.26)
1
2
{
j (ϕ − ϕ )
}1
= xˆ yˆ Re e x y = xˆ yˆ cos(ϕ x − ϕ y ),
2
in agreement with the fact that
1
xy = xˆ cos(ωt + ϕ x ) yˆ cos(ωt + ϕ y ) = xˆ yˆ cos(ϕ x − ϕ y ). (1.9.27)
2
Note that either x̂ or ŷ must be conjugated.
54
Ear, Hearing and Speech
2 Ear, Hearing and Speech

Torben Poulsen
2.1 Introduction
The aim of the present chapter is to give the student a basic understanding of the function of
the ear, the perception of sound and the consequences for speech understanding. The content
covers the basic psychoacoustic aspects of a situation where two persons speak to each other.
The major topics are: the ear and its functional principles, basic psychoacoustics (hearing
threshold, loudness, masking) and speech intelligibility.
2.2 The Ear
Figure 2.2.1 Drawing of the ear. A is the outer ear. B is the middle ear. C is the inner ear. From [1]
The ear may be divided into four main parts: The outer ear, the middle ear, the inner ear and
the nerve connection to the brain. The first three parts (the peripheral parts) are shown in
Figure 2.2.1. Part A being the outer ear, B is the middle ear and C is the inner ear. The sound
will reach the outer ear, progress through the outer ear canal, reach the tympanic membrane
(the ear drum), transmit the movements to the bones in the middle ear, and further transmit
the movements to the fluid in the inner ear. The fluid movements will be transformed to
nerve impulses from the hair cells in the inner ear and the impulses are transmitted to the
brain through the auditory nerve.
55
Ear, hearing and speech
2.2.1 The outer ear

The outer ear consists of the pinna (or the auricle) and the ear canal. The Pinna plays an
important role for our localisation of sounds sources. The special shape of pinna produces
reflections and diffraction so that the signal that reaches the ear will be dependent on the
direction to the sound. The pinna has common features from person to person but there are
big individual differences in the details. Localisation of sound sources is difficult if a hearing
protector or a crash helmet covers the pinna. The outer part of the ear canal is relatively soft
whereas the inner part is stiff and bony. At the end of the ear canal the tympanic membrane is
situated. The length of the ear canal is approximately 25 mm and the diameter is
approximately 7 mm. The area is approximately 1 cm2. These numbers are approximate and
vary from person to person.
The ear canal may be looked upon as a tube that is closed in one end and open in the other.
This will give resonances for frequencies where the length of the ear canal corresponds to 1/4
of the wavelength of the sound. With a length of 25 mm and a speed of sound of 340 m/s the
resonance frequency will be
340 m / s
f res = = 3,4 kHz
4 * 0,025 m
This calculation is correct if the ear canal was a cylindrical tube. Most ear canals will have
one or two bends. This implies that it is usually not possible from the outside to see the
tympanic membrane at the end of the ear canal. It’s necessary to make the canal straighter,
which may be done by pulling pinna upward and backwards.
The tympanic membrane is found at the end of the canal. The membrane is not perpendicular
to the axis of the ear canal but tilted approx. 30 degrees. The tympanic membrane is shaped
like a cone with the top of the cone pointing inwards into the middle ear. The thickness is
approx. 0.1 mm.
2.2.2 The middle ear

The middle ear consists of three small bones: hammer, anvil and stirrup. The Latin names are
also often used: Malleus, Incus and Stapes. These bones are the smallest bones in the human
body. A drawing is shown in Figure 2.2.2. The function of the middle ear is to transmit the
vibrations of the tympanic membrane to the fluid in the inner ear. From Figure 2.2.2 it is seen
that the hammer (Maleus, M) is fixed to the tympanic membrane (1) from the edge and into
the centre of the membrane (the top of the cone). The anvil (Incus, I, 2) connects the hammer
and the stirrup (Stapes, S) and the footplate of the stirrup makes the connection into the inner
ear. This connection is sometimes called the oval window. The footplate rotates around the
point marked (3). The middle ear is filled with air and is connected to the nose cavity (and
thus the atmospheric pressure) through The Eustachian tube (ET, 4). The fluid in the inner
ear is incompressible and an inwards movement of the stirrup will be equalised by a
corresponding outward movement by the round window (5).
56
Figure 2.2.2 Drawing of the middle ear. See text for details. From [2]
Usually the Eustachian tube is closed but opens up when you swallow or yawn. When the
tube is open, the pressure at the two sides of the tympanic membrane is equalised. If the
Eustachian tube becomes blocked (which is typically the case when you catch a cold) the
equalisation will not take place and after some time the oxygen in the middle ear will be
assimilated by the tissue and an under-pressure will build up in the middle ear. This causes
the tympanic membrane to be pressed inwards and thus the sensitivity of the hearing is
reduced.
The chain of middle ear bones forms a lever function that - together with the area ratio
between the tympanic membrane and the footplate of stapes - makes an impedance match
between the air in the outer ear and the liquid in the inner ear. The lever ratio is approx. 1.3
and the area ratio is approx. 14. The total ratio is thus 18, which corresponds to approx.
25 dB.
Two small muscles, tensor tympani (6) and stapedius (7), see Figure 2.2.2, are attached to the
bones and will be activated by the so-called middle ear reflex. The reflex is elicited when the
ear is exposed to sounds above approx. 70 dB SPL whereby the transmission through the
middle ear is reduced. The reduction is about 20 dB at 125 Hz, 10 dB at 1000 Hz and less
than 5 dB at frequencies above 2000 Hz. The middle ear reflex can to some extent protect the
inner ear from excessive exposure. Because the reflex is activated by a signal from the brain
there will be a delay of about 25 to 150 ms before the effect is active. The reflex has
therefore no protective effect on impulsive sounds.
2.2.3 The inner ear

The inner ear consists of a snail-shell shaped structure in the temporal bone called Cochlea.
The cochlea is filled with lymph and is closely connected to the balance organ that contains
the three semicircular canals. There are 2.75 turns in the snail shell and the total length from
the base to the top is 32 mm. A cross section of one of the turns is shown in Figure 2.2.3.
57
This figure shows that the cochlea is divided into three channels (latin: Scala) called scala
vestibuli (1), scala media (2), and scala tympani (3).
Figure 2.2.3 Cross section of a cochlea turn. See text for details. From [1]
There are two connections (windows) from cochlea to the middle ear cavity. The oval
window is the footplate of the stirrup and is connected to Scala Vestibuli (1). The round
window is connected to Scala Tympani (3). The round window prevents an over-pressure to
build up when the oval window moves inwards. Scala Vestibuli and Scala Tympani are
connected at the top of the cochlea with a hole called Helicotrema.
The Basilar membrane (6 in Figure 2.2.3) divides scala tympani from scala media. The width
of the basilar membrane (BM) changes from about 0.1 mm at the base of the cochlea to about
0.5 mm at the top of the cochlea (at helicotrema). The change of the BM-width is thus the
opposite of the width of the snail shell. The function of the BM is very important for the
understanding of the function of the ear.
A structure - called the organ of Corti - is positioned on top of the Basilar Membrane in Scala
Media. The organ of Corti consists of one row of inner hair cells (7 in Figure 2.2.3) and three
rows of outer hair cells (8 in Figure 2.2.3). The designations ‘inner’ and ‘outer’ refer to the
centre axis of the snail shell which is to the left in Figure 2.2.3. The hair cells are special
nerve cells where small hairs protrude from the top of the cells. There are approx. 3000 inner
hair cells and about 12000 outer hair cells. A soft membrane (5 in Figure 2.2.3) covers the
top of the hair cells. The organ of Corti transforms the movements of the Basilar membrane
to nerve impulses that are then transmitted to the hearing centre in the brain.
The inner hair cells are the main sensory cells. Most of the nerve fibres are connected to the
inner hair cells. When sound is applied to the ear, the basilar membrane and the organ of
Corti will vibrate and the hairs on the top of the hair cells will bend back and forth. This will
trigger the (inner) hair cells to produce nerve impulses.
The outer hair cells contain muscle tissue and these cells will amplify the vibration of the
basilar membrane when the ear is exposed to weak sounds so that the vibrations are big
enough for the inner hair cells to react. The amplification function of the outer hair cells is
nonlinear which means that they have an important effect at low sound levels whereas they
58
are of almost no importance at high sound levels. The amplifier function - sometimes called
the cochlear amplifier - is destroyed if the ear is exposed to loud sounds such as gunshots or
heavy industrial noise. This is called a noise induced hearing loss. The amplifier function
also deteriorates with age. This is called an age related hearing loss.
2.2.4 The frequency analyzer at the Basilar membrane

The basilar membrane acts like a frequency analyser. When the ear is exposed to a pure tone
the movement of the basilar membrane will show a certain pattern and the pattern is
connected to a certain position on the basilar membrane. If the frequency is changed, the
pattern will not change but the position of the pattern will move along the basilar membrane.
This is illustrated in Figure 2.2.4 for the frequencies 400 Hz, 1600 Hz and 6400 Hz. The
400 Hz component produce BM-movement close to the top of the cochlea. 6400 Hz produces
a similar pattern but close to the base of the cochlea. Note that a single frequency produces
movements of the basilar membrane over a broad area. This means that even for a single
frequency many hair cells are active at the same time. Note also that the deflection of the BM
is asymmetrical. The envelope of the deflection (shown dotted in Figure 2.2.4) has a steep
slope towards the low frequency side and a much less steep slope towards the high frequency
side. The same different slopes are also found in masking thresholds and it can be shown that
masking is closely related to the basilar membrane movements.
Figure 2.2.4 Movement of the basilar membrane (b) when the ear is exposed to a combination of 400
Hz, 1600 Hz and 6400 Hz (a). O.W.: Owal window (base of cochlea). Hel: Helicotrema (top of
cochlea). From [3]
The non-linear behaviour of the outer hair cells and their influence on the BM movement is
illustrated in Figure 2.2.5. This figure shows the BM-amplitude at a certain position of the
basilar membrane as a function of the stimulus frequency. (Note that this is different from
Figure 2.2.4 where the amplitude is shown as a function of basilar membrane position for
different frequencies). There are at least three nonlinear phenomena illustrated in the figure.
1) At low exposure levels (20 dB) the amplitude is very selective and a ‘high’ amplitude is
achieved only in a very narrow frequency range. For high exposure levels (80 dB) the
59
‘high’ amplitude is achieved at a much wider frequency range. Thus, the filter
bandwidth of the auditory analyser changes with the level of the incoming sound.
2) The frequency where the maximum amplitude is found change with level. At high
levels it is almost one octave below the max-amplitude frequency at low levels.
3) The maximum amplitude grows non-linearly with level. At low levels (20 dB) the
maximum BM-amplitude is about 60 dB (with some arbitrary reference). At an input
level of 80 dB the maximum BM amplitude is about 85 dB. In other words the change
in the outside level from 20 dB to 80 dB, i.e., 60 dB, is reduced (compressed) to a
change in the maximum BM-amplitude of only 25 dB.
These non-linear phenomena are caused by the function of the outer hair cells. The increase
of amplitude at low levels is sometimes called ‘the cochlear amplifier’. In a typical cochlear
hearing loss, the outer hair cells are not functioning correctly or may be destroyed. In other
words: The cochlear amplifier does not work. This will be seen as an elevated hearing
threshold and this is called a hearing loss.
80
Basilar membrane movement, dB
80 dB SPL
60
60 dB
40
40 dB
20
20 dB SPL
0
2 3 5 10 20 30
Frequency, kHz
Figure 2.2.5 Movement of the Basilar membrane at a fixed point for stimulus levels from 20 dB SPL
to 80 dB SPL. Redrawn from [4]
60
2.3 Human hearing

The human hearing can handle a wide range of frequencies and sound pressure levels. The
weakest audible sound level is called the hearing threshold and the sound level of the loudest
sound is called the threshold of discomfort or the threshold of pain.
2.3.1 The hearing threshold
The hearing threshold is frequency dependent, see Figure 2.3.1. At 1000 Hz the threshold is
about 2 dB SPL whereas it is about 25 dB SPL at 100 Hz and about 15 dB at 10000 Hz.
100
80
Sound pressure level, dB
60
40
20
20 50 200 500 2000 5000 20000

10 100 1000 10000
Frequency, Hz
Figure 2.3.1 The binaural hearing threshold in a free field. From [5]
The threshold curve in Figure 2.3.1 is measured under the following conditions:
• Free field (no reflections from walls, floor, ceiling)
• Frontally incoming sound (called frontal incidence)
• signals are single pure tones
• binaural listening (i.e. listening with both ears)
• no background noise
• test subjects between 18 and 25 years of age
• the threshold is determined by means of either the ascending or the bracketing method
The curve is the median value (not the mean) over the subject’s data. The sound pressure
level, which is shown in the figure, is the level in the room at the position of the test subject’s
head but measured without the presence of the test subject. This curve is also called the
absolute threshold (in a free field) and data for the curve may be found in ISO 389-7 [6] and
in ISO 226 [5].
61
In ISO 389-7 also threshold data for narrow band noise in a diffuse sound field are found.
The threshold curve is similar to the curve in Figure 2.3.1 and deviates from the pure tone
curve only by a few dB (-2 to +6) in the frequency range 500 Hz to 16 kHz.
2.3.2 Audiogram
For practical use it is not convenient to measure the hearing threshold in a free or a diffuse
sound field in the way described in the previous section. For practical and clinical purposes,
usually only the deviation from normal hearing is of interest. Such deviations are determined
by means of a calibrated audiometer and the result of the measurement is called an
audiogram.
Frequency, Hz
125 250 500 1k 2k 4k 8k
-10 -10
0 0
Hearing threshold level, dB HL
10 10
20 20
30 30
40 40
50 50
60 60
70 70
80 80
Figure 2.3.2 Audiogram for a typical age related hearing loss.
Figure 2.3.2 shows an audiogram for a person in the frequency range 125 Hz til 8000 Hz.
The zero line indicates the average threshold for young persons and a normal audiogram will
give data points within 10 to 15 dB from the zero line. An elevated hearing threshold (i.e. a
hearing loss) is indicated downwards in an audiogram and the values are given in dB HL.
The term ‘HL’ (hearing level) is used to emphasise that it is the deviation from the average
normal hearing threshold.
The measurements are performed with headphones for each ear separately. The results from
the left ear are shown with '×' and the results from the right ear are shown with '○'.
Sound pressure level, dB SPL, and hearing level, dB HL, is not the same. An example: From
Figure 2.3.1 it can be seen that the hearing threshold at 125 Hz is 22 dB SPL (measured in
the way described previously). If a person has a hearing loss of 5 dB HL at this frequency the
62
threshold would be 27 dB SPL. In an audiogram the 5 dB hearing loss will be shown as a

point 5 dB below the zero line (e.g. right ear, Figure 2.3.2). Another example: At 4000 Hz the
free field threshold is -6 dB (see Figure 2.3.1). A hearing loss of 50 dB HL (e.g. left ear,
Figure 2.3.2) will give a threshold of 44 dB SPL.
In order for the audiometry to give correct results, the audiometer must be calibrated
according to the ISO 389 series of standards. These standards specify the SPL values that
shall be measured in a specific coupler (an artificial ear) when the audiometer is set to 0 dB
HL. The values in the standards are headphone specific, which means that the audiometer
must be recalibrated if the headphone is exchanged with another headphone.
Table 2.3.1 shows reference values for two headphones commonly used in audiometry.
F, Hz 125 250 500 1k 2k 3k 4k 6k 8k 10k 12,5k 14k 16k
TDH 39 45,0 25,5 11,5 7,0 9,0 10,0 9,5 15,5 13,0 - - - -
HDA 200 30,5 18,0 11,0 5,5 4,5 2,5 9,5 17,0 17,5 22,0 28,0 36,0 56,0
Table 2.3.1. Calibration values in dB SPL for a Telephonics TDH 39 earphone and a Sennheiser HDA
200 earphone. The TDH 39 earphone can not be used above 8 kHz. The TDH 39 data are from ISO
389-1 [7]. The HDA 200 data are from ISO 389-5 [8] and ISO 389-8 [9].
2.3.3 Loudness Level

The definition of loudness levels is as follows: For a given sound, A, the loudness level is
defined as the sound pressure level (SPL) of a 1000-Hz tone which is perceived equally loud
as sound A. The unit for loudness level is Phon (or Phone). In order to measure loudness
level a 1 kHz tone is needed and this tone should then be adjusted up and down in level until
it is perceived just as loud as the other sound. When this situation is achieved, the sound
pressure level of the 1 kHz tone is per definition equal to the loudness level in phone. For a
1000-Hz tone the value in dB SPL and in Phone will be the same.
The loudness level for pure tones has been measured for a great number of persons with
normal hearing under the same conditions as for the absolute threshold (Figure 2.3.1). The
result is shown in Figure 2.3.3.
63
140
120
Sound pressure level, dB
100
80
60
40
20
20 50 200 500 2000 5000 20000

10 100 1000 10000
Frequency, Hz
Figure 2.3.3 Equal loudness level contours. Redrawn from [5]
Some examples, see Figure 2.3.3: A 4000-Hz tone at 26 dB SPL will be perceived with the
same loudness as a 1000-Hz tone at 30 dB SPL and thus the loudness level of the 4000 Hz
tone is 30 Phone. A 125-Hz tone at 90 dB SPL will have a loudness level of 80 Phone.
The curves in Figure 2.3.3 are - in principle - valid only for the special measurement situation
where the tones are presented one at a time. They should not be used directly to predict the
perception of more complicated signals such as music and speech because the curves do not
take masking and temporal matters into account. Reflections in a room are not taken into
account either.
Translations of Loudness Level:

Danish: Hørestyrkeniveau (enhed: Phon)
German: Lautstärkepegel (Einheit: Phon)
French: Niveau de Sonie.
2.4 Masking
The term ‘Masking’ is used about the phenomenon that the presence of a given sound (sound
A) can make another sound (sound B) inaudible, in other words A masks B or B is masked
by A. Masking is a very common phenomenon which is experienced almost every day, e.g.
when you need to turn down the radio in order to be able to use the telephone.
The situation described above is also called simultaneous masking because both the masking
signal and the masked signal are present at the same time. This is not the case in backward
and forward masking. Backward and forward refer to time. E.g. forward masking means
masking after a signal has stopped (i.e. forward in time). Simultaneous masking is best
64
described in the frequency domain and is closely related to the movements of the Basilar
membrane in the inner ear.
The masking phenomenon is usually investigated by determining the hearing threshold for a
pure tone when various masking signals are present. The threshold determined in this
situation is called the masked threshold contrary to the absolute threshold.
2.4.1 Complete masking

If the ear is exposed to white noise, the hearing threshold (i.e. masked threshold) will be as
shown in Figure 2.4.1 where also the absolute threshold is shown. The masked threshold is
shown for different levels of the white noise.
Figure 2.4.1 Masking from white noise. The curves show the masked threshold for different spectrum
levels of white noise. From [3]
The masked thresholds are almost independent of frequency up to about 500 Hz. Above
500 Hz the threshold increases by about 10 dB per decade (= 3 dB/octave). A 10-dB change
in the level of the noise will also change the masked threshold by 10 dB.
If a narrow band signal is used instead of the white noise, the masked threshold will be as
shown in Figure 2.4.2. Here the masked threshold is shown for a narrow band signal centred
at 250 Hz, 1 kHz and 4 kHz respectively. Generally the masking curves have steep slopes
(about 100 dB/octave) towards the low frequency side and less steep slopes (about
60 dB/octave) towards the high frequency side.
65
Figure 2.4.2 Masking from narrow band noise. The curves show the masked threshold when the ear is
exposed to narrow band noise (critical band noise) at 250 Hz, 1 kHz and 4 kHz respectively. From
[3]
The masking curves for narrow band noise are very level dependent. This is illustrated in
Figure 2.4.3. The slope at the low frequency side is almost independent of level but the slope
at the high frequency side depends strongly on the level of the narrow band noise. The dotted
lines near the top of the curves indicate experimental difficulties due to interference between
the noise itself and the pure tone used to determine the masked threshold.
Figure 2.4.3 The influence of level on the masked threshold. The slope towards higher frequencies
decreases with increasing level, i.e. masking increases non-linearly with level. From [3]
The masked threshold for narrow band noise is mainly caused by the basilar membrane
motion. The different slopes towards the low and the high frequency side are also seen here
and also the nonlinear level dependency is seen. Compare with Figure 2.2.4.
66
2.4.2 Partial masking

The term ‘Complete masking’ is used when the presence of a given sound (sound A) can
make another sound (sound B) inaudible. Partial masking is a situation where sound A
influences the perception of sound B even though sound B is still audible. The influence is
mainly seen in the loudness of sound B.
An example: When you listen to a standard car-radio while you are driving at, e.g. 100 km/h,
you will adjust the level of the radio to a comfortable level. There will be some background
noise from the engine, the tires, and the wind around the car (at least in ordinary cars). Then,
when you come to a crossing or a traffic light and have to stop you will hear that the radio-
volume is much too high. This is an example of partial masking where the background noise
masks part of the radio signal and when the background noise disappears the masking
disappears too and the radio signal becomes louder than before. (Some modern car radios are
equipped with a speed dependent automatic level control. The example above is therefore not
fully convincing in this situation.)
2.4.3 Forward masking

It has been shown that a strong sound signal can mask another (weak) signal which is
presented after the strong signal. This kind of masking goes forward in time and is therefore
called forward masking. The effect lasts for about 200 ms after the end of the strong signal.
Forward masking is also called post-masking.
2.4.4 Backward masking

It has been shown that a strong sound signal can mask another (weak) signal which appears
before the strong signal. This kind of masking goes back in time and is therefore called
backward masking. The effect is restricted to about 20 ms before the start of the strong
signal.
Backward masking is also called pre-masking.
67
2.5 Loudness
The term ‘loudness’ denotes the subjective perception of strength or powerfulness of the
sound signal. The unit for loudness is Son or Sone. Note that ‘loudness’ and ‘loudness level’
are two different concepts. Translation of terms:
Loudness Loudness Level

Danish Hørestyrke Hørestyrkeniveau
German Lautheit Lautstärkepegel
French Sonie Niveau de Sonie
2.5.1 The loudness curve

The Sone scale was established in order to avoid the confusion between dB SPL values and
the perception of loudness: A 1 kHz tone at 80 dB SPL is not perceived double as loud as the
same tone at 40 dB SPL. Figure 2.5.1 shows the relation between the Sone and the Phone
scales. (Hint: for a 1 kHz tone, phone and dB SPL is the same number). Arbitrarily it has
been decided that one sone should correspond to 40 phones. The curve is based on a great
number of loudness comparisons. The curve is called a loudness curve.
Figure 2.5.1 The loudness curve for a normal hearing test subject (solid line) and for a person with a
cochlear hearing loss (dashed)
68
The straight part of the solid line in Figure 2.5.1 corresponds to Stevens’ power law:
N = 2 ( L − 40 ) / 10
where N is the loudness (in sone) and L is the loudness level ( in phones). The curve shows
that a doubling of the loudness corresponds to a 10-phone increase in loudness level (or a 10-
dB increase in SPL if we are dealing with a 1 kHz tone). For many daily life sounds a rule of
thumb says that a 10-dB increase is needed in order to perceive a doubling of the loudness.
The loudness curve becomes steeper near the hearing threshold. This is also the case for a
person with a cochlear hearing loss (e.g., the very common hearing impairment caused by
age). An example of such a hearing loss is shown by the dashed curve in Figure 2.5.1 where
the threshold (1 kHz) is a little less than 40 dB SPL. The steeper slope means that - near the
threshold - the loudness increases rapidly for small changes in the sound level. This effect is
called loudness recruitment. Recent research have shown that – for this kind of hearing loss –
the loudness at threshold has a value significantly different from nil as indicated in the figure
[10]. In other words, listeners with cochlear hearing loss have softness imperception, rather
than loudness recruitment. Note that at higher sound levels the loudness perception is the
same for both normal and impaired listeners.
2.5.2 Temporal integration

The perception of loudness needs some time to build up. This means that short duration
sounds (less than one second) are perceived as less loud than the same sound with longer
duration. The growth of loudness as a function of duration is called temporal integration. The
growth resembles the exponential growth of a time constant. It has been shown that the time
constant is about 100 ms.
Short sounds - like a pistol shot, fireworks, handclap, etc. - are perceived as weak sounds
although their peak sound pressure levels may be well above 150 dB SPL. This is one of the
reasons why impulsive sounds generally are more dangerous than other sounds.
2.5.3 Measurement of loudness

Many years ago it was thought that a sound level meter with filters corresponding to the ears’
sensitivity (described by the equal loudness level contours (Figure 2.3.3)) could be used to
measure loudness. This is not the case.
Figure 2.5.2 show the characteristics for the commonly used A- and C- filters, but due to
masking and other phenomena these filters will not give a result that corresponds to loudness.
For the determination of loudness, special calculation software is needed. For stationary
sounds two procedures can be found in [11]. For non-stationary sound, loudness calculations
are found in professional Sound Quality calculation software. For research purposes loudness
models (software) can be found on the Internet (e.g. at
http://hearing.psychol.cam.ac.uk/Demos/demos.html )
69
20
10
Weighting characteristic, dB
0
-10
-20
-30
-40
-50 Weighting filters
A-filter
-60
C-filter
-70 D-filter
-80
5 20 50 200 500 2000 5000 20000
10 100 1000 10000
Frequency, Hz
Figure 2.5.2 Filter characteristics for the A, C and D filter. The data for the A and the C filter are
from [12]. The data for the D filter is from [13].
The main effect of the A-filter is that it attenuates the low frequency part of the signal. The
attenuation is e.g. 20 dB at 100 Hz and 30 dB at 50 Hz. Wind noise and other low frequency
components are attenuated by the A-filter and is therefore very practical for many noise
measurement situations.
The C-filter is ‘flat’ in the major part of the audible frequency range. It may me used as an
approximation to a measurement with linear characteristic.
The D-filter is mainly used in connection with evaluation of aircraft noise. The frequency
range around 3 kHz is known to be annoying and therefore this frequency range is given a
higher weight.
70
2.6 The auditory filters

The movements of the basilar membrane in the inner ear constitute a frequency analyser
where the peak of the envelope moves along the basilar membrane as a function of
frequency. See Figure 2.2.4. The width of the envelope peak may be seen as an indication of
the selectivity of the analyser filter and it has been common practice to describe the
frequency selectivity of the ear as a set of filters, a filter bank, which cover the audible
frequency range. It should be noted though that the concept of a filter bank is a very coarse
description and should be seen as a typical engineering approximation to the real situation.
Frequency selectivity is important for the perception of the different frequencies in complex
sound signals such as speech and music. We rely e.g. on our frequency selectivity when we
distinguish different vowels from each other.
The concept of frequency discrimination is different from frequency selectivity. Frequency

discrimination is the ability to hear the difference between two tones that are close in
frequency (one frequency at a time).
2.6.1 Critical bands

The bandwidth of the filters in the filter bank can be determined by means of various
psychoacoustic experiments. Many of these are masking experiments and led to the
formulation of the critical band model. It is outside the scope of the present text to go into
the background and the details of this model.
The results of the investigations are shown in Figure 2.6.1. It is seen that the bandwidth
(Critical Bands) is almost constant at 100 Hz up to a centre frequency of about 500 Hz and
above this frequency the bandwidth increases. The increase in bandwidth above 500 Hz is
similar to the increase in bandwidth for one-third-octave filters.
The critical bandwidth may be calculated from the empirical formula:
CB = 25 + 75 (1 + 1,4 f 2 ) 0,69
where CB is the bandwidth in Hz of the critical band and f is the frequency in kHz (not in
Hz).
71
10000
Bandwidth
Crit. Band
ERB
1000 1/3 octave
Bandwidth, Hz
100
10
20 50 200 500 2000 5000 20000
10 100 1000 10000
Frequency, Hz
Figure 2.6.1 Bandwidth of critical bands and Equivalent Rectangular bandwidth, ERB. The
bandwidth of 1/3-octave filters (straight line) is shown for comparison. The curves are computed from
the formulas given in the text.
If the audible frequency range is ‘filled up’ with consecutive critical bands from the lowest
frequency to the highest frequency, it is seen that 24 critical bands will cover the whole
frequency range. Each of the ‘filters’ has been given a number called Bark. Bark number one
is the band from zero to 100 Hz; Bark number two is the band from 100 Hz to 200 Hz, etc.
Band no. 8 has a centre frequency of 1000 Hz and goes from 920 Hz to 1080 Hz. The band
around 4000 Hz is no. 17 and has a bandwidth of 700 Hz.
The critical bands are not fixed filters similar to the filters in a physical filter bank as the
numbers given above may indicate. The critical bands are a result of the incoming sound
signal and as such much more ‘flexible’ than physical filters would be.
2.6.2 Equivalent Rectangular Bands

The auditory filters have also been determined by means of notched noise measurements
where the threshold of a pure tone is determined in the notch of a broadband noise as a
function of the width of the notch. This leads to the concept of equivalent rectangular
bandwidth, i.e. the bandwidth of a rectangular filter that transmits the same amount of energy
as the auditory filter. The bandwidth of such rectangular filters is shown in Figure 2.6.1 as a
function of centre frequency.
The rectangular bandwidth may be calculated from the empirical formula:
ERB = 24,7 ( 4,37 f + 1)
where ERB is the bandwidth in Hz and f is the centre frequency in kHz.
72
2.7 Speech
A speech signal is produced in the following way. Air is pressed from the lungs up through
the vocal tract, through the mouth cavities and/or the nose cavities and the sound is radiated
from the mouth and the nose. The vocal folds will vibrate when voiced sounds are produced.
2.7.1 Speech production

A schematic illustration of the production of voiced sounds is given in Figure 2.7.1 where the
vocal folds vibrate. The source spectrum is a line spectrum where the distance between the
lines corresponds to the fundamental frequency. The fundamental frequency is around
125 Hz for men, around 250 Hz for woman and around 300 for children, but there are big
individual variations. There are thus more lines in a male spectrum compared to a female.
The source spectrum decreases with the square of the frequency (1/f2). The source spectrum
is transformed by the ‘tube’ consisting of trachea, throat (pharynx) and the mouth. This
structure is simulated in Figure 2.7.1 by a cylindrical tube of length 17 cm.
Figure 2.7.1 The principle of vowel generation. From [14]
The tube has pronounced resonances (where the length of the tube corresponds to the odd
multiples of 1/4 wavelength) indicated by the peaks at 500, 1500 and 2500 Hz. The final
spectrum radiated from the mouth is then the product of the two spectra. The final spectrum
is a line spectrum with characteristic peaks caused by the transfer function. The peaks are
called formants and the formants are positioned differently for each vowel. Table 2.7.1 shows
the formants frequencies (in round numbers) for the three most different vowels. The sounds
are /i/: as in eve, /a/ as in father, /u/ as in moon. There are individual differences from person
to person.
73
/i/ /a/ /u/
1. formant 225 700 250

2. formant 2200 1200 700
3. formant 3000 2500 2200
Table 2.7.1 Formant frequencies in Hz of the vowels /i/, /a/ and /u/.
The unvoiced sounds are produced in many different ways, e.g. by pressing air out through
the teeth /s/, out between the lips and the teeth /f/, by sudden opening of the lips /p/, sudden
opening between tongue and teeth /t/ and between tongue and palate /k/. These sounds are
called unvoiced because the vocal folds do not vibrate but stays open in order for the air to
pass.
2.7.2 Speech spectrum, speech level

A general long-term speech spectrum is shown in Figure 2.7.2 that is based on the average of
18 speech samples from 12 languages.
The spectrum is a one-third octave spectrum which means that the curves are tilted
3 dB/octave compared to the result of a FFT-calculation. (The result of a FFT is a density
spectrum).
It is worth to note that the speech spectrum is almost independent of the language. This is not
surprising when the speech production mechanism is taken into account. The spectrum in
Figure 2.7.2 is based on English (several dialects), Swedish, Danish, German, French
(Canadian), Japanese, Cantonese, Mandarin, Russian, Welsh, Singhalese and Vietnamese. A
total of 392 talkers participated in the investigation.
The spectrum for women falls off below 200 Hz because their fundamental frequency
typically is around 250 Hz. The maximum is found around 500 Hz for both gender and above
500 Hz the two curves are almost identical. The slope above 500 Hz is approximately minus
10 dB per decade (or -3 dB/octave).
74
70
Speech spectrum
Male
Female
60 -3 dB /oct
1/3 octave level
50
40
30
50 200 500 2000 5000 20000
100 1000 10000
Frequency, Hz
Figure 2.7.2 The long-term speech spectrum for male and female speech shown as a 1/3-octave
spectrum. For comparison a line with slope –3 dB per octave (= –10 dB per decade) is shown.
Redrawn from [15]
The average level of male speech is about 65 dB SPL, measured at 1 m in front of the mouth.
For women the level is typically 3 dB lower, i.e. 63 dB. (Compare the number of lines in the
spectrum). During normal speech the level will vary ±15 dB around the mean value.
2.7.3 Speech intelligibility

The speech intelligibility of a transmission system is usually measured by means of a list of
words (or sentences) where the percentage of correctly understood words gives the
intelligibility score. The transmission system could be almost anything, e.g. a telephone line
or a room. The intelligibility depends on the word material (sentences, single words,
numbers, etc.), the speaker, the listener, the scoring method and the quality of the
transmission system.
Often the intelligibility score is given as a function of the signal-to-noise ratio. An example
of this is shown in Figure 2.7.3 for the word-material on the Dantale CD. This CD contains
eight tracks of 25 words each. The words are common Danish single-syllable words that are
distributed phonetically balanced over the eight lists so that the lists can be regarded as
equivalent. The words are recorded on the left channel of the CD and on the right channel a
noise signal is recorded with (almost) the same spectrum as the words. The noise signal is
amplitude modulated in order to make it resemble normal speech. The Dantale CD is
described in [16]
The result in Figure 2.7.3 is obtained with the words and the noise on the Dantale CD with
untrained Danish normal hearing listeners. It is seen that even at a signal-to-noise ratio of
75
0 dB almost all words are understood. It is also seen that an increase of just 10 dB in SNR
can change the situation from impossible to reasonable, e.g. from -15 dB (10%) to - 5 dB
(70%). It is a general finding that such a relatively small improvement of the signal-to-noise
ratio can improve the intelligibility situation dramatically. In other words, if the background
noise in a room is a problem for the understanding of speech in the room, then just a small
reduction of the background noise will be beneficial.
100
90
80
Word score, %
70
60
50
40
30
20
10
0
-25 -20 -15 -10 -5 0
SNR, dB
Figure 2.7.3 Word score for the speech material DANTALE as a function of speech-to-noise ratio
(SNR). Redrawn from [17]
It is time consuming and complicated to measure speech intelligibility with test subjects.
Therefore measurement and calculation methods have been developed for the estimation of
the expected speech intelligibility in a room or on a transmission line.
Articulation Index, AI [18]: Determination of the signal-to-noise ratio in frequency bands

(usually one octave or one-third octave). The SNR values are weighted according to the
importance of the frequency band. The weighted values are added and the result normalised
to give an index between zero and one. The index can then be translated to an expected
intelligibility score for different speech materials.
Speech Intelligibility Index, SII [19]: This method is based on the AI principle, but the
weighting functions are changed and a number of ‘corrections’ to the AI-method are
implemented. One of these is the correction for the change in speech spectrum according to
the vocal effort (shouting, raised voice, low voice).
Speech Transmission Index, STI [20]: In this method the modulation transfer function, MTF,
from the source (the speaker) to the receiver (the listener) is determined. The MTF is
determined for octave bands of noise (125 Hz to 8 kHz) and for a number of modulation
76
frequencies (0,63 Hz to 12,5 Hz). The reduction in modulation is transformed to an

equivalent signal-to-noise ratio and as in the AI method these values are added and
normalised in order to yield an index between zero and one. The index can then be translated
to an expected intelligibility score for different speech materials.
Rapid Speech Transmission Index, RASTI [21]: This is an abbreviated version of STI. Only
the frequency bands 500 Hz and 2 kHz and only nine different modulation frequencies are
used. The result is an index which is used in the same way as in STI.
77
2.8 References
1. Hougaard, S., et al., Sound and Hearing. 2 ed. 1995: Widex.
2. Engström, H. and Engström, B., A short survey of some common or important ear
diseases. 1979: Widex.
3. Zwicker, E. and Fastl, H., Psychoacoustics. Facts and models. 2 ed. 1999: Springer.
4. Kemp, D.T., Developments in cochlear mechanics and techniques for noninvasive

evaluation. Adv Audiol, 1988. 5: p. 27-45.
5. ISO-226, Acoustics - Normal equal-loudness-level contours, in FDIS, May 2002,

N327. 2002, International Standardization Organization: Geneva.
6. ISO-389-7, Acoustics - Reference zero for the calibration of audiometric equipment-

Part 7: Reference threshold of hearing under free-field and diffuse-field listening
conditions. 1996, International Organization for Standardization: Geneva,
Switzerland.
7. ISO-389-1, Acoustics - Reference zero for the calibration of audiometric equipment -

Part 1: Reference equivalent threshold sound pressure levels for pure tones and
supra-aural earphones. 1991, International Organization for Standardisation: Geneva,
Switzerland.

Part 5: Reference equivalent threshold sound pressure levels for pure tones in the
frequency range 8 kHz to 16 kHz“. 1998, International Organization for
Standardization: Geneva, Switzerland.

Part 8: Reference equivalent threshold sound pressure levels for pure tones and
circumaural earphones (ISO/DIS). 2001, International Organization for
Standardization: Geneva, Switzerland.
10. Florentine, M. and Buus, S. Evidence for normal loudness growth near threshold in
cochlear hearing loss. in 19 Danavox Symposium. 2001. Kolding, Denmark. p. xx-yy.
11. ISO-532, Acoustics - Method for calculating loudness level. 1975, International
Organisation for Standardisation: Geneva, Switzerland.
12. IEC-651, Sound level meters. 1979, International Electrotecnical Commission:

Geneva, Switzerland.
13. IEC-537, Frequency weighting for the measurement of aircraft noise (D-weighting).
1976, International Electrotechnical Commission: Geneva, Switzerland.
78
14. Borden, G. and Harris, K., Speech science primer. 1980: Williams & Wilkins.
15. Byrne, D., Ludvigsen, C., and al., e., Long-term average speech spectra ... J. Acoust.
Soc. Am., 1994. 96(no. 4): p. 2110?-2120?
16. Elberling, C., Ludvigsen, C., and Lyregaard, P.E., DANTALE, a new Danish Speech
material. Scandinavian Audiology, 1989. 18: p. 169-175.
17. Keidser, G., Normative data in quiet and in noise for DANTALE - a Danish speech
material. Scandinavian Audiology, 1993. 22: p. 231-236.
18. ANSI-S3.5, American National Standard methods for the calculation of the
Articulation Index. 1969, American National Standards Institute, Inc.: New York.
19. ANSI-S3.5, American National Standard methods for the calculation of the Speech
Intelligibility Index. 1997, American National Standards Institute, Inc.: New York.
20. Steeneken, H. and Houtgast, T., A physical method for measuring speech-
transmission quality. J. Acoust. Soc. Am., 1980. 67: p. 318-326.
21. IEC-268-16, Sound system equipment - Part 16: The objective rating of speech
intelligibility in auditoria by the RASTI method. 1988, International Electrotechnical
Commission.
Further reading:
Plack, C. J. (2005). The sense of hearing. Lawrence Earlbaum Associates. ISBN: 0-8058-
4884-3
Moore, B. C. J. (2003). An introduction to the psychology of hearing. 5th Edition. Academic

press ISBN: 0-12-505628-1
Yost, W. A. (2000). Fundamentals of hearing. An introduction. 4th Edition. Academic press.

ISBN: 0-12-775695-7
About standardized audiological, clinical tests see

http://www.phon.ucl.ac.uk/home/andyf/natasha/
79
3. An introduction to room acoustics
Jens Holger Rindel
3.1 SOUND WAVES IN ROOMS
3.1.1 Standing waves in a rectangular room

A rectangular room has the dimensions lx, ly, and lz. The wave equation can then be written
∂2 p ∂2 p ∂2 p
+ + + k2 p = 0 (3.1.1)
∂ x2 ∂ y2 ∂ z2
where p is the sound pressure and k = ω /c is the angular wave number, ω is the angular
frequency and c is the speed of sound in air. The equation can be solved by separation of the
variables and it is assumed that the solution can be written in the form:
p = X ( x) ⋅ Y ( y ) ⋅ Z ( z ) ⋅ e jω t
Insertion in (3.1.1) and division by p gives
1 ∂ 2 X 1 ∂ 2Y 1 ∂ 2 Z
+ + + k2 = 0
X ∂x 2
Y ∂y 2
Z ∂z 2
This can be separated, and for the x-direction it yields

1 ∂2 X
+ k x2 = 0
X ∂ x2
Similar equations hold for the y- and z-directions. The angular wave number k has been divided
into three
k 2 = k x2 + k y2 + k z2 (3.1.2)
The general solution to the above one-dimensional equation is
X ( x) = C x cos(k x x + ϕ x )
in which the constants Cx and ϕx are determined from the boundary conditions.
The room surfaces are now assumed to be rigid, i.e. the normal component of the particle velocity
is zero at the boundaries
1 ∂p
ux = − = 0 for x = 0 and x = lx
jωρ ∂ x
This means that ϕx = 0 and
π
k x = ⋅ nx where nx = 0, 1, 2, 3, … (3.1.3)
lx
Two similar boundary conditions hold for the y- and z-directions. With these conditions the
solution to (3.1.1) is
⎛ x⎞ ⎛ y⎞ ⎛ z⎞
p = p0 ⋅ cos⎜⎜ π n x ⎟⎟ ⋅ cos⎜ π n y ⎟ ⋅ cos⎜⎜ π n z ⎟⎟ (3.1.4)
⎜ ⎟ lz ⎠
⎝ lx ⎠ ⎝ ly ⎠ ⎝
The time factor ejωt is understood. The amplitude of the sound pressure does not move with time,
so the waves that are solutions to (3.1.4) are called standing waves. They are also called the
modes of the room, and each of them is related to a certain natural frequency (or eigenfrequency)
given by
ω ck c
fn = n = = k x2 + k y2 + k z2
2π 2π 2π
81
2
⎛ nx ⎞ ⎛ n y ⎞ ⎛ nz ⎞
2 2
c
fn = ⎜⎜ ⎟⎟ + ⎜ ⎟ + ⎜⎜ ⎟⎟ (3.1.5)
⎜ ⎟
2 ⎝ lx ⎠ ⎝ l y ⎠ ⎝ lz ⎠
The modes can be divided into three groups:
Axial modes are one-dimensional, only one of nx, ny, nz is > 0.
Tangential modes are two-dimensional, two of nx, ny, nz are > 0.
Oblique modes are three-dimensional, all three of nx, ny, nz are > 0.
Some examples are shown in Fig. 3.1.1. It is observed that the set of numbers (nx, ny, nz) indicate
the number of nodes (places with p = 0) along each coordinate axis.
Figure 3.1.1. Examples of room modes. (2,0,0) is one-dimensional and (2,1,0) is two-
dimensional. The lines are iso-sound pressure amplitude curves.
nx ny nz fn (Hz)
0 1 0 25
1 0 0 30
0 0 1 36
1 1 0 39
0 1 1 43
1 0 1 47
0 2 0 49
1 1 1 53
1 2 0 58
2 0 0 60
0 2 1 61
2 1 0 65
1 2 1 68
2 0 1 70
0 0 2 72
Table 3.1.1. Calculated natural frequencies at low frequencies using (3.1.5) in a rectangular
room with dimensions 5.7 m, 7.0 m, 4.8 m.
82
3.1.2 Transfer function in a room
The transfer function is the frequency response from a source position to a receiver position in a
room. A measured transfer function is shown in Fig. 3.1.2. It fluctuates very much with frequency
and the maxima can be identified as the natural frequencies of the room. The example in Fig.
3.1.2 has the same room dimensions as was used for the calculations in Table 3.1.1.
Figure 3.1.2. Transfer function in a rectangular room. At low frequencies it is possible to identify
the modes by their modal numbers.
3.1.3 Density of natural frequencies

A closer inspection of equation (3.1.5) shows that the natural frequencies of a rectangular room
may be interpreted in a geometrical way. A three-dimensional frequency space is shown in Fig.
3.1.3. The natural frequencies of the one-dimensional modes are marked on each of the axes,
representing the axial modes of the length, the width and the height, respectively. The interesting
observation is now that the points in the grid represent the oblique modes, and the distance to
each point from the origin is the natural frequency of that mode. So, the number of oblique modes
below a certain frequency f is equal to the number of grid points inside the sphere with radius f.
The volume is 1/8 of the sphere with radius f, i.e. (4 π f 3 / 3) / 8 = π f 3 / 6. Each mode occupies a
volume c3 / (8 lx ly lz) = c3 / (8 V). So, the number of oblique modes below f is approximately:
π f 3 8V 4π V f 3
N obl = =
6 c3 3 c3
The tangential modes are found in the plane between two of the axes. If these and the axial modes
are also taken into account, the number of modes with natural frequencies below the frequency f
is:
4π V ⎛ f ⎞ π S ⎛ f ⎞
3 2
L f
N = ⎜ ⎟ + ⎜ ⎟ + (3.1.6)
3 ⎝c⎠ 4 ⎝c⎠ 8 c
V is the volume of the room, S = 2( lx ly + lx lz + ly lz) is the total area of the surfaces, and L = 4 (lx
+ ly + lz) is the total length of all edges. It should be noted that the modal points of the tangential
and axial modes in Fig. 3.1.3 are located on the coordinate planes and axes, respectively.
Therefore we count the tangential points only as halves and those on the axes only as quarters.
At high frequencies the oblique modes dominate, and the first term in (3.1.6) is a good
approximation for any room, not only for rectangular rooms.
83
Figure 3.1.3. Frequency-grid, in which each grid point represents a room mode.
The modal density is the average number of modes per hertz.
dN V πS L
= 4π 3 f 2 + 2
f + (3.1.7)
df c 2c 8c
In Fig. 3.1.4 this is compared to the actual modal density in a room. For high frequencies it is
sufficient to use the first term (oblique modes) for the modal density:
dN V
≅ 4π 3 f 2 (3.1.8)
df c
84
Figure 3.1.4. Modal density as a function of frequency. Actual number of modes per 10 Hz in a
rectangular room and estimated by (3.1.7).
3.2 STATISTICAL ROOM ACOUSTICS
3.2.1 The diffuse sound field

In this chapter the acoustical behaviour of a room is treated from a statistical point of view, based
on energy balance considerations. It is assumed that the modal density is high enough, so the
influence of single modes in the room can be neglected. It is also assumed that the reflection
density is high enough, so the phase relations between individual reflections can be neglected.
This means that the reflections in the room are assumed to be uncorrelated and their contribution
can be added on an energy basis.
The diffuse sound field is defined as a sound field in which:

The energy density is the same everywhere
All directions of sound propagation occur with the same probability
It is obvious that the direct sound field near a sound source is not included in the diffuse sound
field. Neither are the special interference phenomena that are known to give increased energy
density near the room boundaries and corners. The diffuse sound field is an ideal sound field that
does not exist in any room. However, in many cases the diffuse sound field can be a good and
very practical approximation to the real sound field.
85
3.2.2 Incident sound power on a surface
In a plane propagating sound wave the relation between rms sound pressure p1 and sound
intensity I1 is
p12 = I 1 ⋅ ρ c
In a diffuse sound field the rms sound pressure pdiff is the result of sound waves propagating in all
directions, and all having the sound intensity I1. By integration over a sphere with the solid angle
ψ = 4π the rms sound pressure in the diffuse sound field is
= ∫ I1 ⋅ ρ c dψ = 4π ⋅ I 1 ⋅ ρ c
2
pdiff (3.2.1)
ψ = 4π
In the case of a plane wave with the angle of incidence θ relative to the normal of the surface, the
incident sound power per unit area on the surface is
2
p diff
I θ = I 1 cosθ = cosθ (3.2.2)
4π ρ c
where pdiff is the rms sound pressure in the diffuse sound field. This is just the sound intensity in
the plane propagating wave multiplied by the cosine, which is the projection of a unit area as seen
from the angle of incidence, see Fig. 3.2.1.
p1 pdiff
Iθ Iinc
θ
a b
Figure 3.2.1. a: Plane wave at oblique incidence on a surface. b: Diffuse incidence on a surface.
The total incident sound power per unit area is found by integration over all angles of incidence
covering a half sphere in front of the surface, see Fig. 3.2.2. The integration covers the solid angle
ψ = 2π.
2
1 2π π / 2 pdiff
I inc = ∫ Iθ d ψ =
4π ∫0 ∫0 ρ c
cos θ sin θ d θ d ϕ
ψ = 2π
2 2
1 pdiff 1 pdiff 1
∫
1
= ⋅ 2π ⋅ sin θ d(sin θ ) = ⋅ ⋅
4π ρc 0 2 ρc 2
2
p diff
I inc = (3.2.3)
4ρ c
It is noted that this is four times less than in the case of a plane wave of normal incidence.
86
Figure 3.2.2. Definition of angles of incidence in a diffuse sound field.
3.2.3 Equivalent absorption area

The absorption coefficient α is defined as the ratio of the non-reflected sound energy to the
incident sound energy on a surface. It can take values between 0 and 1, and α = 1 means that all
incident sound energy is absorbed in the surface. An example of a surface with absorption
coefficient, α = 1 is an open window.
The product of area and absorption coefficient of a surface material is the equivalent absorption
area of that surface, i.e. the area of open windows giving the same amount of sound absorption as
the actual surface. The equivalent absorption area of a room is
A = ∑ Siα i = S1α1 + S 2α 2 + ..... = Sα m (3.2.4)

i
where S is the total surface area of the room and αm is the mean absorption coefficient. The unit
of A is m2. In general, the equivalent absorption area may also include sound absorption due to
the air and due to persons or other objects in the room.
3.2.4 Energy balance in a room

The total acoustic energy in a room is the sum of potential energy and kinetic energy, or twice the
potential energy, since the time average of the two parts must be equal. The total energy E is the
energy density multiplied by the room volume V:
p2
E = ( w pot + wkin ) V = 2w pot V = V (3.2.5)
ρ c2
Here and in the following, p denotes the rms sound pressure in the diffuse sound field (called pdiff
in section 3.2.2). The energy absorbed in the room is the incident sound power per unit area
(3.2.3) multiplied by the total surface area and the mean absorption coefficient, i.e. the equivalent
absorption area (3.2.4),
p2
Pa ,abs = I inc Sα m = I inc A = A (3.2.6)
4ρ c
If Pa is the sound power of a source in the room, the energy balance equation of the room is
87
dE
Pa − Pa ,abs = (3.2.7a)
dt
p2 V d
Pa − A= ( p2 ) (3.2.7b)
4ρ c ρ c2 d t
With a constant sound source a steady state situation is reached after some time, and the right side
of the equation is zero. So, the absorbed power equals the power emitted from the source, and the
steady state sound pressure in the room is
4 Pa
p s2 = ρc (3.2.8)
A
This equation shows that the sound power of a source can be determined by measuring the sound
pressure generated by the source in a room, provided that the equivalent absorption area of the
room is known. It also shows how the absorption area in a room has a direct influence on the
sound pressure in the room. For some cases it is more convenient to express eq. (3.2.8) in terms
of the sound pressure level Lp and the sound power level LW ,
⎛ 4A ⎞
L p ≅ LW + 10 log⎜ 0 ⎟ (dB) (3.2.9)
⎝ A ⎠
where A0 = 1 m2 is a reference area. The approximation comes from neglecting the term with the
constants and reference values
ρ c Pref 1.204 ⋅ 343 ⋅ 10 −12
10 log = 10 log = 0.14 dB ≅ 0 dB
2
A0 pref 1 ⋅ (20 ⋅ 10 −6 ) 2
3.2.5 Reverberation time. Sabine´s formula

If the sound source is turned off after the sound pressure has reached the stationary value, the first
term in the energy balance equation (3.2.7b) is zero, and the rms sound pressure is now a function
of time:
A
4ρ c
p 2 (t ) +
V d 2
ρ c2 d t
( p (t )) = 0 (3.2.10)
The solution to this equation can be written

cA
− t
p (t ) = p e
2 2
s
4V
(3.2.11)
2
where ps is the mean square sound pressure in the steady state and t = 0 is the time when the
source is turned off. It is seen that the mean square sound pressure, and hence the sound energy,
follows an exponential decay function. On a logarithmic scale the decay is linear, and this is
called the decay curve, see Fig. 3.2.3.
If instead the source is turned on at the time t = 0, the sound build-up in the room follows a
similar exponential curve, also shown in Fig. 3.2.3.
⎛ −
cA
t ⎞
p (t ) =
2 ⎜
p 1− e
2 4V ⎟
(3.2.12)
s
⎜ ⎟
⎝ ⎠
88
Figure 3.2.3. Build-up and decay of sound in a room. Here the source is turned on a t = 0 and
turned off at t = 1 s. Top: linear scale (sound pressure squared). Bottom: logarithmic scale (dB).
The reverberation time T60 is defined as the time it takes for the sound energy in the room to
decay to one millionth of the initial value, i.e. a 60 dB decay of the sound pressure level. Hence,
for t = T60 ,
cA
− T60
−6
p (t ) = p 10
2 2
s = p e 2
s
4V
So, the reverberation time is

4V 55.3 V
T60 = 6 ⋅ ln(10) ⋅ = (3.2.13)
cA cA
This is Sabine’s formula named after Wallace C. Sabine, who introduced the reverberation time
concept around 1896. He was the first to demonstrate that T60 is inversely proportional to the
equivalent absorption area A.
Note: Sabine’s formula is often written as T60 = 0.16 V/A. However, this implies that V must be in
m3 and A in m2.
3.2.6 Stationary sound field in a room. Reverberation distance

A reverberation room is a special room with long reverberation time and a good diffusion. In
such a room the diffuse sound field is a good approximation, and the results for stationary
conditions (3.2.8) and for sound decay (3.2.13) can be applied to measure the sound power of a
sound source:
p s2 55.3 V
Pa = ⋅ (3.2.14)
4 ρ c c T60
The reverberation time and the average sound pressure level in the reverberation room are
measured, and the sound power level is calculated from
89
2
p ref ⋅ 55.3 ⋅ V
LW = L p + 10 log
Pref ⋅ 4 ρ c 2 ⋅ T60
(3.2.15)
V T
= L p + 10 log − 10 log 60 − 14 dB
V0 t0
3
where V0 = 1 m and t0 = 1 s.
In most ordinary rooms the diffuse sound field is not a good approximation. Each of the
following conditions may indicate that the sound field is not diffuse
An uneven distribution of sound absorption on the surfaces, e.g. only one surface is highly
absorbing
A lack of diffusing or sound scattering elements in the room
The ratio of longest to shortest room dimension is higher than three
The volume is very large, say more than 5000 m3
A rather simple modification to the stationary sound field is to separate the direct sound. The
sound power radiated by an omni-directional source is the sound intensity at the distance r in a
spherical sound field multiplied by the surface area of a sphere with radius r
Pa = I r ⋅ 4π r 2 (3.2.16)
Thus, the sound pressure squared of direct sound in the distance r from the source is
Pa
2
pdir = ρc (3.2.17)
4π r 2
The stationary sound is described by (3.2.8)
4 Pa
p s2 = ρc
A
The reverberation distance rrev is defined as the distance where pdir2 = ps2 when an omni-
directional point source is placed in a room. It is a descriptor of the amount of absorption in a
room, since the reverberation distance depends only on the equivalent absorption area
A
rrev = = 0.14 A (3.2.18)
16π
At a distance closer to the source than the reverberation distance, the direct sound field
dominates, and this is called the direct field. At longer distances the reverberant sound field
dominates, and in this so-called far field the stationary, diffuse sound field may be a usable
approximation.
An expression for the combined direct and diffuse sound field can derived by simple addition of
the squared sound pressures of the two sound fields. However, since the direct sound is treated
separately, it should be extracted from the energy balance equation, which was used to describe
the diffuse sound field. To do this, the sound power of the source should be reduced by a factor of
(1 - αm), which is the fraction of the sound power emitted to the room after the first reflection. So,
the squared sound pressure in the total sound field is
⎛ r2 ⎞
2
ptotal = p dir
2
+ p s2 (1 − α m ) = p s2 ⎜⎜ rev2 + 1 − α m ⎟⎟ (3.2.19)
⎝ r ⎠
⎛ 1 4 ⎞
2
ptotal = Pa ⋅ ρ c ⎜⎜ + (1 − α m ) ⎟⎟ (3.2.20)
⎝ 4π r
2
A ⎠
90
Normal sound sources like a speaking person, a loudspeaker or a musical instrument radiate
sound with different intensity in different directions. The directivity factor Q is the ratio of the
intensity in a certain direction to the average intensity,
4π r 2
Q=I⋅ (3.2.21)
Pa
So, the squared sound pressure of the direct sound is
Q ⋅ Pa
2
pdir = ρc (3.2.22)
4π r 2
This leads to a general formula for the sound pressure level as a function of the distance from a
sound source in room.
4A ⎛ r2 ⎞
L p ≅ LW + 10 log 0 + 10 log ⎜⎜ Q rev2 + 1 − α m ⎟⎟ (dB) (3.2.23)
A ⎝ r ⎠
where A0 = 1 m2. In a reverberant room with little sound absorption (say, αm < 0.1) the sound
pressure level in the far field will be approximately as predicted by the diffuse field theory, i.e.
the last term will be close to zero. In the case of a highly directive sound source like a trumpet (Q
>> 1) the direct field can be extended to distances much longer than the reverberation distance. In
the latter situation the last term in (3.2.23) raises the sound pressure level above the diffuse field
value.
Figure 3.2.4. Relative sound pressure level as a function of distance in a room with
approximately diffuse sound field. The source has a directivity factor of one. The parameter on
the curves is A / (1 - αm) in m2.
In large rooms with medium or high sound absorption (say, αm > 0.2) the sound pressure level
will continue to decrease as a function of the distance, because the diffuse field theory is not valid
in such a room. Instead, the slope of the spatial decay curve may be taken as a measure of the
degree of acoustic attenuation in a room. So, in large industrial halls the attenuation in dB per
doubling of the distance may be a better descriptor than the reverberation time.
91
3.3 GEOMETRICAL ROOM ACOUSTICS
3.3.1 Sound rays and a general reverberation formula

In geometrical acoustics rays are used to describe the sound propagation. The concept of rays
implies that the wavelength and the phase of the sound are neglected, and only the direction of
sound energy propagation is treated in geometrical acoustics.
The sound decay shall now be studied by following a plane wave travelling as a ray from wall to
wall, see Fig. 3.3.1. The energy of the wave is gradually decreased due to absorption at the
surfaces, all of which are assumed to have the mean absorption coefficient αm.
The ray representing a plane wave may start in any direction and it is assumed that the decay of
energy in the ray is representative for the decay of energy in the room. The room may have any
shape.
Figure 3.3.1. A plane wave travelling as a ray from wall to wall in a room.
By each reflection the energy is reduced by a factor (1 - αm). The initial sound pressure is p0 and
after n reflections the squared sound pressure is
p 2 (t ) = p02 ⋅ (1 − α m ) n = p02 ⋅ e n ⋅ ln(1−α m ) (3.3.1)
The distance of the ray from one reflection to the next is li and the total distance traveled by the
ray up to the time t is
∑ li = c ⋅ t = n ⋅ l m
i
(3.3.2)
where lm is the mean free path. So, the squared sound pressure is
c
⋅ ln(1−α m ) ⋅ t
p (t ) = p ⋅ e
2 2
0
lm
(3.3.3)
-6
When the squared sound pressure has dropped to 10 of the initial value, the time t is by
definition the reverberation time T60:
c
⋅ ln(1−α m ) ⋅ T60
c
10 −6
=e lm
⋅ ln(1 − α m ) ⋅ T60
⇒ − 6 ⋅ ln(10) =
lm
This leads to an interesting pair of general reverberation formulas:
13.8 ⋅ lm 13.8 ⋅ lm
T60 = ≈ (3.3.4)
− c ⋅ ln(1 − α m ) c ⋅αm
The last approximation is valid if αm < 0.3, i.e. only in rather reverberant rooms. The
approximation comes from:
⎛ 1 ⎞ α2 α3
− ln(1 − α m ) = ln⎜⎜ ⎟⎟ = α + + +L
⎝1 − αm ⎠ 2 3
92
With the assumption that all directions of sound propagation appear with the same probability, it
can be show (Kosten, 1960) that the mean free path in a three-dimensional room is
4V
lm = (3-dimensional) (3.3.5)
S
where V is the volume and S is the total surface area.
Similarly, the mean free path in a two-dimensional room can be derived. This could be the
narrow air space in a double wall, or structure-borne sound in a plate. The height or thickness
must be small compared to the wavelength. In this case the mean free path is
π Sx
lm = (2-dimensional) (3.3.6)
U
where Sx is the area and U is the perimeter. The one-dimensional case is just the sound travelling
back and forth between two parallel surfaces with the distance l = lm.
Insertion of (3.3.5) in the last part of (3.3.4) gives the Sabine formula (3.2.13), whereas insertion
in the first part of (3.3.4) leads to the so-called Eyring’s formula for reverberation time in a room:
55.3 ⋅ V
T60 = (3.3.7)
− c ⋅ S ⋅ ln(1 − α m )
In a reverberant room (αm < 0.3) it gives the same result as Sabine’s formula, but in highly
absorbing rooms Eyring’s formula is theoretically more correct. In practice the absorption
coefficients are not the same for all surfaces and the mean absorption coefficient is calculated as
in (3.2.4):
1
α m = ⋅ ∑ S iα i (3.3.8)
S i
In the extreme case of an anechoic room (αm = 1) Eyring’s formula gives correctly a
reverberation time of zero, whereas Sabine’s formula is obviously wrong, giving the value T60 =
55.3 V/c S. However, in normal rooms with a mixture of different absorption coefficients it is
recommended to use Sabine’s formula.
3.3.2 Sound absorption in the air

A sound wave travelling through the air is attenuated by a factor m, which depends on the
temperature and the relative humidity of the air, see Fig. 3.3.2. The unit of the air attenuation
factor is m - 1. If this attenuation is included in (3.3.3) the squared sound pressure in the decay is
c ct
⋅ ln(1−α m ) ⋅ t ⋅( ln(1−α m ) − m⋅ lm )
−mct
p (t ) = p ⋅ e
2 2
0
lm
⋅e = p ⋅e
2
0
lm
(3.3.9)
The general reverberation formula then becomes
13.8 ⋅ lm 13.8 ⋅ l m
T60 = ≈ (3.3.10)
c ( − ln(1 − α m ) + m ⋅ l m ) c (α m + m ⋅ lm )
In the three-dimensional case with (3.3.5) we then have
55.3 ⋅ V 55.3 ⋅ V
T60 = ≈ (3.3.11)
c (− S ⋅ ln(1 − α m ) + 4 mV ) c ( S ⋅ α m + 4 mV )
These two expressions are the Eyring and the Sabine formula, respectively, with the air
absorption included. By comparison with (3.2.13) it is seen that the equivalent absorption area
including air absorption is
A = ∑ S iα i + 4 mV (3.3.12)
i
Some typical values of m are found later in Table 3.4.3.
93
Figure 3.3.2. The air attenuation factor m as a function of the relative humidity. The air
temperature is 20 °C. (Ref.: Harris1966).
3.3.3 Sound reflections and image sources

The direction of a sound reflection from a large plane surface follows the same geometrical law,
as known from optics, i.e. the angle of reflection is equal to the angle of incidence. This means
that the reflected sound can be interpreted as sound coming from an image source behind the
reflecting surface, see Fig. 3.3.3. This principle can be extended to higher order reflections.
Figure 3.3.3. Reflection in one surface (a) and in two surfaces (b). A is the source and R is the
receiver. First order image sources are indicated by A’ and second order image sources by A’’.
94
Echo is a well-known acoustic phenomenon. It is defined as a single sound reflection that is
clearly audible as separate from the direct sound. The human ear is able to hear a reflection as an
echo if the time delay is approximately 50 ms. The so-called echo-ellipse is shown in Fig. 3.3.4.
Any point E on the ellipse represents a potential reflection with a delay of 50 ms, i.e. the distance
LE + EP = 17 m. Reflections from room surfaces outside the ellipse (as R2 on the figure) are
delayed more than 50 ms and may cause an echo at the receiver point.
Figure 3.3.4. The echo-ellipse in the longitudinal section of an auditorium. L is the source and P
the receiver. (Ref.: Petersen 1984).
3.3.4 Reflection density in a room

The image source principle can easily be applied to higher order reflections in a rectangular
room. An infinite number of image rooms make a grid, and each cell in the grid is an image room
containing an image source. The principle is shown for the two-dimensional case in Fig. 3.3.5.
Figure 3.3.5. Rectangular room with a sound source and image sources, here shown in two
dimensions. Image sources located inside the circle with radius ct will contribute reflections up
to time t.
95
If an impulse sound is emitted the number of reflections that will arrive within the time t can be
calculated as the volume of a sphere with radius ct divided by the room volume V:
3 π (ct )
4 3
N (t ) = (3.3.13)
V
The reflection density is then the number of reflections within a small time interval dt, and by
differentiation:
dN c3 2
= 4π t (3.3.14)
dt V
The reflection density increases with the time squared, so the higher order reflections are
normally so dense in arrival time that it is impossible to distinguish separate reflections. If
(3.3.14) is compared to (3.1.8), it is striking to observe the analogy between reflection density in
the time domain and modal density in the frequency domain.
3.4 ROOM ACOUSTICAL DESIGN
3.4.1 Choice of room dimensions

The room dimensions determine the natural frequencies of a room. A good acoustical design of a
room implies that the transfer function should be as smooth as possible. With reference to Fig.
3.1.2 is clear that the room dimensions of a rectangular room should not be identical, because in a
cubic room many modes will have the same natural frequency, and thus there will be bigger gaps
in the transfer function. This would be very unfortunate, especially at low frequencies in small
rooms for speech, music or acoustic measurements. The dimensions of such rooms should be
designed after calculations of the normal modes below 100 Hz, see also Table 3.1.1.
3.4.2 Reflection control

In room with an audience it is very important to design the room surfaces with respect to the early
reflections. First of all in order to avoid problems with echo and focusing, but also to ensure a
good distribution of reflections to the audience area, see Fig. 3.4.1. In rooms for speech the
ceiling reflections are most important, whereas rooms for music should not give too much
reflection directly from the ceiling. In such room the ceiling should rather give diffuse reflections,
but the side walls are important because lateral reflections contribute to the acoustic of a concert
hall, see Fig. 3.4.2.
Figure 3.4.1. Ceiling reflections in auditoriums. a) concave ceiling causing focusing and uneven
sound distribution. b) plane reflectors causing an even sound distribution. (Ref.: Petersen 1984).
96
Figure 3.4.2. Wall reflections in auditoriums. a) rectangular room, b) fan shape room, c) inverse
fan shape room.
3.4.3 Calculation of reverberation time

Sabine’s formula (3.2.13) is the most well known and simple method for calculation of
reverberation time in a room
55.3 V 0.16 V
T60 = ≅ (3.4.1)
cA A
with volume V in m3 and A in m2. The equivalent absorption area is calculated as in (3.3.12), but
in addition to absorption from surfaces and air, the absorption from persons or other items in the
room should be included, if relevant
A = ∑ S iα i + ∑ n j A j + 4 mV (3.4.2)
i j
Here nj is the number of items, each contributing with an absorption area Aj. Examples of
absorption coefficients of common materials and absorption areas for persons are given in Table
3.4.1 and 3.4.2, respectively. The air attenuation can be taken from Table 3.4.3.
Frequency (Hz)
Material 125 250 500 1000 2000 4000
Brick, bare concrete 0.01 0.02 0.02 0.02 0.03 0.04
Parquet floor on studs 0.16 0.14 0.11 0.08 0.08 0.07
Needle-punch carpet 0.03 0.04 0.06 0.10 0.20 0.35
Window glass 0.35 0.25 0.18 0.12 0.07 0.04
Curtain draped to half 0.10 0.25 0.55 0.65 0.70 0.70
its area, 100 mm air
space
Table 3.4.1. Typical values of the absorption coefficient α for some common materials.
Frequency (Hz)
Persons 125 250 500 1000 2000 4000
Standing, normal 0.12 0.24 0.59 0.98 1.13 1.12
clothing
Standing, with 0.17 0.41 0.91 1.30 1.43 1.47
overcoat
Sitting musician with 0.60 0.95 1.06 1.08 1.08 1.08
instrument
Table 3.4.2. Typical values of absorption area A in m2 for persons.
97
Relative Frequency
humidity (%) 1 kHz 2 kHz 4 kHz 8 kHz
40 0.0011 0.0026 0.0072 0.0237
50 0.0010 0.0024 0.0061 0.0192
60 0.0009 0.0023 0.0056 0.0162
70 0.0009 0.0021 0.0053 0.0143
80 0.0008 0.0020 0.0051 0.0133
Table 3.4.3. Examples of air attenuation factor m (m-1) at a temperature of 20°C.
3.4.4 Reverberation time in non-diffuse rooms

In a room with the sound absorption unequally distributed on the surfaces the assumption of a
diffuse sound field is not fulfilled, and thus Sabine’s formula will not be reliable. The measured
reverberation time may be either shorter or longer than predicted by Sabine’s formula.
A shorter reverberation time will appear in a room in which the first reflections are directed
towards the most absorbing surface. In an auditorium this is typically the floor with the audience,
see Fig. 3.4.1 b.
In a rectangular room without sound scattering surfaces or elements, there is a possibility of

prolonged decay in certain directions. In order to give an idea of the problem it is possible to
calculate the different reverberation times associated to one-dimensional decays in each of the
three main directions using the general reverberation formula (3.3.4).
13.8 ⋅ lm l
T60 ≈ ≈ 0.04 ⋅ m (lm in m) (3.4.3)
c ⋅αm αm
Figure 3.4.3. A rectangular room with indicated absorption coefficients.
As an example the room in Fig. 3.4.3 is considered. The ceiling has a high absorption coefficient
(α = 0.8), but all other surfaces are acoustically hard (α = 0.1).
Volume V = 5 ⋅ 10 ⋅ 20 = 1000 m3
Surface area S = 700 m2
Equivalent absorption area A = 200 ⋅ 0.8 + 500 ⋅ 0.1 = 210 m2
Mean absorption coefficient αm = A / S = 210 / 700 = 0.30
Mean absorption coefficient (height) αm = (0.8 + 0.1) / 2 = 0.45
Mean free path (3-dim.) lm = 4 V / S = 4 ⋅ 1000 / 700 = 5.7 m
Mean free path (2-dim.) lm = π Sx / U = π ⋅ 200 / 60 = 10.5 m
98
The results are shown in Table 3.4.4. A two-dimensional reverberation in the horizontal plane
between the walls has also been calculated (4.2 s). The one-dimensional decays are the extreme
cases with the longest reverberation time being 20 times the shortest one, 8.0 s and 0.4 s,
respectively!
lm (m) αm T60 (s)

Direction
3-dim. (Sabine) 5.7 0.30 0.8
3-dim. (Eyring) 5.7 0.30 0.6
2-dim. (horizontal) 10.5 0.10 4.2
1-dim. (length) 20 0.10 8.0
1-dim. (width) 10 0.10 4.0
1-dim. (height) 5 0.45 0.4
Table 3.4.4. Calculation of the one-dimensional reverberation times of the rectangular room in
Fig. 3.4.3.
The real decay that is measured in the room will be a mixture of these different decays, and the
reverberation time will be considerably longer than predicted from Sabine’s formula. Eyring’s
formula is even worse. The measured decay curve will be bent, and thus the measuring result
depends on which part of the decay curve is considered for the evaluation of reverberation time.
In a room with long reverberation time due to non-diffuse conditions and at least one sound-
absorbing surface, introducing some sound scattering elements in the room can have a significant
effect. It could be furniture or machines on the floor or some diffusers on the walls. This will
make the sound field more diffuse, and the reverberation time will be reduced, i.e. it will come
closer to the Sabine value. In other words: The sound absorption available in the room becomes
more efficient when scattering elements are introduced to the room.
Note. In the one-dimensional case it is strictly not correct to use the arithmetic average of the
absorption coefficients, if one of them is high. By inspection of (3.3.1) it is seen that the mean
absorption coefficient should be calculated from
(1 − α m ) = (1 − α1 )(1 − α 2 ) (3.4.4)
So, if one of the surfaces is reflective and the other is totally absorbing, αm = 1 and hence the
reverberation time is zero.
3.4.5 Optimum reverberation time and acoustic regulation of rooms

The optimum reverberation time depends of the activities in the room. It is important to choose
the room volume and the surface materials with such sound absorbing properties that the
reverberation time can get the right value for the purpose. In workshops with noise sources it is
important to have a reverberation time as short as possible. In schools the classrooms should have
a reverberation time between 0.6 and 0.9 s and independent of the frequency between 100 and
4000 Hz in order to obtain good acoustical conditions for speech. In concert halls the
reverberation time should be around 1.5 to 2.2 s at mid frequencies (500 – 1000 Hz) with the
longer values in the bigger halls. For music the reverberation time may be up to 50% longer at
low frequencies (125 Hz) and somewhat shorter at high frequencies. The latter is unavoidable in a
big hall due to the air attenuation.
99
Use of room Optimum reverberation time, s
(500 – 1000 Hz)
Cinema 0,4 – 1,0
Rock concert 0,8 – 1,1
Lecture 0,8 – 1,2
Theatre 1,0 – 1,2
Opera 1,3 – 1,7
Symphony concert 1,5 – 2,2
Choir concert 1,7 – 2,5
Organ music 2,0 – 3,0
Table 3.4.5. Optimum reverberation time at mid frequencies for various purposes in rooms with
an audience.
3.4.6 Measurement of reverberation time

The reverberation time in a room can be measured with a noise signal or with an impulse. The
traditional method uses white noise emitted by a loudspeaker and a microphone to measure the
sound pressure level as a function of time after the source is switched off. This gives a decay
curve and a typical example is shown in Fig. 3.4.4.
Figure 3.4.4. Typical decay curve measured with noise interrupted at the time t = 0.
From the microphone the signal is led to a frequency filter, which is either an octave filter of a
one-third octave filter. If the sound in the room is sufficiently diffuse and a sufficient large
number of modes are excited the decay curve is close to a straight line between the excitation
level and the background level. The dynamic range is seldom more than around 50 dB and the
whole range of the measured decay curve is not used. The lower part of the decay curve is
influenced by the background noise and the upper part may be influenced by the direct sound,
which gives a steeper start of the curve. So, the part of the decay curve used for evaluation begins
5 dB below the average stationary level and ends normally 35 dB below the same level. The
evaluation range is thus 30 dB and the slope is determined by fitting a straight line or
100
automatically by calculating the slope of a linear regression line. From the slope of the decay
curve in dB per second is calculated the reverberation time, which is the time for a 60 dB drop
following the straight line. The result is sometimes denoted T30 in order to make it clear that the
actually used evaluation range is 30 dB.
If the background noise is too high and a sufficient dynamic range is not available the
reverberation time can instead be measured as T20. In this case the slope of the decay curve is
evaluated between –5 dB and –25 dB below the excitation level.
The reverberation time is measured in a number of source- and receiver positions, and in each
position the decay is determined as an average of a number of excitations. White noise is a
random noise signal and thus the measured decay curves are always a little different.
Sometimes the decay curves are not nice and straight and it is difficult to measure a certain
reverberation time. One reason can be that it is a measurement at low frequencies in a small room
and maybe only two or three modes are excited within the frequency band of the measurement. In
this case there may be interference between the modes causing very irregular decay curves.
Another difficult situation is coupled rooms, i.e. a room divided into sections with different
reverberation times. A typical example is a theatre with a reverberant stage house and a rather
dead auditorium. In this case the decay curve will be bent, i.e. the upper part shows a short
reverberation time and the lower part shows a longer reverberation time. It might be possible to
determine both of these reverberation times, however, the shorter one representing the initial
decay is the most important one, because the subjective evaluation of the reverberation is related
to the initial decay.
3.5 REFERENCES
C.M. Harris (1966). Absorption of sound in air versus humidity and temperature. JASA 40, pp
148-159.
C.W. Kosten (1960). The mean free path in room acoustics. Acoustica 10, pp 245-250.
J. Petersen 1984). Rumakustik (in Danish). SBI-anvisning 137. Danish Building Research
Institute.
W.C. Sabine (1922). Collected papers on acoustics. Dover Publications, Inc. 1964, New York.
101
4 Sound absorbers and their application in room design
Anders Chr. Gade
4.1 Introduction
The reverberation time T60 as defined in Section 3.2.5 is the most important descriptor of the
acoustics of a room. Therefore, calculating predictions of T60 (e.g. according to Equation 3.4.1) is a
very basic part of room acoustical design which in turn calls for the availability of reliable data on
the frequency dependant sound absorption characteristics of materials used for room surface
cladding and for furnishing of rooms (such as furniture, people and machinery).
In Table 3.4.1 absorption coefficients per octave band were listed for some materials generally
found in rooms. The values indicate that some of these, e.g. windows and wooden floors on studs,
primarily absorb low frequency sounds. On the other hand, curtains and persons (see Table 3.4.2)
mainly absorb middle and high frequencies. In order to obtain a well balanced T60 versus frequency
for a given type of room it is therefore important to mix properly different types of materials when
designing the room.
In this chapter we will give a basic introduction to the physical mechanisms involved in sound
absorption and present some types of sound absorption materials well suited for - or specifically
designed for - sound absorption and reverberation control. The absorption properties will be
described in terms of the sound absorption coefficient as defined in Section 1.5.2.
For certain types of rooms, such as schools and work rooms, general demands on reverberation
control exist. Therefore the last section in this chapter is devoted to examples on how sound
absorbing materials can be applied in the design of such rooms.
4.2 The room method for measurement of sound absorption.

In Section 1.5.2, a method for measuring the absorption coefficient, the tube method, was presented
which reveals the absorption coefficient for a single angle of incidence (usually normal incidence as
illustrated to the left in Figure 4.2.1). However, the absorption will normally depend on the
direction of sound incidence1. Materials applied in rooms with a (more or less) diffuse sound field
will be exposed to sound arriving from many different directions as illustrated in Fig. 4.2.1(c).
Therefore we will start this chapter by presenting a method for measurement of sound absorption,
which provides the relevant diffuse field absorption coefficient: the reverberation room method.
Figure 4.2.1 Different conditions for sound incidence on a surface. From [1]
1
The absorption for oblique incidence as illustrated in case (b) in Figure 4.2.1 – or as a function of angle of incidence -
can be measured using various techniques using separation in time or subtraction of incident and reflected sound pulses.
However, these techniques are not always very reliable.
103
The measurement takes place in a reverberation room, with highly irregular or non parallel surfaces
and/or suspended, sound diffusing elements. Hereby it can be assumed that the sound field will
fulfil the requirements for application of the Sabine reverberation equation. Assume the room has a
volume V, total surface area S and that αempty is the absorption coefficient of the room surfaces
(which ideally should all be made from the same, acoustically hard material). In this case equations
3.4.1 and 3.4.2 (disregarding air absorption) yields:
0.16 V
T60,empty = (4.1)
S Room α empty
If now we place a test sample of a material with area Ssample (usually 10 m2) in the room, the
equation changes into:
0.16 V
T60, sample = (4.2)
S sample α sample + ( S Room − S sample )α empty
in which we have considered that an area, Ssample, of the room surface has now been covered by the
sample. Combining equations 4.1 and 4.2 by eliminating S yields for the unknown absorption
coefficient, αsample, of the test sample:
0,16 V ⎡ 1 1 ⎤
α sample = ⎢ − ⎥ + α empty (4.3)
S sample ⎢⎣ T60,sample T60,empty ⎥⎦
The measurement is normally carried out in 1/1 or 1/3 octave bands from 100 to 5000 Hz.
If absorption measurements using the room method is carried out on small sized samples, these
sometimes appear to have a absorption coefficient larger than 1.0, as seen in Figure 4.2.2. Of course
this is not logical, if the absorption power should be related
solely to the physical area of the sample. The phenomenon
is probably due to diffraction of sound around the edges of
the sample, which dominates the behaviour in cases where
the linear dimension of the sample approaches the wave
length of the sound, i.e. the effect is more pronounced at low
frequencies.
Although a complication in documentation of absorption

properties, this phenomenon can be applied successfully in
Fig. 4.2.2 Absorption coefficients practice by providing increased absorption effect, if the
of different materials versus area available absorption material can be provided in smaller
(measured in square feet). From [1]. pieces and spread out over the room surfaces.
4.3 Different types of sound absorbers
In this section the three most common types of sound

absorbing constructions will be described, each with its
own characteristic frequency dependency of the
absorption coefficient as sketched in Fig. 4.3.1.
Fig. 4.3.1 Typical behaviour of absorption versus

frequency for Porous, resonating and membrane
absorbers respectively.
104
4.3.1 Porous absorbers
Porous absorbers are present in rooms in the form of textiles like curtains, carpets and furniture
upholstery, porous mortar in (unpainted !) brick walls and not least as a wide variety of dedicated
sound absorbing products for suspended ceilings.
Figure 4.3.2 Left: Standing wave pattern formed by an incident and a reflected sound wave in front
of a porous material of a certain thickness flush mounted on a heavy and hard surface. Right:
Absorption versus frequency of a thin, porous sheet placed in front of a hard surface. From [1].
Porous materials are characterized by having an open structure of e.g. of fibres glued or woven
together which is accessible by the air. Thus, air can be pressed through the material more or less
easily depending on the flow resistance (determined e.g. by how densely a fabric is woven – try for
yourself by blowing through clothing or curtains !). The absorption properties are caused by viscous
friction between the moving air molecules in the sound waves and the often huge internal surface
area of the structure whereby the (kinetic) sound energy is converted into heat.
If a porous sheet of a certain thickness is placed

flush on a rigid surface and hit by an normal
incidence sound wave a standing wave pattern
will be created with pressure amplitude as
indicated to the left in Fig. 4.3.2. As seen from
Figure 1.2.10 (c), with a rigid termination, the
particle velocity and so the kinetic energy of the
sound field will be high where the pressure
amplitude (the potential energy) is low. In other
words, for the absorber to be efficient (with
normal incidence of the sound wave), the
thickness of the porous layer need to be at least
λ/4, so that friction takes place where the
energy is primarily kinetic. In other words, for a
given thickness of the material, there is a lower
limiting frequency below which the absorption
drops off because the material can no longer
“reach” the region of high kinetic energy. On
Fig. 4.3.3 Absorption coefficients for mineral the other hand, as the absorber is not absorbing
wool (glasswool) with thickness as parameter the potential energy anyway, one can save
(a) and with wall distance as parameter (b). material and just place a thin sheet (but still
105
with a suitable flow resistance) at a certain distance from the rigid wall (like a curtain in front of a
window). In the case of normal incidence, applying a thin sheet will cause the absorption to drop
again at a higher frequency where the distance between sheet and hard wall equals λ/2; but with
diffuse field incidence this dip will not be very pronounced. Diffuse field incidence also causes the
absorbers to be effective (α > 0,8) if just the thickness/distance is > λ/8.
Fig. 4.3.3 shows how the absorption coefficient varies with frequency for mineral wool mats of
different thickness (upper graph) and different distances to the rigid wall (lower graph). It is seen
that more low frequencies are absorbed as the thickness or the wall distance increases.
Mineral wool consists of thin fibres pressed and glued together. The fibres are made from melted
glass (Glasswool) or stone (Rockwool) much like “Candy Floss”. Mineral wool is used as porous
sound absorbers, very often in the form of tiles which can be mounted in a suspended ceiling
system. Such ceilings will often be placed below ventilation ducts and other technical installations,
whereby a large distance (typically between 20 cm and one metre) is ensured to the hard concrete
deck behind. Hereby the ceiling can absorb efficiently over a wide frequency range – as well as hide
the installations. Mineral wool ceiling tiles are normally given a carefully controlled layer of special
paint from the factory to make them look like normal (white) plaster ceilings as much as possible.
However, if one tried to repaint them, the porous properties and so the absorption normally
disappears.
4.3.2 Membrane absorbers

A membrane absorber is characterized by consisting of a non porous sheet or panel placed at a
certain distance from a hard backing whereby an air filled cavity is formed. This system can
resonate at frequency determined by the mass per unit area of the plate, m, and the spring function
of the enclosed air, which is determined by the depth, L, of the cavity:
1 ρ c2
f0 = (4.4)
2π m L
However, Equation 4.4 only apply if the plate is completely limp. Normally, the resonance
frequency is also determined by the plate stiffness and mode of plate vibration, of which a few are
illustrated in Fig. 4.3.4, with p and q being integers determining the shape of the two dimensional
oscillation pattern of the plate.
Fig. 4.3.4 Different modes of vibration in a stiff plate.
In this case the resonance frequency can be described as follows:

2
π 4 ⎡⎛ p ⎞ ⎛ q ⎞ ⎤
2 2
1 ρc2 Eh3
fr = + ⎢⎜ ⎟ + ⎜ ⎟ ⎥ (4.5)
2π m L m ⎢⎣⎝ a ⎠ ⎝ b ⎠ ⎦⎥ 12 (1 −ν 2 )
in which a and b are the dimensions of the plate (or the distance between studs supporting the
plate), h is the thickness while E and ν are the Young’s modulus and the Poisson ratio respectively.
106
From this formula it is seen that a resonance frequency is determined completely by the stiffness if
the depth of the cavity is infinitely deep – as is the case e.g. with a single pane window.
Fig. 4.3.5 show absorption versus frequency

for two different thickness of plywood placed
45 mm from a hard backing – with and
without mineral wool in the cavity. As
expected it is seen that the thicker and heavier
plate result in the lowest resonance frequency
as expected from equations 4.4 and 4.5.
Besides, it is observed that the mineral wool
inlay, which increases the internal damping of
the construction causes a significant
Fig. 4.3.5 absorption versus frequency of improvement in the absorption around the
membrane absorber for two different plate resonance frequency and also causes the
thicknesses and with and without mineral resonance frequency to become lower.
wool in the cavity. From [1]
Membrane absorbers are often found in rooms in the form of wooden floors on joists or as gypsum
board or wood panel walls. The effect is a controlled low frequency T60 value as opposed to rooms
made entirely from heavy concrete or masonry which causes the sound to be “dark” and blurred at
low frequencies.
Fig. 4.3.6 Example of membrane absorbers attached to the concrete side wall in the multi purpose
hall (Kolding Teater). Besides controlling low frequency reverberation, the panels also provide
some diffusion of the sound.
107
4.3.3 Resonator absorbers
In stead of having a plate forming the mass of the resonating system, the mass can be oscillating air
in an opening between a closed cavity and the open atmosphere. Also in this case, the enclosed air
Fig. 4.3.7 Single resonator (left) and resonating panel (right). From [1].
in the cavity provides the spring function. An example of such a single resonator, called a Helmholz
resonator, is illustrated in Figure 4.3.7. The resonance frequency (which can be experienced by
blowing across the opening of a bottle) is given by:
c S
f0 = (4.6)
2π V ( l + δ )
with S being the area of the opening, V being the enclosed volume, l the length of the neck and δ a
correction to the neck length which is due to the fact that the oscillating air mass - often moving
with very high velocity - is not confined to the physical length of the neck; but some of the air
outside both ends of the neck will be moving as well.
Resonators like the build in “bottle” in the left side of Fig. 4.3.7 are not very practical, as the
frequency range of the absorption is normally very limited around the sharp resonance frequency.
However, if a perforated panel is placed in front of a cavity as seen to the right in Fig. 4.3.7, then
this construction can be regarded as a large number of single resonators put together, and the
physical proportions in this case often causes a much more useful frequency range of absorption.
For the resonance frequency of the panel we have:
c P
f0 = (4.7)
2π L ( l + δ )
which is almost identical with Equation 4.6 except for the opening area being replaced by the
degree of perforation, P, of the panel and the volume V being changed into the depth of the cavity L.
If the holes are circular with diameter d, we have for the end correction: δ ≈ 0.8 d. Resonating
panels will often have a higher resonance frequency and absorb efficiently in a wider frequency
range than the membrane absorbers.
Regarding damping, the viscous damping can be significant if the hole/slit dimensions are small;
but often the absorption can be optimised by placing a thin layer of mineral wool or glass felt
(called vlies) in the cavity. Like in the case of the membrane absorber, it is important to adjust the
damping to achieve optimal absorption.
108
Perforated panels are found in the form of perforated gypsum board or steel plates (used e.g. for
suspended ceilings)2, or as panels made of wooden boards with slits between the individual boards
as illustrated to the left in Figure 4.3.8. Other possibilities are walls made from perforated tiles,
which make use of the cavity already present in a double masonry wall as shown to the right in the
same Figure.
Fig. 4.3.8 Resonating panel constructions in practice. Left: Wooden boards separated by
controlled gaps in front of a former window niche filled with mineral wool. The panel controls low
frequency reverberation in a former power plant building made from heavy masonry converted into
a concert hall (Værket, Randers). Right: Perforated bricks on the rear wall in a sports and multi
purpose hall. By making this wall absorbing, echoes back to the stage placed more than 50 m away
are avoided (Frihedshallen, Sønderborg).
4.4 Application of sound absorbers in room acoustic design
The main purpose of introducing absorption for reverberation control in rooms is to reduce noise
levels (see Fig. 3.2.4) and in some cases to increase intelligibility. The Danish Building Law
(Bygningsreglementet af 1995, BR95) [2] contains demands on maximum T60 values in school class
rooms, day care institutions and apartment buildings, whereas the Danish Working Environment
Agency have issued rules for industrial buildings and offices [3]. These current Danish rules are
2
It should be added that in many cases with perforated gypsum or steel plates used as suspended ceilings, the
combinations of perforation and cavity depth causes the absorber to act more like a porous absorber but with reduced
performance at high frequencies due to the panel shielding off the porous layer to some degree.
109
briefly listed in Figure 4.4.1. Recommendable values for other types of rooms – including auditoria
and concert halls were listed in Table 3.4.5. Special standards exist for design of cinemas and studio
control rooms and listening rooms. In Denmark, no rules exist for other public spaces like traffic
terminals, sports arenas and restaurants - although the acoustic conditions in these places are often
horrible. However, acoustic concerns a generally included in modern design of these spaces as well.
Fig. 4.4.1 Listing of Danish rules regarding maximum values of reverberation time in buildings.
(The values listed for single person offices and corridors in office buildings just reflect common
design practice.)
As indicated in Figure 4.4.1 the rules for large industrial halls as well as open plan areas in offices
and schools are specified in terms of a required minimum absorption area. The reason for this is that
often calculation as well as measurement of T60 is often questionable in these rooms.
In most cases the ceiling is the most obvious surface to treat with absorption, as it constitutes a large
area which is normally available apart from a few light or ventilation fixtures and because here the
often delicate absorption materials are not subject to mechanical damage.
Fig. 4.4.2 Examples of acoustic treatment mounted in ceiling in industrial halls. Left: suspended
ceiling of mineral wool tiles with integrated light fixtures. Right: Vertical Mineral wool baffles.
110
In Figure 4.4.2 are shown two examples of acoustic treatment of ceilings. To the left a normal
suspended ceiling of mineral wool tiles with integrating lighting and ventilation. This type of
ceiling is often found in offices, schools, shops etc.. The vertical mineral wool baffles shown to the
right can be a solution when the ceiling is already heavily occupied by technical installations.
In rooms where practically all the absorption is placed in the ceiling, the reverberation time
basically becomes a function of the room height as shown in Figure 4.4.3. In high rooms, it is not
always sufficient to place the absorption in the ceiling surface alone; but also available wall areas
must be used as illustrated by the mineral wool tiles to the right in Figure 4.4.3.
Fig. 4.4.3 Simplified calculation of T60 in room with all absorption placed on the ceiling
surface(left) shows the need for additional absorption on walls in tall rooms (right).
In many public places like traffic terminals, department stores, sports halls etc., the room acoustic
absorption treatment is not only done with the purpose of reducing noise but also to ensure proper
intelligibility of speech (often emitted through loudspeakers). In Figure 4.4.4 is illustrated how a
long room decay can cover (mask) the weak phonems illustrated schematically as vertical bars. In
speech the consonant sounds are often the weaker elements; but they contain most of the
information. Therefore, a long reverberation can seriously deteriorate intelligibility.
Fig. 4.4.4 Schematic illustration of the influence of reverberation on the intelligibility of speech.
111
In rooms dedicated for speech like auditoria, class rooms and theatres, the room acoustic design not
only consists of reverberation control by absorption treating of the room surfaces. In these rooms
also the design of the room geometry is important to ensure proper propagation of sound from the
source to the listeners through reflection of the sound waves off non absorbing room surfaces. And
in order to support intelligibility, these reflections must arrive not long (up to 40 ms) after the direct
sound.
Even in normal sized class rooms this concern about supporting reflections may be applied by
leaving a central part of the ceiling reflective (given that enough other surface areas can be found to
provide the required reverberation control). Thus, Fig. 4.4.5 illustrates such a case in which the
ODEON programme was used to balance the application of absorbing and reflective part of the
ceiling for a school project and to predict reverberation time and the intelligibility in terms of the
Speech Transmission Index mentioned in Section 2.7.
Fig. 4.4.5 Illustrations from the room acoustic simulation programme ODEON of a class room
design with a partly absorbing (dark) and reflective (lighter grey) ceiling.
References
[1] Z. Maekawa and P. Lord: Environmental and Architectural Acoustics. E & FN Spon,
London, 1994.
[2] Bygningsreglementet; Bygge- og Boligstyrelsen 1995. Publications from the Danish

National Agency for Enterprise and Construction (in Danish) can be found at:
http://www.ebst.dk/pub_lydforhold/0/8/0
[3] AT anvisning nr.1.1.0.1, november 1995; Akustik i arbejdsrum

(Acoustics in work places) http://www.at.dk/sw5110.asp
112
5. An introduction to sound insulation
Jens Holger Rindel
5.1 THE SOUND TRANSMISSION LOSS
5.1.1 Definition
A sound wave incident on a wall or any other surface separating two adjacent rooms partly
reflects back to the source room, partly dissipates as heat within the material of the wall, partly
propagates to other connecting structures, and partly transmits into the receiving room.
The power incident on the wall is P1 and the power transmitted into the receiving room is P2.
The sound transmission coefficient τ is defined as the ratio of transmitted to incident sound
power
P2
τ = (5.1.1)
P1
However, the sound transmission coefficients are typically very small numbers, and it is more
convenient to use the sound transmission loss R with the unit deciBel (dB). It is defined as
P 1
R = 10 log 1 = 10 log = − 10 log τ (dB) (5.1.2)
P2 τ
Another name for the same term is the sound reduction index.
5.1.2 Sound insulation between two rooms
Figure 5.1.1. Airborne sound transmission from source room (1) to receiving room (2)
The most common case is the sound insulation between two rooms. With the assumption of
diffuse sound fields in both rooms it is possible to derive a simple relation between the
transmission loss and the sound pressure levels in the two rooms. The rooms are called the
source room and the receiving room, respectively. In the first room is a sound source that
generates the average sound pressure p1. The sound power incident on the wall is, see eq.
(3.2.6)
113
p12 S
P1 = I inc S = (5.1.3)
4ρ c
The area of the wall is S. In the receiving room the average sound pressure p2 is generated from
the sound power P2 radiated into the room, see eq. (3.2.8)
4 P2
p22 = ρc (5.1.4)
A2
Here A2 denotes the absorption area in the receiving room. Insertion in the definition (5.1.2)
gives
p2 S S
R = 10 log 21 = L1 − L2 + 10 log (dB) (5.1.5)
p2 A2 A2
Here L1 and L2 are the sound pressure levels in the source and receiving room, respectively.
This important result is the basis for transmission loss measurements.
5.1.3 Measurement of sound insulation

Sound insulation is measured in one-third octave bands covering the frequency range from 100
Hz to 3150 Hz. In recent years the international standards for measurement of sound insulation
have been revised and it is recommended to extend the frequency range down to 50 Hz and up
to 5000 Hz. One reason for this is that the low frequencies 50 – 100 Hz are very important for
the subjective evaluation of the sound insulation properties of lightweight constructions. In
recent years lightweight constructions have been more commonly used in new building
technology, whereas heavy constructions have traditionally been used for sound insulation.
The sound pressure levels are measured as the average of a number of microphone positions or
as the average from microphones slowly moving on a circular path. The results are averaged
over two different source positions. More details are given in ISO 140 Part 3 and 4.
In addition to the two sound pressure levels it is also necessary to measure the reverberation
time in the receiving room in order to calculate the absorption area. Sabine’s equation is used
for this, see eq. (3.2.13)
55.3 V2
A2 = (5.1.6)
c T2
Only under special laboratory conditions it is possible to measure the transmission loss of a
wall without influence from other transmission paths. In a normal building the sound will not
only be transmitted through the separating construction, but the flanking constructions will also
influence the result, see later in section 5.5.4.
For measurements of sound insulation in buildings the apparent sound transmission loss is
S
R′ = L1 − L2 + 10 log (dB) (5.1.7)
A2
The apostrophe after the symbol indicates that flanking transmission can be assumed to
influence the result.
5.1.4 Multi-element partitions and apertures

A partition is often divided into elements with different sound insulation properties, e.g. a wall
with a door. Each element is described by the area Si and the transmission coefficient τi . If the
sound intensity incident on the surfaces of the source room is denoted Iinc the total incident
sound power on the partition is
114
n
P1 = ∑S
i =1
i I inc = S I inc
The total area is called S. The total sound power transmitted through the partition is
n
P2 = ∑τ
i =1
i S i I inc
Thus, the transmission coefficient of the partition is

P 1 n
τ res = 2 = ∑τ i Si (5.1.8)
P1 S i =1
The same result can also be written in terms of the transmission losses Ri of each element
⎛1 n ⎞
Rres = − 10 log τ res = − 10 log⎜ ∑ S i 10 − 0,1 Ri ⎟ (5.1.9)
⎝ S i =1 ⎠
In the simple case on only two elements the graph in Fig. 5.1.2 may be used.
Figure 5.1.2. Graph for estimating the transmission loss of a multi-element partition
An aperture in a wall is a special example of an element with different transmission properties.

As an approximation it can be assumed that the transmission coefficient of the aperture is 1. If
also the area of the aperture Sap is very small compared to the total area, this leads to the
following result for the resulting transmission loss of the wall with aperture:
⎛ S ap ⎞
⎛1
( )
⎞
Rres = − 10 log⎜ S1 10 −0,1 R1 + S ap ⎟ ≅ − 10 log⎜⎜10 −0,1 R1 +
S ⎟⎠
⎟ (5.1.10)
⎝S ⎠ ⎝
Fig. 5.1.3 can illustrate the result. It is seen that the relative area of the aperture defines an
upper limit of the sound insulation that can be achieved.
115
Figure 5.1.3. Graph for estimating the transmission loss of a construction with an aperture
116
5.2 SINGLE LEAF CONSTRUCTIONS
5.2.1 Sound transmission through a solid material

The solid material is supposed to have the shape of a large plate with thickness h. The material
is characterised by the density ρm and the speed of longitudinal waves cL . The surface of the
material defines two transition planes where the sound waves change from one medium to
another. It is assumed that the medium on either side is air with the density ρ and the speed of
sound c (also longitudinal waves). The symbols and notation are explained in Fig. 5.2.1.
pi p1 p2 pt
pr p4 p3
Figure 5.2.1. Thick wall with incident, reflected and transmitted sound waves
The sound pressure is equal on either side of the two transition planes:
p i + p r = p1 + p 4
(5.2.1)
pt = p 2 + p3
Also the particle velocity is equal on either side of the two transition planes:
u i − u r = u1 − u 4
(5.2.2)
ut = u 2 − u3
The characteristic impedance in the surrounding medium (air) is denoted Z0 and that in solid
material is denoted Zm. Thus the ratio of sound pressure to particle velocity in each of the plane
propagating waves is:
pi pr pt
= = = Z0 = ρc
ui ur ut
(5.2.3)
p1 p2 p3 p4
= = = = Z m = ρ m cL
u1 u2 u3 u4
Using (5.2.3) in (5.2.2) leads to:
Z0
pi − p r = ( p1 − p 4 )
Zm
(5.2.4)
Z0
pt = ( p 2 − p3 )
Zm
Assuming propagation from one side of the material to the other without losses means that there
is only a phase difference between the pressure at the two intersections:
117
p2 = p1 e − j km h
(5.2.5)
p 4 = p3 e − j k m h
Here km = ω /cL is the angular wave number for longitudinal sound propagation in the solid
material.
From the above equations (5.2.1), (5.2.4) and (5.2.5) can be derived the ratio between the sound
pressures pi and pt and thus the transmission loss can be expressed by:
2
⎛ ⎛ Z0 Zm ⎞
2
⎞
R0 = 10 log
pi ⎜
= 10 log cos (k m h) + ⎜⎜
2 1 + ⎟⎟ sin 2 (k m h) ⎟ (5.2.6)
pt ⎜ 4 ⎝ Zm Z0 ⎠ ⎟
⎝ ⎠
R0, dB
Fig. 5.2.2. Transmission loss at normal incidence of sound on a 600 mm thick concrete wall.
At high frequencies some dips can be observed in the transmission loss curve. They occur at
frequencies where the thickness is equal to half a wavelength in the solid material, or a multiple
of half wavelengths. However, the dips are very narrow and they are mainly of theoretical
interest.
Two special cases can be studied. First the case of a thin wall: Zm >> Z0 and kmh << 1
⎛ ⎛ Z ⎞2 ⎞ ⎛ ⎛ ω ρ h ⎞2 ⎞
⎜
R0 ≅ 10 log 1 + ⎜⎜ m 2 ⎟
⎟ sin (k m h) ≅ 10 log⎜1 + ⎜⎜ m
⎟ ⎟ (5.2.7)
⎜ ⎝ 2 Z 0 ⎟⎠ ⎟ ⎜ ⎝ 2 ρ c ⎟⎠ ⎟
⎝ ⎠ ⎝ ⎠
The other special case is a very thick wall: Zm >> Z0 and kmh >> 1
2
⎛ Z ⎞ ⎛ρ c ⎞
R0 ≅ 10 log⎜⎜ m ⎟⎟ ≅ 20 log⎜⎜ m L ⎟⎟ (5.2.8)
⎝ 2 Z0 ⎠ ⎝ 2ρc ⎠
118
The cross-over frequency from (5.2.7) to (5.2.8) is the frequency fh at which kmh = 1:
cL
fh = (5.2.9)
2π h
This is the frequency at which the thickness is approximately one sixth of the longitudinal
wavelength λL in the material:
cL λL
h = =
2π f 2π
The result for the thin wall is the so-called mass law, which will de derived in a different way in
the next section. The result for a very thick wall (5.2.8) means that there is an upper limit on the
sound insulation that can be achieved by a single-leaf construction, and this limit depends on
the density of the material. For wood it is 68 dB, for concrete 80 dB and for steel 94 dB. (These
numbers should be reduced by 5 dB in the case of random incidence instead of normal
incidence, see section 5.2.3).
5.2.2 The mass law
pi vn
vt = vn / cos θ
pr pt
Figure 5.2.3. Thin wall with sound pressures and particle velocities
A thin wall with the mass per unit area m is considered, see Fig. 5.2.3. The application of
Newton’s second law (force = mass ⋅ acceleration) gives:
dv
Δ p = pi + p r − pt = m n = j ω m v n (5.2.10)
dt
where vn is the velocity of the wall vibrations (in the direction normal to the wall). The
separation impedance Zw is introduced:
Δp
Zw = = jω m (5.2.11)
vn
The separation impedance will be more complicated if the bending stiffness of the wall is also
taken into account, see below.
The particle velocities in the sound waves are called u with the same indices as the
corresponding sound pressures. Due to the continuity requirement the normal component of the
velocity on both sides of the wall is:
vn = ut cos θ = (ui − u r ) cos θ (5.2.12)
which leads to
119
vn
pt = pi − p r = Z0 (5.2.13)
cos θ
The sound transmission loss Rθ at a certain angle of incidence θ is:
2 2
p Z cos θ
Rθ = 10 log i = 10 log 1 + w (dB) (5.2.14)
pt 2 Z0
In the special case of normal sound incidence (θ = 0) the insertion of (5.2.11) gives the
important mass law of sound insulation:
2
ωm ⎛π f m ⎞
R0 = 10 log 1 + j ≅ 20 log⎜⎜ ⎟⎟ (dB) (5.2.15)
2 ρc ⎝ ρc ⎠
Since m = ρmh this result is the same as derived above in (5.2.7).
5.2.3 Sound insulation at random incidence

The transmission coefficient at the angle of incidence θ is from (5.2.14)
1
τ (θ ) = 2
(5.2.16)
⎛ωm⎞
1 + ⎜⎜ ⎟⎟ cos 2 θ
⎝ 2 ρ c ⎠
Random incidence means that the sound field on the source side of the partition is
approximately a diffuse sound field. In a diffuse sound field the incident sound power P1 on a
surface is found by integration over the solid angle ψ = 2 π assuming the same sound intensity
I1 in all directions. The principle is the same as used in section 3.2.2. Since, in each direction
the transmitted sound power is equal to the incident sound power multiplied by the transmission
coefficient, the ratio between transmitted and incident power is:
π /2
τ =
P2
=
∫ψ = 2π
τ (θ ) I1S d ψ
=
∫
0
τ (θ ) cos θ sin θ d θ
π /2
P1 ∫ψ = 2π
I1 S d ψ
∫
0
cos θ sin θ d θ
π /2 1
τ = 2 ∫ τ (θ ) cos θ sin θ d θ = ∫ τ (θ ) d(cos
2
θ)
0 0
( )
2
d(cos 2 θ ) ⎛ 2ρ c ⎞
1
τ = ∫0 1 + (ω m 2 ρ c )2 cos 2 θ = ⎜⎜ ⎟⎟ ln 1 + (ω m 2 ρ c )2
⎝ ω m ⎠
R = − 10 log τ = R0 − 10 log(0,23R0 ) (dB) (5.2.17)

This is the theoretical result for random incidence, and for typical values (R0 between 30 and 60
dB) it means that R is 8 to 11 dB lower than R0. However, in real life this is not true and it can
be shown that the result is related to partitions of infinite size. Taking the finite size into
account the result is approximately:
R ≅ R0 − 5 dB (5.2.18)
This is in good agreement with measuring results on real walls.
120
5.2.4 The critical frequency
The bending stiffness per unit length of a plate with thickness h is:
E h3
B = (5.2.19)
12 (1 − ν 2 )
where E is Young’s modulus of the material and ν is Poisson’s ratio. (ν ≅ 0.3 for most rigid
materials).
The speed of propagation of bending waves in a plate with bending stiffness per unit width B
and mass per unit area m is (see section 6.3.3):
B f
cb = ω4 = c (5.2.20)
m fc
Here fc is introduced as the critical frequency. It is defined as the frequency at which the speed
of bending waves equals the speed of sound in air, cb = c.
The critical frequency is:

c2 m
fc = (5.2.21)
2π B
A sound wave with the angle of incidence θ propagates across the wall with the phase speed
c / sin θ , i.e. the phase speed is in general higher than c, see Fig. 5.2.4. If the bending wave
speed happens to be equal to the phase speed of the incident sound wave, this is called
coincidence:
cb = c / sin θ
Figure 5.2.4. Thin wall with bending wave and indication of speed of propagation along the
wall
The coincidence leads to a significant dip in the sound transmission loss. The coincidence dip
will be at a frequency higher than or equal to the critical frequency:
121
f co = f c sin 2 θ (5.2.22)
The separation impedance (5.2.11) is replaced by:
⎛ ⎛ f ⎞2 ⎞
Z w = jω m ⎜1 − ⎜⎜ ⎟⎟ sin 4 θ ⎟ (5.2.23)
⎜ ⎝ fc ⎠ ⎟
⎝ ⎠
Insertion in the general equation (5.2.14) leads to the sound transmission loss at a certain angle
of incidence:
Rθ = R0 + 20 log cos θ + 20 log 1 − ( f f c ) sin 4 θ
2
(dB) (5.2.24)
5.2.5 A general model of sound insulation of single constructions

The general model of sound insulation is based on mass law as given in (5.2.15). However, the
following results are valid for sound insulation between rooms with approximately diffuse
sound fields. In the frequency range below the critical frequency, f < fc:
R ≅ R0 + 20 log 1 − ( f f c ) − 5 dB
2
(5.2.25)
In the frequency range above the critical frequency, f ≥ fc:
2η f
R ≅ R0 + 10 log (dB) (5.2.26)
π fc
where η is the loss factor (see section 6.2.2.3).
The upper limit for sound insulation of a single-leaf construction is, according to (5.2.8):
⎛ρ c ⎞
R ≤ 20 log⎜⎜ m L ⎟⎟ − 5 dB (5.2.27)
⎝ 2ρc ⎠
A sketch of the transmission loss as a function of frequency is shown in Fig. 5.2.5
R, dB
fc
Frequency (log)
Figure 5.2.5. Sound insulation of a single-leaf construction, fc is the critical frequency and the
upper limit is the dotted line.
122
5.3 DOUBLE LEAF CONSTRUCTIONS
5.3.1 Sound transmission through a double construction
m1 d m2
pi p1 p2 pt
pr p4 p3
v1 v2
Fig. 5.3.1. A double construction with indication of sound pressures and particle velocities
A double construction with two plates in the distance d is considered, see Fig. 5.3.1. The
separation impedance of the two plates is denoted Z1 and Z2, respectively. As for the single
construction in (5.2.10) the movement of each wall is:
pi + p r − ( p1 − p 4 ) = Z 1v1
(5.3.1)
p 2 + p3 − pt = Z 2 v 2
The velocity of each wall equals the particle velocity on either side:
1
v1 = ui − u r = ( pi − p r )
Z0
1
v1 = u1 − u 4 = ( p1 − p4 )
Z0
(5.3.2)
1
v2 = u 2 − u3 = ( p 2 − p3 )
Z0
1
v2 = ut = pt
Z0
Assuming propagation from one side of the cavity to the other without losses means that there
is only a phase difference between the pressure at the two intersections:
− jk d
p2 = p1 e
(5.3.3)
p4 = p3 e − j kd
From the above equations (5.3.1), (5.3.2) and (5.3.3) can be derived the ratio between the sound
pressures pi and pt and thus the transmission loss can be expressed by:
123
2
p
R0 = 10 log i
pt
2
(5.3.4)
⎛ Z + Z2 ⎞ ⎛ Z + Z 2 Z1 Z 2 ⎞
= 10 log ⎜⎜1 + 1 ⎟⎟ cos(kd ) + j ⎜⎜1 + 1 + ⎟ sin(kd )
⎝ 2Z 0 ⎠ ⎝ 2Z 0 2 Z 02 ⎟⎠
If only the mass of each wall is taken into account the separation impedances are:
Z1 = jω m1
(5.3.5)
Z 2 = j ω m2
Neglecting the smaller parts and inserting Z0 = ρ c together with (5.3.5) yields:
⎡⎛ ω (m + m ) 2
⎞ ⎛ ω (m1 + m2 ) ω 2 m1m2 ⎞ ⎤
2
R0 ≅ 10 log ⎢⎜⎜ 1 2
sin(kd ) ⎟⎟ + ⎜⎜ cos(kd ) − sin(kd ) ⎟⎟ ⎥ (5.3.6)
⎢⎣⎝ 2ρ c ⎠ ⎝ 2ρ c 2( ρ c) 2 ⎠ ⎥⎦
This result will be discussed and simplified below.
5.3.2 The mass-air-mass resonance frequency

The transmission loss is minimum when the last term is zero, i.e.
m1 + m2 ρ c
tg( kd ) = (5.3.7)
m1m2 ω
For a cavity that is narrow compared to the wave length (kd << 1) we get:
ωd m1 + m2 ρ c
tg( kd ) ≈ kd = =
c m1m2 ω
The solution is the mass-air-mass resonance frequency f0 = ω 0 / 2 π
c ρ⎛ 1 1 ⎞
f0 = ⎜⎜ + ⎟ (5.3.8)
2π d ⎝ m1 m2 ⎟⎠
If the depth d of the cavity is comparable to the wavelength there are many solutions to (5.3.7)
and they are approximately kd = n π. The dips in the sound insulation occur at frequencies at
which the cavity depth equals one or more half wavelenghts: d = n λ /2.
However, more important than these dips is the shift from low- to high-frequency behaviour of
the air cavity. The cross-over frequency has no particular physical meaning, but it is the
frequency fd at which kd = 1:
c
fd = (5.3.9)
2π d
This is quite similar to the result (5.2.9) found for the sound transmission through a solid
material. Only, in this case the transmission is through air. The spring-like behaviour of the air
cavity changes from that of a simple spring below the cross-over frequency to that of a
transmission channel at higher frequencies.
5.3.3 A general model of sound insulation of double constructions

The result (5.3.6) can be simplified in different way depending on the frequency range. In the
frequency range below the resonance frequency, f < f0:
⎛ ω (m1 + m 2 ) ⎞
R0 ≈ 20 log ⎜⎜ ⎟⎟ = R(1+ 2 ) (5.3.10)
⎝ 2ρ c ⎠
124
This means that the construction behaves as a single construction with the mass per unit area
(m1 + m2). In the frequency range above the resonance frequency, f0 < f < fd:
⎛ ω 3 m1 m 2 d ⎞
R0 ≈ 20 log ⎜⎜ ⎟ ≈ R1 + R2 + 20 log(2kd ) (5.3.11)
⎟
⎝ 2ρ c ⎠
2 3
In this a much better sound insulation can be obtained, and it depends on the product of the
three parameters m1, m2 and d. At frequencies above fd where the cavity is wide compared to the
wavelength, sin (kd) is replaced by its maximum value 1, and for f ≥ fd:
⎛ ω 2 m1 m 2 ⎞
R0 ≈ 20 log ⎜⎜ 2 ⎟
⎟ ≈ R1 + R2 + 6 dB (5.3.12)
⎝ 2 ( ρ c) ⎠
In this high-frequency range, d is no longer an important parameter.
A sketch of the transmission loss as a function of frequency is shown in Fig. 5.3.2.
R, dB
f0 fd
Frequency (log)
Figure 5.3.2. Sound insulation of a double-leaf construction, f0 is the resonance frequency and
fd is the cross-over frequency of the cavity.
R, dB
f0 fd fc1 fc2
Frequency (log)
Figure 5.3.3. Sound insulation of an asymmetric double-leaf construction with two thin plates
having different critical frequencies, fc1 and fc2, respectively.
125
5.4 FLANKING TRANSMISSION
Fig. 5.4.1. Direct transmission and three flanking transmission paths via the floor.
The transmission of sound from a source room to a receiver room can be via flanking
constructions like the floor, the ceiling or the façade. When all relevant transmission paths are
considered the sound insulation is described by the apparent sound transmission loss:
P1 S
R′ = 10 log = L1 − L2 + 10 log (dB) (5.4.1)
P2 + P3 A2
where P2 is the sound power transmitted through the partition wall to the receiver room and P3
is the sound power radiated to the receiver room from the flanking surfaces and other flanking
paths:
P3 = ∑ PF ,i (5.4.2)
i
Each single flanking transmission path i can be characterised by the flanking transmission loss,
RF,i :
P
RF ,i = 10 log 1 (dB) (5.4.3)
PF ,i
It is convenient to keep the incident sound power P1 on the partition wall as a reference for all
the flanking transmission losses. In this way it is very simple to add all the contributions
together, and the apparent transmission loss is calculated from:
⎛ − 0 ,1R ⎞
R′ = − 10 log⎜10 −0,1R + ∑ 10 F ,i ⎟ (dB) (5.4.4)
⎝ i ⎠
In the typical case of horizontal transmission through a wall the will be 12 flanking paths,
namely three possible paths for each of the four surrounding flanking constructions, see Fig.
5.4.1.
126
5.5 ENCLOSURES
A noise source is supposed to radiate the sound power Pa. The noise source is totally covered
by an enclosure with surface area S, absorption coefficient α on the inside, and the enclosure is
made from a plate with transmission loss R or transmission coefficient τ. The average sound
pressure in the enclosure pencl can be estimated, if a diffuse sound field is assumed:
4 Pa
2
pencl = ρc (5.5.1)
αS
The sound power incident on the inner surface of the enclosure is (still with the assumption of a
diffuse sound field):
2
pencl S
Pinc = (5.5.2)
4ρ c
The sound power transmitted through the enclosure is then:
τ
Pout = τ Pinc = Pa (5.5.3)
α
The insertion loss of the enclosure is the difference in radiated sound power level without and
with the enclosure:
P α
ΔL = 10 log a = 10 log = R + 10 log α (dB) (5.5.4)
Pout τ
This result cannot be considered to be very accurate. Especially the assumption of a diffuse
sound field inside the enclosure is doubtful. However, the result is not bad as a rough estimate
for the design of an enclosure. It is clearly seen from (5.5.4) that both transmission loss and
absorption coefficient are important for an efficient reduction of noise by an enclosure.
5.6 IMPACT SOUND INSULATION

The noise generated from footsteps on floors is characterised by the impact noise level. It is
measured according to ISO 140 Part 6 and 7 by a standardised tapping machine. The main data
for the tapping machine are:
• The noise is generated by steel hammers with a fall height of 40 mm

• Each steel hammer has a mass of 500 g
• The number of taps per second is 10.
In the source room the tapping machine is placed on the floor in a number of positions. In the
room below - or any other room in the building – the calibrated sound pressure level L2 is
measured. The reverberation time in the receiving room must also be measured in order to
calculate the absorption area A2. The impact sound pressure level is the sound pressure level in
dB re 20 μPa that would be measured if the absorption area is A0 = 10 m2:
A
Ln = L2 + 10 log 2 (dB) A0 = 10 m 2 (5.6.1)
A0
The frequency range is the same as for airborne sound insulation, i.e. the 16 one-third octave
bands from 100 Hz to 3150 Hz. However, it is recommended to extend the frequency range
down to 50 Hz, especially in the case of lightweight floor constructions.
127
Fig. 5.6.1. Principle of measuring the impact sound pressure level from a floor to a receiving
room (2)
5.7 SINGLE-NUMBER RATING OF SOUND INSULATION
5.7.1 The weighted sound reduction index

The single-number rating of sound insulation is practical for several purposes:
• to characterise the measuring result of a building construction,

• for quick comparison of the sound insulation obtained with different constructions, and
• to specify requirements for sound insulation.
The weighted sound reduction index Rw is based on a standardised reference curve that is
defined in one-third octaves in the frequency range 100 Hz – 3150 Hz. The reference curve is
made from three straight lines with a slope of 9 dB per octave from 100 to 400 Hz, 3 dB per
octave from 400 to 1250 Hz, and 0 dB per octave from 1250 to 3150 Hz.
128
The measured transmission loss is compared to the reference curve, and the sum of
unfavourable deviations is calculated. An unfavourable deviation is the deviation between the
reference curve and the measured curve if the measured sound insulation is lower than the value
of the reference curve.
The reference curve is shifted up or down in steps of 1 dB, and the correct position of the
reference curve is found when the sum of unfavourable deviations is as large as possible, but do
not exceed 32 dB. The value of the reference curve at 500 Hz is taken as the single-number
value of the measuring result. The method is also shown in Fig. 5.7.1.
Transmisson Loss, R, dB
Frequency, Hz
Fig. 5.7.1. Determination of the weighted sound reduction index. M is the measured curve, V1 is
the reference curve in position 52 dB, and V2 is the shifted reference curve. The result is Rw =
60 dB.
5.7.2 The weighted impact sound pressure level

The weighted impact sound pressure level Ln,w is very similar to the weighted sound reduction
index. It is based on a standardised reference curve that is defined in one-third octaves in the
frequency range 100 Hz – 3150 Hz. The reference curve is made from three straight lines with a
slope of 0 dB per octave from 100 to 315 Hz, -3 dB per octave from 315 to 1000 Hz, and -9 dB
per octave from 1000 to 3150 Hz.
The measured impact sound pressure level is compared to the reference curve, and the sum of
unfavourable deviations is calculated. An unfavourable deviation is the deviation between the
129
reference curve and the measured curve if the measured impact sound pressure level is higher
than the value of the reference curve.
The reference curve is shifted up or down in steps of 1 dB, and the correct position of the
reference curve is found when the sum of unfavourable deviations is as large as possible, but do
not exceed 32 dB. The value of the reference curve at 500 Hz is taken as the single-number
value of the measuring result. The method is also shown in Fig. 5.7.2.
Impact Sound Pressure Level, Ln,w, dB
Frequency, Hz
Fig. 5.7.2. Determination of the weighted impact sound pressure level. M is the measured
curve, V1 is the reference curve in position 60 dB, and V2 is the shifted reference curve. The
result is Ln,w = 47 dB.
130
5.8 REQUIREMENTS FOR SOUND INSULATION
The Danish requirements for new buildings are laid down in “Bygningsreglement 1995” (BR-
95) and in “Bygningsreglement for småhuse 1998” (BR-S 98).
For dwellings in multi-storey houses and for hotels the main requirements are:
• The airborne sound insulation shall be R´w ≥ 52 dB in horizontal directions and R´w ≥ 53 dB
in vertical directions.
• The impact sound pressure level shall be Lń,w ≤ 58 dB.
• Between rooms for common service or commercial use and dwellings the airborne sound
insulation shall be R´w ≥ 60 dB and the impact sound pressure level shall be Lń,w ≤ 48 dB.
For row-houses or semi-detached houses the main requirements are:
• The airborne sound insulation shall be R´w ≥ 55 dB.

• The impact sound pressure level shall be Lń,w ≤ 53 dB.
In schools the main requirements are:
• Between classrooms the airborne sound insulation shall be R´w ≥ 48 dB in horizontal

directions and R´w ≥ 51 dB in vertical directions.
• The impact sound pressure level in classrooms shall be Lń,w ≤ 63 dB.
• From rooms for music or workshops to classrooms the airborne sound insulation shall be
R´w ≥ 60 dB and the impact sound pressure level shall be Lń,w ≤ 53 dB.
The sound insulation of facades is not specified directly, but in buildings where then outdoor
traffic noise exceeds LAeq, 24 ≥ 55 dB, the indoor noise in living rooms shall not exceed LAeq, 24 ≤
30 dB.
5.9 REFERENCES
ISO 140-3 (1995): Acoustics. Measurement of sound insulation in buildings and of building
elements. Part 3: Laboratory measurements of airborne sound insulation of building elements.
elements. Part 4: Field measurements of airborne sound insulation between rooms.
elements. Part 6: Laboratory measurements of impact sound insulation of floors.
elements. Part 7: Field measurements of impact sound insulation of floors.
131
ISO 717-1 (1996): Acoustics. Rating of sound insulation in buildings and of building elements.
Part 1: Airborne sound insulation.
ISO 717-2 (1996): Acoustics. Rating of sound insulation in buildings and of building elements.
Part 2: Impact sound insulation.
BR-95. (1995). Bygningsreglement (Building regulations, in Danish). Bygge-og Boligstyrelsen,

Copenhagen.
BR-S 98. (1998). Bygningsreglement for småhuse (Building regulations for small houses, in
Danish). Bygge-og Boligstyrelsen, Copenhagen.
132
6 MECHANICAL VIBRATION AND STRUCTUREBORNE SOUND
Mogens Ohlrich
6.1 INTRODUCTION
Audio frequency vibration of mechanical systems and waves in solid structures form an
integral part of engineering acoustics in describing the dynamic phenomena in solids and
fluids, and their interaction. This subject, referred to as phenomena of structureborne sound
or vibro-acoustics, is important because sound or noise is very often generated directly by
mechanical vibration of solid bodies or by waves transmitted in solid structures, and
eventually radiated into the fluid as audible sound. Examples are musical sound from a string
instrument or noise from a pump in a central heating system.
Vibration of simple resonant systems (resonators) is characterised by mass and stiffness
properties and by some form of damping mechanism, which dissipate vibrational energy. The
simplest description of dynamic behaviour applies to resonators that can be modelled as a
(minimal) combination of discrete or ‘lumped’ elements. If the response of the resonator
primarily occurs in only one direction, ie in a single motion coordinate, then the system is said
to have a single degree of freedom (sdof). Figure 6.1.1 shows examples of sdof-resonators.
The mathematical description of the vibration of such systems is governed by an ordinary
second-order differential equation. This is usually derived from a force balance of the mass
element. Solution of the equation shows that such systems have a single preferred ‘natural’
frequency of vibration, which can exist in the absence of external excitation.
Figure 6.1.1 Examples of single degree of freedom resonators. After ref. [1].
Vibration of more complex systems requires more than one motion coordinate for a
complete description. For example, in the case of a loudspeaker three degrees of freedom are
required for describing the designed translational motion of the ‘piston cone’ and its
unintentional rocking motions, which can occur in two planes. In general such motions will be
governed by three coupled, second-order differential equations. However, by using a special
set of coordinates these equations can be uncoupled and solved independently, as is the case
133
for the sdof-resonator.
Vibration of different phase, ie, structural wave motion, can occur when the wavelength
of vibration in a solid structure is less than one of its typical dimensions. If this is the case it is
natural to threat the system as a continuous one. The response of such a system is governed by
a partial differential equation, because the response depends upon both time and a spatial
position coordinate that specifies the location at which the response is to be determined.
6.1.1 SOURCES OF VIBRATION

There are many types of excitation mechanisms that generate vibration and waves in solid
structures. Such sources are associated with nature or they involve the employment of
machines in the broadest sense, that is, devices that do work, ranging from a miniature loud-
speaker in a hearing aid to a combustion engine of a truck, say. The sources can be classified
by their temporal variations for which there are two types, transient and continuous that
includes time variation of either deterministic (periodic) or random nature.
Examples of sources of vibration are shown in Figure 6.1.2. Transient sources
representing local impact are very common both as a single impact and in repetition, in which
case the excitation time-history becomes periodic. The hammer impact symbolizes a variety
of excitation mechanisms such as musical percussion (drums, xylophones), impulsive sources
of vibration and noise in buildings (foot-falls, door slamming), impacts in production
machinery (punch presses, forge hammers) and periodic impacts in combustion engines
(valves, piston slab). Figure 6.1.2b illustrates force excitation caused by an unbalanced
rotating mass; such excitation is often of a harmonic (pure tone) nature. Other sources of
vibration and noise are random variation of surface roughness, eg in wheel/surface contacts,
or distributed excitation of a structure, eg caused by a sound field.
Figure 6.1.2 Examples of sources that generate vibration and structure-borne sound.
6.1.2 MEASUREMENT QUANTITIES

Investigations of vibration in solid structures are usually carried out by measuring a local
quantity at a specific position on the structure. Distribution of vibration over a larger area can
be determined by measurements in a number of discrete positions. The local measurement
quantity is either a motion (displacement, velocity or acceleration) or a force. Both types of
variables are vectors, and thus assigned to a certain orientation or direction.
Vibratory motion is usually measured uni-directional with a small transducer of the
accelerometer-type that is fastened to the structure’s surface. The accelerometer is based upon
the piezo-electric principle with an output signal proportional to the acceleration a = a(t) of
the vibrating surface. Accelerometers are available with different sensitivities. The velocity v
or displacement ξ of the vibratory motion is obtained by integration of the acceleration signal.
134
A localised (point) force F = F(t) is mostly measured with a piezo-electric force
transducer, which produces an output proportional to the force. The measurement is carried
out by inserting the transducer between a source (eg a vibration exciter) and the measurement
object. This arrangement is mostly used for measuring the dynamic properties of structures,
for example, the impedances or the mobilities.
6.1.3 LINEAR MECHANICAL SYSTEMS

The dynamic properties of a physical system depend upon its mass and stiffness distribution
and damping losses. These properties are attempted described by mathematical models in the
form of one or more differential equations of motion. The system is said to be linear if the
dependent response variables are of first order. When this is the case, one can use the very
important superposition principle. This means that the response contributions from
independent excitations can be superimposed or summed as vectors.
Herein we assume that systems considered are linear, which is often the case when
vibration or waves have small amplitudes. System dynamics can therefore be described by
linear differential equations. These can be based either on a discrete model or on a continuous
model. In the discrete model the properties of system components are described by discrete
(‘lumped’) quantities, represented by ideal masses, massless springs and dampers, see Figure
6.1.3. The physical properties of the continuous model are functions of the spatial coordinates.
Dynamic properties of the system are therefore described by partial differential equations.
Figure 6.1.3 Lumped model of a physical system, where the physical properties are represented by
ideal discrete elements of point masses, massless springs and dampers.
The choice between the two models depends upon a number of factors such as frequency
range of interest, structural shape and forms of excitation. However, the actual decision of the
type of model is usually not strictly scientific, but is often based on intuition and practical
experience. In this note we shall focus mainly on the analysis of discrete models, whereas
only a brief summary will be given of wave motion in continuous structures (structure-borne
sound).
Figure 6.1.4a shows the basic lumped elements; the quantity s represents the spring
constant (stiffness), m is the mass and r is the damping constant of a viscous damper; for
translatory motion these quantities have units of [N/m], [kg] and [kg/s], respectively. The
viscous damper represents a velocity proportional resistance that results in energy losses.
Symbolically, the viscous damping is thought caused by motion of a piston in a fluid-filled
cylinder.
The properties of the elements are independent of time t, and there is a linear relation
between forces Fi = Fi(t) and changes in, respectively, displacement ξ = ξ(t) , velocity v = v(t)
135
and acceleration a = a(t) over the terminals of the elements. Thus, for the ideal spring there is
proportionality between force and deformation according to Hooke’s law. The viscous
damping force is proportional to the velocity of the ‘deformation’ in the massless damper.
Figure 6.1.4 (a) Force-response-relations for ideal lumped elements.(b) Excitation (action) and
reaction by compression of spring.
Note that both motion and force variables are vector quantities, as shown by the example
in Figure 6.1.4b . Both quantities are defined as positive in the direction of the vector; the
motion variables are thus defined as positive in the x-direction. In Figure 6.1.4a, the positive
force F required for accelerating the mass m is therefore F = ma , which is Newton’s second
law of motion in its simplest form.
6.2 SIMPLE MECHANICAL RESONATORS
Figure 6.2.1a shows a model of a single degree of freedom system that is connected to a rigid
foundation. The system consists of a mass m , a spring of spring constant s , and a velocity
proportional viscous damper of damping constant r.
Figure 6.2.1 (a) Viscously damped simple resonator driven by an external force F ; (b) diagram which
shows the forces acting on the mass m .
136
6.2.1 EQUATION OF MOTION FOR SIMPLE RESONATOR
The system is assumed excited by a time-varying external force F = F(t) and it is understood
that the system can vibrate only translatory, to and fro, in the direction of the force, that is, in
the horizontal plane in this example. The motion of the mass from its equilibrium position is
denoted by the displacement ξ = ξ(t) , and this is taken positive towards the right-hand side.
The vibration response caused by the external force is uniquely defined by the
instantaneous value ξ . This displacement of the mass results in a compression of the spring
that produces a restoring, elastic spring force
Fs = − sξ . (6.2.1)
Thus, the reaction on the mass that is caused by the spring force, acts in the opposite direction
of the displacement imposed by the external force. If viscous damping is assumed as
illustrated by the parallel-coupled dashpot in Figure 6.2.1 then this element will exert a
corresponding restoring damping force
dξ
Fr = − r , (6.2.2)
dt
that is, a force which is also directed opposite to that of the motion of the mass and in
proportion to its vibration velocity v = dξ /dt .
The vector sum of forces that act on the mass, that is, F + Fs + Fr = F − sξ − rv , thus
serves to accelerate the mass. So, according to Newton’s second law of motion, this sum must
be equal to the product of mass m and acceleration a = d2ξ/dt2 , ie
d 2ξ
∑ i F = ma = m
dt 2
. (6.2.3)
The equation of motion for the system therefore becomes

d 2ξ dξ
m 2 +r + sξ = F . (6.2.4a)
dt dt
This equation is often written in a reduced form as
d 2ξ r dξ F
2
+ + ω 02ξ = , (6.2.4b)
dt m dt m
where ω0 is the natural angular frequency in [rad/s] of the corresponding undamped system
(r = 0), defined as
s
ω0 = . (6.2.5)
m
In the literature the fraction r/m in eq. (6.2.4b) is often replaced either by 2δ or by 2ζω0
where δ is the damping coefficient and ζ is the non-dimensional viscous damping ratio.
Their definitions are respectively
r r r
δ = and ζ = = . (6.2.6a,b)
2m 2mω 0 2 ms
Moreover, from Figure 6.2.1b it is seen that the total force Ff acting on the rigid
foundation is equal to the sum of the spring force and the damping force, that is,
dξ
Ff = r + sξ . (6.2.7)
dt
137
6.2.2 FORCED HARMONIC RESPONSE OF SIMPLE RESONATOR
Let us assume that the excitation force F in eq. (6.2.4) varies harmonically with time as
F = |F1|cosωt with angular frequency ω. After a certain built-up of vibration the mass will
then also execute stationary, harmonic vibration with the same angular frequency ω. Herein
we shall only deal with the stationary vibration of the system, since it is assumed that the
initial built-up of vibration caused by ‘starting’ the force has completely decayed because of
damping effects, see Figure 6.2.2.
Figure 6.2.2 Time history of vibration built-up in the case of harmonic force excitation of a simple,
damped resonator when ω < ω0 . The vibration built-up response is succeeded by a stationary
vibration at the angular frequency ω of the excitation.
6.2.2.1 Undamped system

Initially, we shall disregard the damping of the considered system by setting r = 0 . Thus for
harmonic excitation the equation of motion (6.2.4) reduces to
d 2ξ F
2
+ ω 02ξ = 1 cos ωt . (6.2.8)
dt m
The complete solution for ξ = ξ(t) of such a differential equation has the well-known form
ξ = ξ1 cos ωt + solutions to the homogeneous equation (6.2.9)

where the first term represents the stationary harmonic vibration and the second term
represents the above-mentioned phenomenon of vibration built-up or decay.
The displacement amplitude ξ1 of the stationary vibration is obtained directly from eq.
(6.2.8) by substituting the assumed solution ξ = ξ1 cosωt :
F
( )
− ω 2 + ω 02 ξ1 cos ωt = 1 cos ωt ,
m
which gives
F 1 F 1 1
ξ1 = 1 = 1 = ξ stat (6.2.10 a,b,c)
m ω0 − ω
2 2
s 1− ω ω0
2 2
1 − ω 2 ω 02
where eq. (6.2.10b) follows from eq. (6.2.5). Furthermore, the quantity ξstat represents the so-
called static displacement, which is the compression or extension of the spring caused by the
force F = |F1|cosωt when ω = 0:
F
ξ stat = 1 . (6.2.11)
s
The stationary part of the solution (6.2.9), which describes the forced harmonic motion of the
138
resonator, is thus given by
1
ξ = ξ1 cos ωt = ξ stat cos ωt . (6.2.12)
1 − ω 2 ω 02
The fraction 1/(1 − ω2/ω02) represents the variation of the vibration amplitude with respect to
the excitation frequency ω and it is sometimes referred to as the response amplification factor;
this quantity also reveals the phase relation between the displacement response and excitation
force. Figure 6.2.3 shows the variation of this quantity ξ1/ξstat with angular frequency; in
Figure 6.2.3b the same quantity is shown as absolute value (modulus) and phase.
From the figure it can be seen that the vibration amplitude grows towards infinity when
the excitation frequency ω approaches the undamped natural frequency ω0 of the system; this
excitation condition is called resonant excitation, and the frequency at which ω = ω0 is the
resonance frequency. At ω = ω0 , the response ξ1 is also seen to undergo a change in sign,
which corresponds to a phase change of π radians. Physically, this simply means that the
quantities ξ1 and |F1| are in-phase at low frequencies, that is, for ω < ω0 where the system
behaves spring-like, whereas they are in anti-phase for ω > ω0 where the response is lagging
the harmonic force excitation by 180 degrees because of the system mass (inertia).
Figure 6.2.3 (a) Relative displacement response ξ1/ξstat for an undamped simple resonator; (b) the
same response function plotted as modulus and phase.
For this undamped case the force Ff that is transmitted to the foundation is caused by
the spring force and is given by Ff = sξ , which follows from eq. (6.2.7) for r= 0. The disturb-
ance force on the foundation thus follows directly by substituting the solution eq. (6.2.12)
1
F f = F f cos ωt = F1 cos ωt . (6.2.13)
1 − ω 2 ω 02
This force ratio Ff /|F1| has the same frequency variation as the motion ratio ξ1/ξstat shown in
Figure 6.2.3. For excitation frequencies below the natural frequency of the system, that is for
ω < ω0 , the mass has a negligible influence. This means that the excitation force is in
equilibrium with the spring force, which is transmitted unchanged to the foundation. Thus, if
the force on the foundation is to be reduced by vibration isolation it is required that natural
frequency of the system is designed in such a way that ω0 << ω/√2 is fulfilled. For a set
139
excitation frequency and system mass this is accomplished by selecting a ‘soft’ spring
element with an appropriately small spring constant s .
6.2.2.2 Viscously damped system

The influence of damping is now being considered. When damping losses are assumed to be
of the viscous type as in Figure 6.2.1 then eq. (6.2.4) applies.
By using complex notation the harmonic excitation force F(t) = |F1|cosωt can be
expressed as F(t) = Re{F1eiωt}, where F1 is the complex amplitude of the force. The solution
of the equation of motion is assumed to be of the same form ξ(t) = Re{ξ1eiωt}, where
ξ1 = |ξ1|eiφ is the complex amplitude of the harmonic displacement with φ being the phase
angle between the displacement response and the driving force. Physical quantities are of
course always real, and it is therefore necessary to take the real part of the mathematical
solution when we want the time variation of the physical motion. This yields
ξ (t ) = Re{ξ1e iωt } = ξ1 cos(ωt + ϕ ) . (6.2.14)

By performing in eq. (6.2.4a) substitutions of F(t) ≡ F1eiωt and ξ(t) ≡ ξ1eiωt result in the
solution for the stationary, harmonic vibration1 :
(− ω )
m + iωr + s ξ1e iωt = F1e iωt
2
(6.2.15)
F1 F1
⇔ ξ1 = = . (6.2.16a, b)
( s − ω m) + iωr
2
m(ω 0 − ω 2 ) + iωr
2
Hereby, the problem is basically solved. (If the time variation of the response is sought then
this is obtained by substitution in eq. (6.2.14).) Furthermore, since the squared modulus is
given by ξ1ξ1* = |ξ1|2 , we get
2
2 F1
ξ1 = . (6.2.16c)
m 2 (ω 02 − ω 2 ) 2 + ω 2 r 2
Thus, |ξ1| is obtained by simply taking the square-root of the expression (6.2.16c).
The force transmitted to the foundation follows similarly from eq. (6.2.7)
F f e iωt = (iωr + s )ξ1e iωt , (6.2.17)
which by substituting eq. (6.2.16b) gives
2
F1 ( s + iωr ) 2 F1 ( s 2 + ω 2 r 2 )
Ff = , and Ff = . (6.2.18a,b)
m(ω 02 − ω 2 ) + iωr m 2 (ω 02 − ω 2 ) 2 + ω 2 r 2
1
) Here the symbol Re{··} is left out. This does not result in any trouble as long as one is strictly dealing
with field quantities (displacement, velocity, force etc). However, when dealing with energy or power quantities,
one must only include the real part of the field quantity. The time variation eiωt is also often left out in the
analyses, but it is of course to be recalled and taken into account when necessary.
140
Solution in sum form. The solution (6.2.16) for the complex displacement can also be written
in terms of its real and imaginary parts
ξ1 = ξ re + iξ im . (6.2.19a)
In the following we shall assume that the arbitrary phase of F1 is set equal to zero by a
suitable choice of time-reference (t = 0); this means that the force amplitude is assumed to be
real, ie F1 = |F1|. Thus, by transforming the denominator in eq. (6.2.16b) to a real quantity this
yields
m(ω 02 − ω 2 ) F1 (−ωr ) F1
ξ1 = 2 2 + i 2 2 . (6.2.19b)
m (ω 0 − ω ) + ω r
2 2 2 2
m (ω 0 − ω 2 ) 2 + ω 2 r 2
The frequency variations of this solution are sketched in Figure 6.2.4a. Shown is the real and
imaginary parts of the displacement response of the viscously damped resonator when this is
driven by a harmonic force of constant amplitude F1 . The damping is seen to limit the
displacement response in the frequency range around ω~ω0 where the response ξ1 is
controlled largely by its imaginary part ξim .
Figure 6.2.4 Frequency variation of displacement ξ1 for a viscously damped simple resonator driven
by a harmonic force of constant amplitude. (a) Real and imaginary parts; (b) Modulus and phase.
Solution in product form. The solution for the complex displacement response eq. (6.2.16) or
(6.2.19) is often written in the alternative ‘product form’
ξ1 = ξ1 e iϕ (6.2.20a)
where the modulus |ξ1| and phase angle φ as usual are determined from eq. (6.2.19):
2
ξ1 = ξ re2 + ξ im2 and tan ϕ = ξ im ξ re .
The squared modulus of the displacement is already given by eq. (6.2.16c), whereas the phase
angle is found directly from eq. (6.2.19b), ie
− ωr
tan φ = . (6.2.20b)
m(ω 02 − ω 2 )
141
Note that the phase angle becomes φ = −π/2 at resonant excitation. As previously, the actual
physical time variation of the vibration response follows from eq. (6.2.14)
ξ (t ) = Re{ξ1e iωt } = Re{ ξ1 e iϕ e iωt } = ξ1 cos(ωt + ϕ )
F1
⇔ ξ (t ) = cos(ωt + ϕ ) . (6.2.20c)
m 2 (ω 02 − ω 2 ) 2 + ω 2 r 2
Figure 6.2.4b shows how the modulus and phase of the displacement varies with frequency
for harmonic force excitation. This type of graph is the most commonly used form of
presentation for frequency response functions.
The vibration velocity v(t) of the resonator is often of interest and this follows simply by
taking the time derivative of the displacement response, eq. (6.2.16) or (6.2.20):
dξ (t )
v(t ) = = − ω ξ1 sin(ωt + ϕ ) ,
dt
or (6.2.21)
⎧d
( ⎫
) { }
v(t ) = Re⎨ ξ1e iωt ⎬ = Re v1e iωt , where v1 = iωξ1 .
⎩ dt ⎭
So, with respect to the complex amplitudes a differentiation is simply archived by a
multiplication with iω ; evidently integration is performed by a division by iω. Moreover, the
acceleration a(t) of the motion is obtained similarly by the time derivative of velocity or by
the second derivative of displacement.
Non-dimensional form. It is often convenient to introduce non-dimensional parameters that

enable solutions for a class of systems to be presented in a general form. For simple
resonators the frequency ratio Ω is readily used as frequency parameter
Ω = ω /ω0 . (6.2.22)
By substituting this as well as the dimensionless viscous damping ratio ζ into eqs. (6.2.16c)
and (6.2.20b) we obtain the general expressions for the displacement ratio |ξ1|/ξstat and for the
phase angle φ :
2
ξ1 1 − 2Ωζ
= and tan ϕ = , (6.2.23a,b)
ξ stat
2
(1 − Ω ) + 4Ω 2ζ 2
2 2
1 - Ω2
here, it is recalled that the static displacement is ξstat = |F1|/s . Amplitude and phase
characteristics for the displacement ratio (6.2.23), are shown logarithmically in Figure 6.2.5a
for different values of damping ratio ζ . It is clearly seen that the damping has a dominant
influence on the response in the frequency range Ω ~ 1, which is close to the natural
frequency of the system.
Similar expressions for the force ratio Ff / |F1| are obtained by substituting the non-
dimensional parameters in eq. (6.2.18). Amplitude and phase characteristics for this ratio
between transmitted force and driving force are shown in Figure 6.2.5b.
142
In forced harmonic vibration the displacement response of the system reaches its
maximum value |ξmax| at, say, Ωr = ωr /ω0 where ωr is the resonance frequency. The actual
value of Ωr is determined by differentiating eq. (6.2.23a) with respect to Ω and by setting
the obtained expression equal to zero. This gives the value
Ω ≡ Ωr = 1 − 2ζ 2 (6.2.24a)
⇔ Ωr ≅ 1 − ζ 2 , when 2ζ 2 << 1 ; (6.2.24b)
in the last approximate expression use have been made of the truncated series:
(1−x)½ ≅ 1 − x/2 provided that x << 1. The maximum displacement thus occurs at an angular
frequency, which is slightly lower than the angular natural frequency of the undamped
system. By substituting eq. (6.2.24a) in (6.2.23a) we get
2
ξ max 1
= . (6.2.25)
ξ stat
2
4ζ (1 − ζ 2 )
2
Figure 6.2.5 Amplitude and phase characteristics for: (a) Displacement ratio ξ1/ξstat , and (b) Force
ratio Ff /|F1| . From ref. [2].
However, when the damping is small (ζ << 0.05) the resonance frequency will nearly
coincide with the natural frequency ω0 of the undamped system, that is, ωr ≅ ω0 ; the
maximum displacement thus becomes
143
1 F
ξ max ≅ ξ stat = 1 . (6.2.26)
2ζ ω0r
The displacement at resonance is thus equal to ξstat divided by 2ζ .
Similarly, the vibration velocity of the system can be shown to take its maximum value
|vmax| at ω = ω0 , that is, at Ω =1 . Since |v| = ω |ξ| this yields
1 F
ξ max ≅ ω 0ξ stat = 1 . (6.2.27)
2ζ r
Relations for maximum acceleration can be derived in the same manner.
Finally, the modulus and phase of the frequency response functions for displacement and
velocity, respectively, are sketched in log-log format in Figure 6.2.6.
Figure 6.2.6 Logarithmic plots of the frequency response functions of a simple resonator represented
as displacement and velocity. A unit force excitation is assumed.
Characteristic properties. As apparent from previous discussions the dynamic properties of

the resonator are predominantly spring-like at low frequencies (Ω << 1) and predominantly
mass-like at high frequencies (Ω >> 1) ; the asymptotes shown in Figure 6.2.6 actually
represent the dynamic properties of the individual elements s, m and r under the action of the
force F1. The dynamic properties of the resonator (ie, the combined system) are therefore
characterised as being:
F1
• Stiffness controlled for Ω << 1 , where ξ1 ≈ ,
s
F1
• Damping controlled at Ω ≅ 1 , where ξ1 ≈ , (6.2.28)
ωr
F1
• Mass controlled for Ω >> 1 , where ξ1 ≈ .
mω 2
These asymptotic values for the displacement response |ξ1| follow directly from eq. (6.2.16c).
Similar relations can be determined for velocity and acceleration.
144
6.2.2.3 Structurally damped systems
So far we have only considered damping of the viscous type. A second type is structural
damping, which is proportional to changes in elastic deformation, like the displacement of a
spring. Such structural damping is therefore appropriately modelled by assigning the inherent
losses to the spring element. For harmonic motion this can be represented by a complex
stiffness s = s(1 + iη) where η is the damping loss factor and s is the real part of the
complex spring constant. The loss factor thus defines the phase lag (hysteresis) between
harmonic driving force and spring displacement. By using the loss factor the equation of
motion for a single mass-spring resonator becomes
d 2ξ
m 2 + s (1 + iη )ξ = F1 e iωt , (6.2.29)
dt
which, similar to eq. (6.2.15), has the solution ξ(t) = Re{ξ1eiωt}, where ξ1 = |ξ1|eiφ is the
complex amplitude:
F1 F1
ξ1 = = . (6.2.30)
( s − ω m ) + i sη
2
m(ω 0 − ω 2 ) + isη
2
This ‘complex stiffness’ approach is very convenient, because the equation of motion
can be formulated initially without regard to damping and finally the spring constant is
replaced by its complex value s = s(1 + iη) .
Now, comparing eq. (6.2.30) with (6.2.15) shows that sη corresponds to ωr . The
equivalent damping ‘constant’ req for a structurally damped spring thus becomes frequency
dependent, and so does the equivalent damping ratio ζeq , ie
req = req (ω ) = sη / ω and ζ eq = ζ eq (ω ) = ηω 0 /(2ω ) . (6.2.31a,b)
Alternatively, the loss factor of a parallel combination of an ideal spring and a viscous damper
of constant r may be expressed as η = rω/s . Note also that the equivalent damping ratio eq.
(6.2.31b) becomes ζeq = η/2 at resonance. This relation may be used as an approximation
for other frequencies that are close to resonance.
6.2.3 FREQUENCY RESPONSE FUNCTIONS

The frequency response of a system is defined as the ratio of complex amplitudes of two
quantities representing the response to a certain excitation. This broad characterisation by the
term ‘frequency response’ is often imprecise because the response quantity can be either
displacement or one of its time derivatives: velocity and acceleration. It is therefore
customary to assign specific names and symbols to the various types of frequency response
functions.
6.2.3.1 Receptance
So far we have been dealing with ratios of response over force. When the system response is
characterised by its displacement the complex frequency response is called the receptance
H(ω) . So, this is defined as
ξ e iωt
H (ω ) = H (ω ) e iϕ (ω ) = 1 iωt , (6.2.32)
F1e
145
where the notation with angular frequency dependence, H(ω), implies that the quantity is a
continuous function of ω ; its amplitude spectrum |H(ω)| and phase spectrum φ(ω) can be
determined from
2 Im{H (ω )}
H (ω ) = H (ω ) H ∗ (ω ) and tan ϕ (ω ) = . (6.2.33)
Re{H (ω )}
The definition eq. (6.2.32) states that ξ1eiωt = H(ω)F1eiωt , which means that the time variation
of the displacement for harmonic excitation is
ξ (t ) = Re{H (ω ) F1 e iωt } = H (ω ) F1 cos(ωt + ϕ (ω ) ) , (6.2.34)
where the force amplitude is assumed to be real.

Receptances of the discrete elements: spring s, damper r and mass m , follow
respectively from the fundamental relations between harmonic force and the associated
motion for such elements
1 1 −1
H s (ω ) = , H r (ω ) = and H m (ω ) = 2 . (6.2.35a,b,c)
s iωr ω m
Since the ideal spring and damper are massless it is assumed in the definition of their
receptances that one of their terminals is blocked and that a harmonic force drives the other,
free end.
It is sometimes useful to use the reciprocal of the receptance function; this is called
dynamic stiffness [3].
6.2.3.2 Mobility and Impedance

The velocity response is often of interest in vibro-acoustics, for instance, because the radiated
sound power from a vibrating structure is proportional to its surface velocity. The complex
ratio between response velocity and driving force is called the mobility Y(ω) (or sometimes
admittance) and is defined as
v e iωt
Y (ω ) = Y (ω ) e iθ (ω ) = 1 iωt , (6.2.36)
F1e
There is, of course, a very simple relation between mobility and receptance since the complex
velocity amplitude is v1 = iωξ1 , ie
Y (ω ) = iωH (ω ) . (6.2.37)
The mobilities of the ideal components are therefore easily determined either from the
fundamental relations or directly from eq. (6.2.35). Thus
iω 1 1
Ys (ω ) = , Yr (ω ) = and Ym (ω ) = . (6.2.38)
s r iωm
The reciprocal of a mobility function is named the impedance Z(ω)
1
Z (ω ) = . (6.2.39)
Y (ω )
146
These different frequency response functions are summarized in Table 6.2.1 together
with corresponding functions that involve acceleration response. The latter is called
accelerance and its reciprocal, the apparent mass. The accelerance is sometimes used because
acceleration is the response quantity that is usually measured directly.
Table 6.2.1 Definition of frequency response functions R/F and F/R , where F is the force and R is
the response that represents either displacement, velocity or acceleration.
---------------------------------------------------------------------------------------------------------
Response Name of frequency response function
quantity
R R/F F/R
---------------------------------------------------------------------------------------------------------
Displacement ξ Receptance H(ω) Dynamic stiffness S(ω)
---------------------------------------------------------------------------------------------------------
Velocity v Mobility Y(ω) Impedance Z(ω)
---------------------------------------------------------------------------------------------------------
Acceleration a Acceleration A(ω) Apparent mass M(ω)
---------------------------------------------------------------------------------------------------------
6.2.4 FORCED VIBRATION CAUSED BY MOTION EXCITATION

Vibratory disturbances like motion excitation is very common and occurs, for example, in
transportation of any kind, in machinery and in certain cases also in buildings. In all these
examples and in vibration isolation of delicate equipment from disturbing environments, the
‘foundation’ has a given motion ξf = ξf (t) as shown in Figure 6.2.7. Thus we want to find the
imposed/generated motion ξ = ξ(t) of the mass.
Figure 6.2.7 Motion excitation of a damped simple resonator.
There are two motion coordinates, but despite of this the system has only one degree of
freedom, because the motion of the system is uniquely described by a so-called generalized
coordinate q = q(t) ; in this case by the motion differences
. . .
q = ξ f −ξ and q = ξ f − ξ . (6.2.40)
.
The quantities q and q describe, respectively, the compression (or elongation) of the spring
and the velocity difference over the damper. Since the total force on the mass in Figure 6.2.7
readily can be written down, is it not necessary to use q explicitly. From eq. (6.2.3) follows
directly
147
. . ..
∑F i = s (ξ f − ξ ) + r (ξ f − ξ ) = m ξ . (6.2.41)
This gives the equation of motion
.. . .
m ξ + r ξ + sξ = r ξ f + sξ f . (6.2.42)
It is seen that there is a clear analogy between this expression and eq. (6.2.4a), if the right-
hand-side of eq. (6.2.42) simply is interpreted as a special ‘forcing function’.
In the case of steady-state harmonic motion excitation ξf eiωt , the solution to eq. (6.2.42)
can be assumed to be ξ ≡ ξ1eiωt ; by substituting these quantities we obtain the solution for the
complex amplitude of the displacement ξ = Re {ξ1eiωt }
s + iωr
ξ1 = ξ f . (6.2.43)
m(ω 0 − ω 2 ) + iωr
2
This expression has the same form as eq. (6.2.18a). In motion excitation the ratio between
displacements is thus identical to the ratio between forces in the case of force excitation
(Figure 6.2.1). The frequency variation of ξ1 /ξf is therefore exactly identical to that of Ff /F1
shown in Figure 6.2.5b.
This finishes the analysis of simple sdof mechanical resonators. A treatment of free
vibration of such systems and an analysis of more complicated multi-degree of freedom
systems is outside the scope of this introductory note on discrete systems. We will therefore
proceed with a brief introduction of continuous structures.
6.3 VIBRATION AND WAVES IN CONTINUOUS SYSTEMS
Distributed solid structures become ‘dynamically elastic’ and exhibit wave-type vibratory
behaviour as the frequency is increased to an extent, where the wavelength become
comparable to, or less than, the physical dimensions of the structure. Although discrete
models can be used for analysing wave motion at the lower frequencies, it becomes expedient
to use wave-type analysis in problems where the wavelength is short. Thus, a brief
introduction will be given to vibration and wave motion in continuous systems. Only systems
of one and two dimensions will be considered here, because most engineering structures have
at least one dimension, which is small in comparison with the relevant structural wavelength
of vibration. In the audible frequency range this is the case for basic engineering components,
such as strings, rods, beams, membranes, plates, shells, pipes etc.
Equations of motion that describe different wave types and vibro-acoustic phenomena
have been formulated for many types of continuous structures [4,5,6]. Usually each wave type
is treated separately, although wave conversion between different types generally occurs at
most structural discontinuities, such as edges, corners and cross sectional changes.
The most important wave types in structures are considered to be (a) longitudinal waves,
(b) shear or torsional waves and (c) bending waves, which are also called flexural waves, see
Figure 6.3.1. In the following an introduction of these waves in plane structures will be given.
148
Figure 6.3.1 Different wave types: (a) Longitudinal wave (the lateral deformations are exaggerated),
(b) Torsional wave and (c) Bending wave. After ref. [7].
6.3.1 LONGITUDINAL WAVES

Longitudinal waves in one-dimensional structures like rods and beams are compression-type
waves that are similar to plane sound waves in a fluid. The local structural deformation in
connection with longitudinal wave motion is primarily in the direction of wave propagation,
although there is also a small lateral deformation normal to the structural surface. However,
this deformation is generally so small that it can be neglected as a radiator of sound to the
surrounding fluid. It should also be mentioned that the impedance of longitudinal waves in
solids generally is very high.
The equation of motion for longitudinal waves in an undamped beam can be written in a
compact form; the longitudinal displacement in the wave motion will be denoted by
u = u(x, t), where x represents its spatial dependence. If we assume purely harmonic
excitation and harmonic wave motion u = u(x)eiωt this reads
L{ u ( x)} − ω 2 m′u ( x) = F ′( x) , (6.3.1)
where L{····} is a differential operator that describes the force gradient in the beam, m' is its
mass per unit length and F '(x) is an external force excitation per unit length. For
longitudinal waves the operator is given by −ES d2/dx2 , where E in [N/m2] is Young’s
modulus of elasticity of the beam material and S is the cross sectional area of the beam.
Two field variables are required for describing the longitudinal wave motion; these are
the already mentioned displacement u = u(x)eiωt – or its time-derivative, the velocity
v = iωu(x)eiωt = v(x)eiωt – and the internal force F = F(x)eiωt associated with the wave
motion. This is given by
∂u
F = − ES . (6.3.2)
∂x
Moreover, the wave speed cl2 of a freely propagation longitudinal wave in the beam is
149
E
cl 2 = , (6.3.3)
ρ
where ρ is the material mass density; index 2 on cl2 indicates that the structure has two
surfaces that are small compared with the wavelength of the motion. The corresponding wave
speed in a flat, homogenous plate is slightly higher (by about 5%):
E
cl1 = , (6.3.4)
ρ ( 1 −ν 2 )
where ν is Poisson’s ratio, which is a material constant that expresses the ratio between
deformations in the lateral and length-wise directions of the structure. For common solid
material ν ≈ 0.3 , and for rubber-like materials ν ≈ 0.5 .
A listing of material properties and wave speeds are given in Table 6.3.1. Note that the
wave speed in metals is about 3000 to 5000 m/s, that is, a magnitude higher than for sound in
air. Furthermore, the mass density for metals is seen to be up to 7000 times higher than for
air. This means that the characteristic impedance (ρcl) for compression waves in solid
structures is much higher than for air; for example, the characteristic impedance for steel is
105 times higher than in air, but only 27 times higher than the impedance in water.
Table 6.3.1 Material properties and wave speeds (phase speeds) for solid structures. After ref. [8].
6.3.2 SHEAR WAVES

In this wave type only shear deformations occur, but no volume changes. Moreover, the
direction of the ‘particle’ motion is perpendicular to the direction of propagation. Shear waves
are of importance in plates that are built-up of several layers of material with different
properties, eg sandwich honeycomb panels.
150
The equation of motion for shear waves is governed by a second order partial differential
equation [5] of a general form similar to that of longitudinal waves; the details shall not be
given here, though. The wave speed cs for shear waves in a plate is given by
G E
cs = = , (6.3.5)
ρ ρ 2(1 + ν )
where G is the shear modulus of the material. From the right-hand-side of this equation it is
clear that there is a unique relation between Young’s modulus E and the shear modulus G, ie
E
G = . (6.3.6)
2(1 + ν )
Shear waves in rods are called torsional waves. This type of wave motion that involves
twisting of the cross section of the rod was shown in Figure 6.3.1b. If the rod has a circular
cross section then the wave speed is as given by eq. (6.3.5); otherwise the wave speed will be
lower.
The two wave types discussed so far have high characteristic impedances. These waves
may therefore be important for the wave transmission over large distances (eg in buildings
and ships) and in wave conversion to bending waves, which is the dominant wave type when
it comes to sound radiation to the surrounding fluid media, being air or water.
6.3.3 BENDING WAVES

Bending waves in beams and plates are characterised by the motion being perpendicular to
both the direction of propagation, and the surface of the structure, see Figure 6.3.1c. Bending
waves do therefore play a dominant role in sound radiation from structures. The reasons for
this are that the wave motion has a good ‘match’ to the adjacent air, and that bending waves
are easily generated, because of their low characteristic impedance.
The equation of motion for bending waves in an undamped beam can be written in the
previous compact form, but with the transverse displacement of the bending wave motion
being denoted by w=w(x, t). If we again assume purely harmonic excitation and harmonic
wave motion w=w(x)eiωt , we get
L{w( x) } − ω 2 m′ w( x) = F ′( x) , (6.3.7)
where the differential operator L{···} that describes the shear force gradient in the beam now
takes the form B d4/dx4 . Here, B is the bending stiffness of the beam, m' is its mass per unit
length and F'(x) is an external force excitation per unit length. The operator is of fourth order,
and four field variables are thus required for describing the bending wave motion. There are
two motion variables, the transverse displacement w = w(x)eiωt and the angular displacement
β = β(x)eiωt , which is the first spatial derivative of w , ie dw /dx . Two force variables are
associated with the wave motion, the internal shear force Fy = Fy(x)eiωt and the internal
bending moment Mz = Mz(x)eiωt ; these are given by
∂3w ∂2w
Fy = B 3 and Mz = − B 2 . (6.3.8)
∂x ∂x
Moreover, the wave speed cb of a freely propagation bending wave in the beam is
151
1
⎛B⎞
4
cb = ω ⎜ ⎟ ,
1
2
(6.3.9)
⎝ m′ ⎠
which is seen to depend upon frequency; this special phenomenon is called dispersion. Such
dependence results in complicated sound radiation properties for plates and built-up
structures. The wave speed or phase speed is furthermore noticed to depend upon the bending
stiffness and the mass per unit length.
The phase speed of bending waves in a thin homogeneous beam with a rectangular
cross-section and of thickness h in the direction of the motion, is given by
cb ≅ 1.8 cl 2 h f , (6.3.10)
where f is the frequency (in Hz) and cl2 is given by eq. (6.3.3).
Moreover, the phase speed in a thin homogeneous plate of thickness h is given by
cb ≅ 1.8 cl1 h f , (6.3.11)
where cl1 is given by eq. (6.3.4).
6.3.4 INPUT MOBILITY OF INFINITE SYSTEMS

Finally, in this brief introduction it is appropriate to list some input mobilities for point force
excitation. Or more specifically, input mobilities relating translational velocity v eiωt to
translational force F eiωt , both at the same point and in the same coordinate (direction). The
corresponding point impedances are the reciprocal of the given point mobilities.
6.3.4.1 Beam or rod

Longitudinal vibration. In the case of a semi-infinite (s∞) beam driven axially at the end, the
input mobility is
1
Ys ∞ = . (6.3.12)
m′ cl 2
where m' is mass per unit length and cl2 is given by eq. (6.3.3).
Bending vibration. The input mobility of a semi-infinite beam driven at the end is
1− i
Ys ∞ = . (6.3.13)
m′ cb
where m' is mass per unit length and cb is given by eq. (6.3.9), or by eq. (6.3.10), provided
that the beam is of rectangular cross-section and is vibrating in the direction in which the
beam thickness h is measured.
The input mobility of an infinite beam driven in the ‘middle’ is given by
1− i
Y∞ = . (6.3.14)
4m′ cb
Note that this is four times lower than the input mobility of the semi-infinite beam, eq.
(6.3.13).
6.3.4.2 Plate
Bending vibration. The input mobility of a semi-infinite plate driven normal to its surface and
at the end (edge) is
152
1
Ys ∞ = , (6.3.15)
3.5 B′ m′′
where m'' is the mass per unit area, and for a homogeneous plate of thickness h the bending
stiffness B' is
E h3
B′ = . (6.3.16)
12(1 − v 2 )
It is noted that this input mobility, eq. (6.3.15), is purely real, provided that the plate is
undamped as is assumed here.
The input mobility of an infinite plate driven in the ‘middle’ is also real and is given by:
1
Y∞ = . (6.3.17)
8 B′ m′′
Other point mobilities relating angular velocity to moment excitation, as well as cross
mobilities, are given in ref. [5].
6.4 VIBRATION ISOLATION AND POWER TRANSMISSION
Vibration isolation is one of the most effective ways of reducing the transmission of audio
frequency vibration from a disturbing source (machine, apparatus, etc) to a connected
receiver structure. This is generally accomplished by ‘disconnecting’ the transmission paths
between the two systems. In practice vibration isolation is done by inserting resilient
mechanical connections or rubber elements that are much more compliant (ie, dynamically
soft), than both the source structure and the receiving structure. Such vibration isolators have
spring-like properties and are often made of vulcanised rubber elements, metal springs or
combinations thereof. The isolation principle is depicted in Figure 4.1a, and Figure 4.1b
shows an example of measured mobilities of a rubber isolator, engine source and elastic
receiver.
Figure 6.4.1 (a) Vibration isolated diesel engine on elastic ship foundation; (b) Mobilities of isolator,
engine and ship foundation. From ref. [9].
The principle of vibration isolation has already been described in Chapter 6.2. Thus, in
the case of a harmonically driven simple source of mass m resting on a spring s attached to
153
an idealised rigid foundation, it was found that vibration isolation is achieved when the
angular natural frequency ω0 of the system is somewhat lower than the frequency component
ω of the excitation force.
6.4.1 ESTIMATION OF SPRING STIFFNESS AND NATURAL FREQUENCY

It is often easy2 to determine the important quantities (m, s and ω 0 = s / m ) for uncritical
arrangements of simple machinery sources that are mounted on vibration isolators (springs).
Usually the mass m of the machine is known. For a vertically loaded spring the static force
F0 = mg from the mass results in a static deflection (compression) of the spring of magnitude
ξ0 = F0/s . These two relations enable the determination of the static stiffness of the spring, ie
mg
s = , (6.4.1)
ξ0
2
where g (= 9.81m/s ) is the gravitational acceleration. The designed natural frequency of the
system can therefore be determined by a very simple formula:
s g ω 0.5
ω0 = = [rad/s] , ⇔ f0 = 0 ≅ [Hz] . (6.4.2)
m ξ0 2π ξ0
If the spring element is slender and rod-like with a cross sectional area S , length
(height) d and made from a material of Young’s modulus E , then the static spring constant
s can be calculated from
s = ES/d . (6.4.3)
Note that the dynamic stiffness of rubber-like material generally differs from this value of
static spring constant or stiffness s. This will be treated in more details in Section 6.4.4.
Figure 6.4.2 Static deflection of spring, which in the unloaded condition has the length d .
It was mentioned previously that the vibration isolation can be improved by reducing ω0,
that is to say, by increasing the static deflection ξ0 . This can be accomplished by reducing s,
but this results in a more laterally unstable arrangement. As a compromise for a number of
practical source cases it is therefore often ‘common’ to choose values in the approximate
range of 0.004 m < ξ0 < 0.01 m, which corresponds to 8 > f0 > 5 Hz.
2
) It should be recalled, however, that the simple oscillator model is a coarse simplification of the
reality, where an extended rigid body on springs will have six degrees of freedom and thus six natural
frequencies, eg see ref. [2].
154
6.4.2 TRANSMISSION OF POWER IN RIGIDLY COUPLED SYSTEMS
In contrast to the idealised model of a simple source on a rigid foundation we shall now
examine the more realistic case of source and foundation or receiving structure of finite
mobilities or impedances. It is reasonable to expect that the dynamic properties of the source
and receiver will effect the vibration isolation that is achievable in practice.
For reference we shall initially address the situation where the vibration source is rigidly
connected to the receiving structure, and it is assumed that source and receiver are connected
via a single motion coordinate (or terminal). First consider the source in a free uncoupled state
in which the vibration activity of the source can be characterised by its free terminal velocity
vfree and its ability to transmit power by its terminal mobility YS , see Figure 6.4.3a . These
source quantities are suitably combined into a single descriptor [10] called the terminal source
strength |Jterm| :
v 2free
J term = , (6.4.4)
YS
where v 2free is the time-average mean-square value of the free velocity vfree= vfree(t). This
source strength |Jterm|, with units of power [W], is useful when comparing different vibratory
sources.
Figure 6.4.3 Systems with a single coupling coordinate: (a) Free vibration source, (b) Source coupled
rigidly to receiving structure, (c) Reaction forces on systems.
In the analysis that follows we assume harmonic vibration vfree ≡ vfree eiωt . The source is
now being connected to a receiving structure, which is characterised by the input mobility YR .
This loading of the source causes the free velocity to change to vR , because of the force
reaction (−F) on the source, ie
vR = v free − YS F . (6.4.5)
Since per definition vR = YRF, we find directly for the rigid coupled system:
F = (YS + YR ) v free vR = YR (YS + YR ) v free .
−1 −1
and (6.4.6a,b)
155
The force and velocity at the coupling point have hereby been determined for this case of
rigid coupling.
The power that is transmitted to the receiving structure is given by the well-known
relations:
P = 1
2 {
Re FvR∗ }= 1
2 F Re{ YR } =
2
1
2 vR Re{ Z R }.
2
(6.4.7)
By substituting the expressions from eq. (6.4.6) herein yields
2 Re{ YR } 2 Re{ YR }
P = 1
2 v free 2
= 1
2 vR 2
. (6.4.8)
YS + YR YR
For further evaluation of the transmitted power this can be written in a convenient
alternative form. Introducing the terminal source strength |Jterm | , eq. (6.4.4), and a power
coupling factor CP yields
P = J term C P , (6.4.9)
where
YS YR cos ϕ R
CP = cos ϕ R = , (6.4.10)
YS + YR
2
YS YR + YR YS + 2 cos θ
and φR is the phase angle of the receiver mobility and θ = φR − φS is the phase difference
between receiver and source mobilities. This takes values in the interval: 0 ≤ |θ| ≤ π . The
power coupling factor is noted to be symmetric with respect to the logarithm of the mobility
ratio |YR |/|YS | . For further details see ref. [10, 11].
6.4.3 VIBRATION ISOLATED SOURCE

The effect of a vibration isolator is now considered. The source is connected to the receiver
via a vibration isolator as schematically shown in Figure 6.4.4a. For simplicity it is assumed
that the isolator can be modelled as an ideal spring with a spring constant s . Thus, because
the spring is assumed massless, this implies that the force on the left-hand-side of the spring
Figure 6.4.4 (a) Block diagram of vibration isolation of a source with a single coupling coordinate.
(b) Diagram that shows the forces on the system elements.
156
is identical to the force on the receiver, F1 = FR . The velocities are different, of course, and
similar to before given by
v1 = v free − YS F1 and v′R = YR F1 . (6.4.11a,b)
The force and the velocities are related according to Hooke’s law as
⎛v v′ ⎞
F1 = s ⎜ 1 − R ⎟ , (6.4.12)
⎝ iω iω ⎠
which, together with eq. (6.4.11) give
F1 = (iω s + YS + YR ) v free v′R = YR (iω s + YS + YR ) v free . (6.4.13a,b)

−1 −1
and
By substituting F1 into the general relation, eq. (6.4.7b), gives the transmitted power to the
receiver
2 Re{ YR }
P′ = 12 v free 2
. (6.4.14)
iω s + YS + YR
So, this gives
P′ = J term C P′ , (6.4.15)
where
YS YR
C P′ = 2
cos ϕ R . (6.4.16)
iω s + YS + YR
These results for the vibration-isolated source have to be compared with those for the
rigid coupled case in order to realistically evaluate the influence of the vibration isolator. This
influence is most suitably described by the effectiveness Eiso = Eiso(ω) of the vibration
isolator, also called its insertion loss. This is defined as the ratio between the squared
magnitudes of the receiver velocities before and after the installation of the vibration isolator -
or for that matter - as the ratio of the corresponding injected powers. Eqs. (6.4.6b) and
(6.4.13b) thus give
2 2
vR iω / s
Eiso = = 1+ . (6.4.17)
v′R
2
YS + YR
From this equation it is evident that a high effectiveness (ie, large number) requires that
the isolator mobility iω/s ≡ YI is much higher (ie, much more mobile or compliant) than the
sum of the source and receiver mobilities, that is,
YI = iω/s >> YS + YR . (6.4.18)
Such a large value of inequality is not easily accomplished over the broad audible frequency
range, because lightly damped resonance in elastic source and receiving structures will occur
and limit the effectiveness of the isolator. Furthermore, at high frequencies the mass of the
isolator can no longer be ignored and resonance occur in the isolator itself, which also limit
the effectiveness. In the case of a symmetric vibration isolator, such modal behaviour can be
accounted for in a prediction by replacing iω/s in eq. (6.4.17) with the actual mobility of the
isolator YI , see also ref. [12].
157
At first, the definition of the isolator effectiveness in eq. (6.4.17) does not seem to apply
to the ideal case of a rigid (immoveable) foundation that was assumed in Chapter 6.2.
However, this is not so, because Eiso might as well be defined as the ratios of forces acting on
the receiver, whether this is moving or not. This follows from the fact that velocities and
forces are related via the receiver mobility. So, for the general elastic receiver the
effectiveness also reads Eiso = |FR|2 / |FR' |2 , where the dash refers to the case with the source
resiliently connected to the receiver. Hence, by substituting the derived expressions for the
corresponding forces, eq. (6.4.6a) and (6.4.13a), respectively, we obtain exactly eq. (6.4.17).
Example 6.4.1 The isolation effectiveness Eiso is to be determined for a harmonically driven mass-spring
resonator, which is connected to a rigid foundation, similar to the systems in Figure 6.2.1 or 6.4.2. The
undamped natural frequency of the resonator is ω 0 = s / m , where m is its mass and s is the spring stiffness. It
is here assumed that the system is structurally damped and that this is accounted for by taken the spring stiffness
to be complex s = s(1+iη).
The source, being the mass m , has the mobility YS = (iωm)–1 and the mobility of the receiver in the form
of a rigid foundation is YR = 0 . Substituting these into eq. (6.4.15) gives
2 2 2 2
m ⎛ω ⎞ ⎛ω ⎞
Eiso = 1 − ω 2
≅ 1 − ⎜⎜ ⎟⎟ + iη ⎜⎜ ⎟⎟ ; (6.4.19)
s (1 + iη ) ⎝ ω0 ⎠ ⎝ ω0 ⎠
in the last approximation it is assumed that η << 1, so that 1 + η2 ≈ 1. By comparison it is seen that eq.(6.4.19)
is equal to the reciprocal of the results for |Ff|2 / |F1|2 in Figure 6.2.5b. This can also be deduced from eq.
(6.2.18), if the damping constant r is replaced by the equivalent constant req for a structurally damped spring
req = sη/ω .
Figure 6.4.5 shows an example of measured and predicted values of the isolation
effectiveness for a complicated vibration source (the diesel engine in Figure 6.4.1a), which is
resiliently mounted on an elastic foundation. The source is mounted on ten multi-directional
isolators; note that these isolators have a much higher mobility than the isolator example
shown in Figure 6.4.1b. The effectiveness is seen to be rather good, about 25 dB on average.
Also shown are two course estimations based upon, respectively, a simple mass-spring-mass
model (LF-prediction of resemblance to eq. (6.4.19)), and a simple mono-coupled model,
where measured isolator mobility and average point mobilities of source and receiver have
Figure 6.4.5 Effectiveness of vibration isolation 10 log Eiso of a multi-coupled machinery source on
an elastic receiving structure.
158
been used in eq. (6.4.17). Despite of the coarse simplifications in these models, a reasonable
agreement with measurement is found in the frequency range up to 800 Hz.
Another example of predicted isolation effectiveness is shown in Figure 6.4.6. Here, a
105 m tall building structure is mounted on large, flat rubber pads that allow thermal
expansion or contraction of the huge building. Calculations were carried out in order to
estimate their isolation effectiveness against structureborne sound transmission from
disturbing underground rail traffic. It is apparent from Figure 6.4.6 that these thermal
expansion devises are not very useful as vibration isolators; their static deformation is simply
too small – in other words – the stiffness of the isolators is too high. At the fundamental
natural frequency of the system vibration amplification is observed and in the frequency range
above 90 Hz the effectiveness is seen to become very small at certain frequencies. These
correspond to the natural frequencies of the foundation columns (≈ ‘source’), on which the
rubber pads and building structure rest.
Figure 6.4.6 Isolation effectiveness of rubber expansion devises that support a tall building.
6.4.4 DESIGN CONSIDERATIONS FOR RESILIENT ELEMENTS

It was mentioned in Section 6.4.1 that the dynamic stiffness of rubber-like material generally
differs from the static spring stiffness s determined by static measurement. When such
isolators are used it is therefore necessary to insert the dynamic stiffness value sdyn instead of
s in the equation for the natural frequency, eg eq. (6.4.2a).
6.4.4.1 Rubber-like materials

The dynamic stiffness of rubber isolators depends upon a number parameters. An important
parameter is the rubber hardness, which is usually characterised in °Shore A of hardness. The
typical hardness-range of commercial rubber isolators is from about 40°Shore A (for soft
isolators) to 80°Shore A , which is rather hard. Table 6.4.1 presents a coarse guide that shows
approximate, empirical values for the relation between rubber hardness, static Young’s
modulus E and dynamic Young’ modulus Edyn , or more specifically their ratio Edyn /E .
Thus, for a slender rubber isolator (of static stiffness given by eq. (6.4.3)), the
appropriate dynamic stiffness sdyn becomes
sdyn = Edyn S/d . (6.4.20)
159
However, this is generally not the final estimate, because the stiffness of rubber isolators also
depends upon another important parameter, which is basically the compactness of the isolator.
Generally, the stiffness of a short rubber block is found to be much higher than the stiffness of
a long slender sample. (Note, that this effect of course is accounted for when estimations are
based on a static load-deflection test, ie on eq. (6.4.1).)
Table 6.4.1 Approximate values for the relation between rubber hardness, static Young’s modulus E
and dynamic Young’s modulus Edyn. The results apply to natural rubber.
---------------------------------------------------------------------------------------------------------
Rubber hardness Static Young’s modulus E Ratio: Edyn /E
°Shore A 106 N/m2 --
---------------------------------------------------------------------------------------------------------
40 1.5 1.2
---------------------------------------------------------------------------------------------------------
50 2.5 1.4
---------------------------------------------------------------------------------------------------------
60 4.0 1.8
---------------------------------------------------------------------------------------------------------
70 6.0 2.2
---------------------------------------------------------------------------------------------------------
The stiffness expressed by eq. (6.4.18) therefore has to be corrected for the ‘bulkiness’ of
the rubber isolator. This can be characterised by an area ratio (or shape factor) RS =Sconst /Sfree ,
in which the area Sconstr represents the total constrained or loaded area of the isolator, and
Sfree is the total free surface area of the isolator. Figure 6.4.7 shows the stiffness correction
factor Cs to be used for a given area ratio RS . Thus, sdyn is to be multiplied with Cs to give
the actual, corrected dynamic stiffness.
Figure 6.4.7 Stiffness correction Cs to be used as a function of the area ratio RS of the vibration
isolator. After ref. [13].
6.4.4.2 Metal and other elastic solids

As oppose to the rubber-like materials, the static and dynamic elastic properties for most
engineering materials are found to be practical identical. For a given elastic material this
means that its Young’s modulus E ≅ Edyn and its shear modulus G ≅ Gdyn . Furthermore,
160
since ν ≈ 0.3 for most solid materials, we have E ≈ 3G .
Resilient elements of metal may take many different forms. Usually they are extended,
continuous components with distributed mass and stiffness, and basically they are designed to
achieve a specified small stiffness at low frequencies. However, at mid and high frequencies
such a resilient element can support different wave types, and resonances will occur in the
resilient element because it is of finite size. This will diminish the isolator effectiveness,
unless damping and/or rubber elements are incorporated into the final design of the resilient
element.
The most common resilient element of metal is probably the helical spring, which is
often made of harden steel. The static and low frequency stiffness in the axial direction of the
spring is
G d4
s= , (6.4.21)
8 n D3
where G is the shear modulus of the material, D is the average diameter of the spring, d is
the diameter of the coil and n is the number of coils or windings.
Other types of resilient elements are leaf springs, which may be thin metal beams or
plates. One example is a so-called cantilever beam, which is rigidly built-in at the receiver-
end and is completely free at the other end, where it supports the source to be isolated. For a
beam with constant thickness h and constant rectangular cross-section S the spring stiffness
is
E S h2
s = , (6.4.22)
4 L3
in which E is Young’s modulus and L is the length of the beam. However, usually the
source will be bolted to the beam and this will hinder angular motion at its ‘free’ end. Thereby
the spring stiffness of the resilient element will increase by a factor of four, to become
s = E S h2/L3 . This clearly illustrates the importance of the boundary conditions at mounting
positions.
161
6. 5 REFERENCES
1. Resonators by M. Heckl, Chapter 20 in ‘Modern methods in analytical acoustics: Lecture

Notes’ (ed. D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl and F. G.
Leppington), Springer-Verlag, London Ltd. 1994.
2. C. M. Harris and C. E. Crede: Shock and vibration handbook, 2nd ed. McGraw-Hill, New
York 1976.
3. D. J. Mead: Passive vibration control. John Wiley & Sons, Chichester 1999.
4. E. Skudrzyk: Simple and complex vibratory systems. Penn State University Press 1968.
5. L. Cremer , M. Heckl and E. E. Ungar: Structure-borne sound, 2nd ed., Springer Verlag,
Berlin 1988.
6. L. Cremer und M. Heckl: Körperschall, 2nd ed. Springer Verlag, Berlin 1996.
7. Vibration of one- and two-dimensional continuous systems by M. Heckl, Chapter 64 in

‘Encyclopedia of acoustics’ (ed. M.J. Crocker), John Wiley & Sons, Ney York 1997.
8. F. J. Fahy: Sound and structural vibration. Academic Press, London 1985.
9. M. Ohlrich and F. Jacobsen: Isolation of structural vibration from machinery.

Proceedings of Nordic Acoustical Meeting, NAM-82, Stockholm, 1982, pp. 309-312.
10. M. Ohlrich: Vibrational source strength as a prerequisite for response prediction by SEA.
NOVEM 2000, Proceedings of Intern. Conf. on Noise & Vibration Pre-design and
Characterisation using Energy Methods, Lyon, 2000, on CD-ROM, pp.12.
11. J. M. Mondot and B. Petersson: Characterisation of structure-borne sound sources: The

source descriptor and the coupling function. Journal of Sound and Vibration 114, 1987,
pp. 507-518.
12. E. E. Ungar and C. W. Dietrich: High-frequency vibration isolation. Journal of Sound

and Vibration 4(2), 1966, pp. 224-241.
13. VDI 2062 Blatt 2 Vibration isolation: Resilient elements (In German), 1976.
162
LIST OF SYMBOLS
a radius of sphere [m]; acceleration [m/s2]

A equivalent absorption area [m2]; accelerance [m/Ns2]
A0 reference area [m2]
B bending stiffness per unit length [Nm]; bending stiffness [Nm2]
B´ bending stiffness per unit width [Nm]
c speed of sound [m/s]
cb speed of bending waves [m/s]
cL speed of longitudinal waves [m/s]
CP power coupling factor [dimensionless]
d length [m]
D directivity [dimensionless]
DI directivity index [dB]
E total acoustic energy [J]; Young’s modulus of elasticity [N/ m2]
Eiso vibration isolation effectiveness; insertion loss [dimensionless]
f frequency [Hz]
f0 resonance frequency [Hz]
fc critical frequency [Hz]
F force [N]
G shear modulus [N/m2]
h distance [m]; plate thickness [m]
H receptance [m/N]
H1 Struve function
I sound intensity [W/m2]
Iref reference sound intensity [W/m2]
Ix component of sound intensity [W/m2]
Jm Bessel fuction
|Jterm| terminal source strength [W]
k wavenumber [m-1]
K stiffness constant [N/m]
Ks adiabatic bulk modulus [N/m2]
l length [m]
lm mean free path [m]
L loudness level [phone]; total length of edges [m]; length [m]
LA A-weighted sound pressure level [dB re pref]
LAeq equivalent A-weighted sound pressure level [dB re pref]
LAE sound exposure level [dB re pref]
LC C-weighted sound pressure level [dB re pref]
Leq equivalent sound pressure level [dB re pref]
LI sound intensity level [dB re Iref]
LLin sound pressure level measured without frequency weighting [dB re pref]
Ln impact sound pressure level [dB re pref]
Lp sound pressure level [dB re pref]
LW sound power level [dB re Pref]
m air attenuation factor [m-1]; mass [kg]; mass per unit area [kg/m2]
m´ mass per unit length [kg/m]
163
m´´ mass per unit area [kg/m2]
M mass [kg]
n natural number [dimensionless]
N loudness [sone]; number of modes [dimensionless]
p sound pressure [Pa]
pA(t) instantaneous A-weighted sound pressure [Pa]
pref reference sound pressure [Pa]
prms rms value of sound pressure [Pa]
p0 static pressure [Pa]
P power [W]
Pa sound power [W]
Pref reference sound power [W]
q volume velocity associated with a fictive surface [m3/s]; generalised coordinate [m]
Q volume velocity of source [m3/s]; directivity factor [dimensionless]
r radial distance in spherical coordinate system [m]; damping constant of viscous
damper [kg/s]
rrev reverberation distance in a room [m]
R gas constant [m2s-2K-1]; reflection factor [dimensionless]; transmission loss [dB]
R0 transmission loss at normal incidence [dB]
s standing wave ratio [dimensionless]; spring constant [N/m]
S surface area [m2]; cross sectional area [m2]
t time [s]
T absolute temperature [K]; averaging time [s]
T60 reverberation time [s]
u longitudinal displacement [m]
u particle velocity [m/s]
ux component of the particle velocity [m/s]
U velocity [m/s]
v velocity [m/s]
V volume [m3]
w transverse displacement [m]
wkin kinetic energy density [J/m3]
wpot potential energy density [J/m3]
x, y, z Cartesian coordinates [m]
Za acoustic impedance [kg m-4s-1]
Za, r acoustic radiation impedance [kg m-4s-1]
Zm mechanical impedance [kg/s]
Zm, r mechanical radiation impedance [kg/s]
Zw separation impedance [kg m-2s-1]
Y mobility (mechanical admittance) [s/kg]
α absorption coefficient [dimensionless]

αm mean absorption coefficient [dimensionless]
β angular displacement [radian]
γ ratio of specific heats [dimensionless]
δ damping coefficient [s-1]; end correction [m]
∆L insertion loss [dB]
164
∆V volume displacement [m3]
ζ viscous damping ratio [dimensionless]
η loss factor [dimensionless]
θ polar angle in spherical coordinate system [dimensionless]
λ wavelength [m]
ν Poisson’s ratio [dimensionless]
ξ displacement [m]
ρ density [kgm-3]
τ time constant [s]; transmission coefficient [dimensionless]
φ phase angle [radian]; azimuth angle in spherical coordinate system [radian]
ω angular frequency [radian/s]
Ω frequency ratio [dimensionless]
^ indicates complex representation of a harmonic variable
165
166
INDEX
Absorption area, 87, 97, 110 Bending stiffness, 121

Absorption coefficient, 29, 35, 87, 103 of beam, 151
Absorption, 29, 35 Bending waves on structures, 1, 2
Accelerance, 147 see also Wave types
Acceleration, 27, 53, 134, 137, 142, 147 Bessel function, 43
Accelerometer, 134 Boundary conditions, 4
Acoustic filters, 28
Acoustic impedance, 28 Cancellation of sound, 9
Acoustic properties of materials, 29, 35 Cartesian coordinate system, 3
Acoustic two-port, 29 Cavity, sound field in, 3, 30, 106
Adiabatic bulk modulus, 3 C-filter
Adiabatic process, 2, 3 see C-weighting
Admittance, 27 Characteristic impedance
A-filter see Impedance
see A-weighting Cochlea, 57
AI principle, 76 Coincidence, 121
Air attenuation factor, 93, 98 Combinations of monopoles, 38, 40
Amplitude, 5, 6, 15, 52 Complex amplitude, 6, 52
Analogous electrical circuit, 29 Complex exponential representation, 52
Angular displacement, 151 Complex stiffness, 145
Angular frequency, 5 Compliance
Antinode see Stiffness
see Node Condenser microphone, 21
Antiphase, 9, 40 Conservation of mass, 2
see also Quadrature Conservation of sound energy, 32
Aperture, 115 Consonant (intelligibility) 111
Apparent mass, 147 Constant percentage filters, 15
Apparent sound transmission loss, 114, 127 Constructive interference, 9, 20
Audible frequency range, 5 Continuous structure, 148
Audiogram, 62 Converging waves, 13
Averaging time Crest factor, 26
see Integration time Critical band, 66, 71, 72
A-weighting, 24, 69, 70 Critical frequency, 121
Axial modes, 82 Cross-over frequency, 119, 124
C-weighting, 24, 69, 70
Background noise, correction for, 18, 20
Backward masking, 67 Damping coefficient, 137
Baffle, effect of, 38, 42 Damping constant, 135
Bandpass filters, 15 Damping loss factor, 145
Bark, 72 Damping force, 137
Basilar membrane, 58, 59, 66, 71 Danish Building Law, 109
Beam, 148 Danish Working Environment Agency, 109
Bel, 18 Dantale, 75
Bending moment, 151 dB HL, 62, 63
167
Decade, 16 Equivalent rectangular bandwidth, 72
Decay curve, 100 Equivalent sound pressure level, 24
Decibels, 18 Equivalent viscous damping ratio, 145
Density of the medium, 2, 3 ERB, 72
Destructive interference, 9, 20 Euler’s equation of motion, 4, 6
Detection of a pure tone in noise, 17 Excursion of a loudspeaker membrane, 48
Deterministic signal, 17 Exponential averaging
D-filter see Time averaging
see D-weighting Eyring’s formula, 93, 99
Diatonic scale, 16
Differentiation with respect to time, Far field, 90
6, 53, 142, 149 Far field approximation, 4, 14, 34, 42
Diffraction, 2 FFT analysers, 15, 17
Diffuse sound field, 85, 98, 103 Field variables, 149
Dipole, 39 Filter, 15
Dipole strength, 41 Filter bank analysers, 15
Direct field, 90 Flanking transmission, 114, 127
Directivity, 43 Flanking transmission loss, 127
Directivity factor, 48, 91 Flexural waves
Directivity index, 49 see Wave types
Dispersion, 2, 152 Fluctuating noise, 24
Displacement, 53, 134, 138, 147 see also Intermittent noise
Displacement ratio, 142 Focusing, 96
Displacement response, 139, 141, 143 Force, 27, 135, 138, 140
Diverging waves, 13 Force transducer, 135
Double construction, 123 Formant, 74
D-weighting, 23, 70 Forward masking, 64, 67
Dwellings (reverberation control) 110 Free field, 61, 63
Dynamic stiffness, 146, 159 Free-field correction, 22
Free-field method
Echo, 95 see Sound power determination
Echo-ellipse, 95 Free-field microphones, 22
Electret microphone, 21 Free terminal velocity, 155
Enclosure Frequency, 5
see Cavity, sound field in Frequency analysis, 15
Energy balance equation, 87 Frequency discrimination, 71
Energy density in sound field Frequency response of microphone, 22
kinetic, 32 Frequency selectivity, 71
potential, 32 Frequency weighting filters, 24
Energy of a signal, 26 Fundamental frequency, 9
Engine exhaust system, 37
Equally tempered scale, 16 Gas constant, 3
Equation of motion for Gauss’s theorem, 32
simple resonator, 137, 138 Generalised coordinate, 147
continuous structures, 148 Ground effect, 39
Equilibrium position, 137
Equivalent integration time, 24 Harmonic sound field, 6, 52
168
Harmonics, 9 Linear averaging
Hearing level, 62 see Time averaging
Hearing threshold, 55, 60, 61, 62, 65, 69 Linear frequency weighting, 24
Helicotrema, 58, 59 Linearised wave equation, 2
Helmholtz equation, 6 Linearity, 3
Helmholtz resonator, 30 Liquids, sound in, 4
Hooke’s law, 27, 136 Locally plane waves, 4, 34
Logarithmic frequency scale, 15
Image sources, 38, 42, 94 Longitudinal waves, 1
Impact sound pressure level, 128 see also Wave types
Impedance Loss factor, 122
acoustic, 28 Loudness, 55, 63, 64, 67, 68, 69
characteristic, 7, 31 Loudness level, 63, 64, 69
mechanical, 27, 146 Loudspeakers, 47
radiation, 29, 35, 37, 46 Lumped elements, 135, 136
specific acoustic, 29 Lumped parameter models, 29
Incident sound intensity, 35
Incident sound power, 86 Masking, 55, 59, 64, 65, 66, 67, 69
Incoherent signals Mass, 135
see Uncorrelated signals Mass density, 150
Industry (reverberation control) 110 Mass law, 120
Independent sources, 17, 20 Material properties, 150
Inhomogeneous medium, 2 see also Acoustic properties of
Inner ear, 55, 56, 57, 65, 71 materials
Input impedance, 29 Mean absorption coefficient, 87, 99
Input point mobility Mean free path, 92
see Mobility Mean square value, 15, 16
Insertion loss, 128, 157 Mechanical admittance, 27
Instantaneous energy density, 32 see also Mobility
Instantaneous sound intensity, 32 Mechanical oscillator, 28
Integration time, 24 Mechanical resonators, 136
see also Time averaging see also Mechanical oscillator
Intelligibility, 76, 109 Mechanical systems, 135
Intensity Membrane absorber, 106
see Sound intensity Middle ear, 55, 56, 57, 58
Interface between two fluids, 11 Mobility, 146, 152, 155
Interference effects, 2, 9, 20, 39 see also Mechanical admittance
Intermittent noise, 25 Mobility, input for semi-infinite or infinite
Inverse distance law, 13, 20 beam or rod, 152
Isolation effectiveness, 157, 161 plate, 152
Modal density, 84, 96
Junctions between coupled pipes, 29 Modes, 81
Monopole, 37
Kinetic energy Motion excitation, 147
see Energy density MTF, 76
Musical tones, 16
Levels, 18
169
Natural angular frequency, 137 Piston in a baffle, radiation from, 41
Natural frequency, 81 Pitch, 9, 16
see also Resonance frequency Plane waves, 4
Nearfield characteristics, 14 Plate, 148
Newton’s second law of motion, 4, 27, 136, Point dipole, 40
137 Point source
Node, 9, 82 see Monopole
Noise Poisson’s ratio, 106, 150, 161
see Random noise Porous absorber, 105
Noise event, 26 Potential energy
Nominal centre frequencies, 16 see Energy density
Normal ambient conditions, 3 Power coupling factor, 156
Number of modes, 83 Power transmission, 155
Pressure microphone, 22
Oblique modes, 82 Pressure node
Octave bands, 16, 103 see Node
ODEON programme, 112 Psychoacoustics, 55, 71
Office spaces (reverberation control) 110 Pulsating sphere, 36
Omnidirectionality, 22, 43 Pure-tone source
see also Directivity, Monopole see Sinusoidal source
One-dimensional wave equation, 4
One-third octave bands, 16 Quadrature, 14
Orders of magnitude of perturbations, 1 see also Antiphase
Oscillating sphere
see Dipole Radiation impedance
Outdoor sound propagation see Impedance
see Ground effect Radiation of sound, 36
Overtones Random errors
see Harmonics see Statistical uncertainty
Random incidence microphone, 22
Parseval’s formula, 17 Random noise, 17
Partial masking, 67 Rapid Speech Transmission Index, 77
Partials RASTI
see Harmonics see Rapid Speech Transmission
Particle displacement, 2 Index
Particle velocity, 2, 71 Ratio of specific heats, 3
Partitioning into frequency bands, 17 Rayleigh’s integral, 42
Pascal, 3 Reactive sound field, 14
Peak level, 26 Receiving structure, 153, 156
Phase, 5, 6, 52 Receptance, 145, 146
Phase speed Reciprocity principle, 38
see Wave speed Reduction index, 113
Phon, 63, 64 Reference sound intensity, 20
Phone scale, 68 Reference sound power, 20
Phonems, 111 Reference sound pressure, 19
Pink noise, 17 Reference velocity, 20
see also White noise Reflection, 2, 8, 94
170
Reflection density, 96 non-dimensional form, 142
Reflection factor, 10, 36 Son, 68
Refraction, 2 Sone scale, 68
Resilient element, 159, 161 Sound absorption, 103
Resonance, 9, 27, 30 Sound exposure level, 26
Resonance frequency, 9, 27, 30, 106, 124, Sound intensity, 32
139 Sound intensity in a plane wave, 33
see also Natural frequency Sound intensity level, 20
Resonant excitation, 138 Sound level meter, 21
Resonator absorber, 108 Sound power determination, 34
Reverberation distance, 90 Sound power level, 20
Reverberation room, 89, 103 Sound pressure level, 18
Reverberation time, 89, 97, 98, 103, 109 Sound pressure, 1
Rigid surface, reflection from, 8, 39 Sound reflection
Rms value see Reflection
see Root mean square value Source spectrum, 73
Rms sound pressure, 15 Source strength, 37, 41, 155
Rod, 148 Source structure, 153
Root mean square value, 15 Sources of vibration, 134, 153
Rubber hardness, 159, 160 Specific acoustic impedance
Rubber isolator, 159 see Impedance
Spectral density, 17
Sabine’s formula, 89, 97, 98, 99 Speech intelligibility, 55, 75, 76
Scattering, 2, 99 Speech intelligibility index, 76, 79, 112
Schools (reverberation control) 110 Speech level, 74, 75
SEL Speech spectrum, 74, 75, 76
see Sound exposure level Speech Transmission Index, 76, 112
Semitone, 16 Speed of sound, 3, 4
Sensitivity of auditory system, 24 Spherical coordinate system, 12
Separation impedance, 119, 122 Spherical sound waves, 13
Shadow see also Monopole
see Diffraction Spherical symmetry, 12
Shear modulus, 151 see also Monopole
Shear force, 151 SPL
Shear waves see Sound pressure level
see Wave types Spring constant, 135, 136, 154
Sign convention, 6, 27 see also Stiffness
SII, 76 Standing wave pattern, 9
Silencers, 29 Standing wave ratio, 2
Simple source see also Standing wave tube
see Monopole Standing wave tube, 35
Simultaneous masking, 64 Standing waves, 81
Single degree of freedom system, 136, 147 Stapes, 56
Sinusoidal source, 5, 6 Static pressure, 1, 3
Solution in Static stiffness, 154
product form, 141 Stationary signals, 18
sum form, 141 Statistical models of sound fields, 31
171
Statistical uncertainty in measurements, 24
STI Uncorrelated signals, 16, 17
see Speech Transmission Index Undamped simple resonator, 139
Stiffness, 28 Undamped system, 137, 138, 143
see also Spring constant Unvoiced, 74
Stiffness correction, 160
Stochastic signals Velocity, 27, 53, 134, 142, 144, 147
see Random noise Vibrating sphere
Structureborne sound, 133 see Pulsating sphere, Dipole
Structural damping, 145 Vibration isolation, 139, 153, 156
Struve function, 46 Vibration isolator, 153
Subwoofer Vibro-acoustics, 133, 148
see Loudspeakers Viscous damper, 135, 136
Sum of harmonic signals, 20, 52 Viscous damping ratio, 137, 142
Suspended ceiling, 109 Viscous friction, 105
Viscously damped system, 140
Tangential modes, 82 Voiced, 73
Tapping machine, 128 Volume acceleration, 48
Temperature fluctuations in sound field, 2 Volume displacement, 3
Temperature, influence on the speed of Volume velocity, 28, 37
sound, 3
Temporal integration, 69 Water, 11, 18
Thick wall, 118 see also Liquids, sound in
Time average of a product, 54 Wavelength, 5
Time averaging Wavenumber, 5
exponential, 24 Wave speed for
linear, 24 longitudinal waves, 149
Time constant, 24 shear waves, 150
Time derivative bending waves, 152
see Differentiation with respect to Wave types, structural
time longitudinal waves, 148
Time integration, 53 shear or torsional waves, 148, 150
see also Time averaging bending or flexural waves, 148, 151
Time weighting Weighted impact sound pressure level, 130
see Time averaging Weighted sound reduction index, 129
Time-averaged energy density, 33 White noise, 17, 21, 100
Time-averaged sound intensity, 33
Transfer function, 83 Young’s modulus of elasticity, 106,149,160
Transmission between fluids, 11
Transmission coefficient, 113
Transmission loss, 113, 114
Transmitted force, 142
Transversal waves, 1
Transverse displacement in beams, 151
Two-port, 29
Typical values of sound power levels, 31
Typical values of sound pressure levels, 19
172

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FUNDAMENTALS OF ACOUSTICS

AND NOISE CONTROL

Finn Jacobsen, Torben Poulsen, Jens Holger Rindel,

Printed at Ørsted • DTU, Technical University of Denmark

1 An elementary introduction to acoustics....................................................................1

1.1 Introduction ....................................................................................................................1

1.2 Fundamental acoustic concepts......................................................................................1

1.2.1 Plane sound waves .............................................................................................4

1.3 Acoustic measurements................................................................................................15

1.3.1 Frequency analysis ...........................................................................................15

1.4 The concept of impedance............................................................................................27

1.5.1 The energy in a sound field ..............................................................................32

1.6 Radiation of sound .......................................................................................................36

1.6.1 Point sources ....................................................................................................36

1.7 References ....................................................................................................................49

1.9 Appendix: Complex notation .......................................................................................51

2 Ear, Hearing and Speech ...........................................................................................55

2.1 Introduction ..................................................................................................................55

2.2 The ear..........................................................................................................................55

2.2.1 The outer ear.....................................................................................................56

2.3 Human hearing .............................................................................................................61

2.3.1 The hearing threshold.......................................................................................61

2.4.1 Complete masking............................................................................................65

2.5.1 The loudness curve...........................................................................................68

2.6 The auditory filters .......................................................................................................71

2.6.1 Critical bands....................................................................................................71

2.7 Speech ..........................................................................................................................73

2.7.1 Speech production ............................................................................................73

2.8 References ....................................................................................................................78

3 An introduction to room acoustics............................................................................81

3.1 Sound waves in rooms..................................................................................................81

3.1.1 Standing waves in a rectangular room .............................................................81

3.2 Statistical room acoustics .............................................................................................85

3.2.1 The diffuse sound field.....................................................................................85

3.3 Geometrical room acoustics .........................................................................................92

3.3.1 Sound rays and a general reverberation formula..............................................92

3.4 Room acoustical design................................................................................................96

3.4.1 Choice of room dimensions..............................................................................96

3.5 References ..................................................................................................................101

4 Sound absorbers and their application in room design ........................................113

4.1 Introduction ................................................................................................................103

4.2 The room method for measurement of sound absorption ..........................................103

4.3 Different types of sound absorbers.............................................................................104

4.3.1 Porous absorbers ............................................................................................105

4.4 Application of sound absorbers in room acoustic design...........................................109

4.5 References ..................................................................................................................112

5 An introduction to sound insulation .......................................................................113

5.1 The sound transmission loss.......................................................................................113

5.1.1 Definition .......................................................................................................113

5.2 Single leaf constructions ............................................................................................117

5.2.1 Sound transmission through a solid material .................................................117

5.3 Double leaf constructions...........................................................................................123

5.3.1 Sound transmission through a double construction........................................123

5.4 Flanking transmission ................................................................................................126

5.5 Enclosures ..................................................................................................................127

5.6 Impact sound insulation .............................................................................................127

5.7 Single-number rating of sound insulation ..................................................................128

5.7.1 The weighted sound reduction index .............................................................128

5.8 Requirements for sound insulation.............................................................................131

5.9 References ..................................................................................................................131