Acoustic Quantities

LMS Test.Lab

16A

Copyright Siemens Industry Software NV

 Table of Contents

Chapter 1 Acoustic quantities ....................................................................................... 5

Section 1.1 Sound power (P) .................................................................................. 5
Section 1.2 Sound pressure ..................................................................................... 5
Section 1.3 Sound (Acoustic) intensity ................................................................... 6
Section 1.4 Free field .............................................................................................. 7
Section 1.5 Particle velocity ................................................................................... 7
Section 1.6 Acoustic impedance (Z) ....................................................................... 8

Chapter 2 Reference conditions .................................................................................... 9

Section 2.1 Sound power level Lw ......................................................................... 9
Section 2.2 Particle velocity level Lv ..................................................................... 9
Section 2.3 Sound (Acoustic) intensity level LI ................................................... 10
Section 2.4 Sound pressure level LP..................................................................... 10

Chapter 3 Octave bands............................................................................................... 11

Section 3.1 Octave filter midband and edge frequencies ...................................... 11
Section 3.2 Octave filter shapes ............................................................................ 14
Section 3.2.1 Octave Filtering Options in Test.Lab ................................................. 15

Chapter 4 Acoustic weighting ..................................................................................... 17

Section 4.1 Frequency weighting.......................................................................... 17

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 Chapter 1 Acoustic quantities

Chapter 1 Acoustic quantities

In This Chapter
Sound power (P) .................................................................5
Sound pressure ...................................................................5
Sound (Acoustic) intensity .................................................6
Free field ............................................................................7
Particle velocity ..................................................................7
Acoustic impedance (Z) .....................................................8

Section 1.1 Sound power (P)

The amount of noise emitted from a source depends on the sound power of that
source. The sound power is a basic characteristic of a noise source, providing an
absolute parameter that can be used for comparison. This differs from the
sound pressure levels it gives rise to, which depend on a number of external
factors.

The total sound power PI of a source surrounded by N measurement surfaces is

given by:

The power of a sound source is expressed in Joules per second, or Watts. The
sound power can also be represented by the letter W.

Section 1.2 Sound pressure

The effect of the sound power emanating from a source is the level of sound
pressure. Sound pressure is what the ear detects as noise, the level of which
depends to a great extent on the acoustic environment and the distance from the
source. The sound pressure is defined as the difference between the actual and
ambient pressure.

This is a scalar quantity that can be derived from measured sound pressure
spectra or autopower spectra either at one specific frequency (spectral line), or
integrated over a certain frequency band. Sound pressure measurements can be
obtained at each measurement point, and are independent of the measurement
direction (X,Y, or Z). The units are Pascal (Pa) or N/m2.

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 Chapter 1 Acoustic quantities

Section 1.3 Sound (Acoustic) intensity

An important quantity to be derived from the sound power is sound intensity.

The sound intensity of a sound wave describes the direction and net flow of
acoustic energy through an area.

Sound intensity is a vector, orientated in 3D-space with the fundamental units of

W/m2, (power transmitted per unit area).

The area is represented as a vector in 3D space with a length equal to the

amount of geometrical area, and a direction perpendicular to the measurement
surface. As such, the vector product (Ii.Si) represents the flow of acoustic
energy in a direction perpendicular to a surface. This is the usual direction in
which intensity is measured. If the acoustic intensity vector lies within the
surface itself, the transmitted sound power equals zero.

Intensity is also the time-averaged rate of energy flow per unit area.

As such, if the energy is flowing back and forth resulting in zero net energy
flow then there will be zero intensity.

Normal sound intensity

This is the component of the sound intensity vector normal to the measurement
surface.

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 Chapter 1 Acoustic quantities

Section 1.4 Free field

This term refers to an idealized situation where the sound flows directly out
from the source and both pressure and intensi ty levels drop with increasing
distance from the source according to the inverse square law.

Diffuse field

In a diffuse field the sound is reflected many times such that the net intensity
can be zero.

Section 1.5 Particle velocity

Pressure variations give rise to movements of the air particles. It is the product
of pressure and particle velocity that results in the intensity. In a medium
with mean flow therefore

where:

p= sound pressure (Pa)

= particle velocity (m/s)

The particle velocity of a medium is defined as the average velocity of a volume

element of that medium. This volume element must be large enough to contain
millions of molecules so that it may be thought of as a continuous fluid, yet
small enough so that acoustic variables such as pressure, density and velocity
may be considered to be constant throughout the volume element.

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 Chapter 1 Acoustic quantities

Equation 2-4 can be used to compute the particle velocity, once the acoustic
intensity and the sound pressure have been measured. Particle velocity is a
vector in 3D-space expressed in units of (m/s).

In a diffuse field the pressure and velocity phase vary at random giving rise to a
net intensity of zero. Under certain circumstances (i.e. plane progressive waves
in a free field), the particle velocity can also be calculated from the pressure and
the impedance of the medium (rc).

where:

pe= effective sound pressure (Pa)

= mass density of the medium (kg/m3)

c= velocity of sound in the medium (m/s)

By combining equations 2-4 and 2-5 it can be seen that in a free field a
relationship exists enabling the acoustic intensity to be determined from the
effective pressure of a plane wave.

Section 1.6 Acoustic impedance (Z)

This is defined as the product of the mass density of a medium and the velocity
of sound in that medium.

where:

= mass density (kg/m3)

c = velocity of sound in the medium (m/s)

8 LMS Test.Lab Acoustic Quantities

 Chapter 2 Reference conditions

Chapter 2 Reference conditions

In This Chapter
Sound power level Lw........................................................9
Particle velocity level Lv ....................................................9
Sound (Acoustic) intensity level LI....................................10
Sound pressure level LP .....................................................10
It is a common practise to define standards for acoustic intensity, pressure, etc...
at an air temperature of 20°C and a standard atmospheric pressure of 1023 hPa
(1 bar). Under these conditions

the density of air = 1.21 (kg/m3)

the velocity of sound in air c = 343 (m/s)

the acoustic impedance = 415 rayls (kg/m2s)

dB scale

Since the range of pressure levels that can be detected is large and the ear
responds logarithmically to a stimulus, it is practical to express acoustic
parameters as a logarithmic ratio of a measured value to a reference value.
Hence the use of the decibel scales for which the reference values for intensity,
pressure and power are defined below.

Section 2.1 Sound power level Lw

This is defined as the logarithmic measure of the absolute (unsigned) value of

the sound power generated by a source.

The reference sound power is P0 = 10-12 (W)

Section 2.2 Particle velocity level Lv

This is defined as the logarithmic measure of the particle velocity.

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 Chapter 2 Reference conditions

The reference particle velocity is v0 = 50 10-9 (m/s)

Section 2.3 Sound (Acoustic) intensity level LI

This is the logarithmic measure of the absolute value of the intensity vector.

The commonly used reference standard intensity for airborne sounds is Io=
10-12 (W/m2)

Normal acoustic intensity level (LI)

This is the logarithmic measure of the absolute value of the normal intensity
vector.

Section 2.4 Sound pressure level LP

This is defined as

p is the rms value of the acoustic pressure (in Pa)

The above reference values for intensity and power correspond to an effective
rms reference pressure of

po = 0.00002 (Pa)

= 20 mPa

This sound pressure level of 20 mPa is known as the standardized normal

hearing threshold and represents the quietest sound at 1000Hz that can be heard
by the average person.

10 LMS Test.Lab Acoustic Quantities

 Chapter 3 Octave bands

Chapter 3 Octave bands

In This Chapter
Octave filter midband and edge frequencies ......................11
Octave filter shapes ............................................................14

Section 3.1 Octave filter midband and edge frequencies

There are two accepted methods to determine the midband frequencies of the
octave bands:
 the base-2 method: subsequent center frequencies have a ratio of 21/b with 1/b
the bandwidth designator (e.g. b=3 for 1/3 octave band).
Edge frequencies are derived from the center frequency by multiplying or
dividing with 21/(2b).
The reference frequency is fr=1000 Hz. Center frequencies are given by:
 fcn=fr*2n/b for b odd
 fcn=fr*2(2n+1)/(2b) for b even
 the base-10 method: subsequent center frequencies have a ratio of (103/10)1/b
with 1/b the bandwidth designator (e.g. b=3 for 1/3 octave band).
Edge frequencies are derived from the center frequency by multiplying or
dividing with (103/10)1/(2b).
The reference frequency is 1000 Hz.
 fcn=fr*(103/10)n/b for b odd

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 Chapter 3 Octave bands

 fcn =fr*(103/10)(2n+1)/(2b) for b even

Note: Current IEC 61260:1995 and ANSI S1.11-2004 standards accept both
base-10 and base-2, but recommend base-10. Some standards (e.g. ISO
266-1997) are based on base-10 but mention that base-2 may be used as an
acceptable approximation because the differences are small (103/10 =
1.995262).

Note: Apart from the exact midband frequencies as mentioned above, the
designation of the band will be expressed in ‘nominal’ midband frequencies
(typically rounded numbers, also specified in the standards for full and 1/3
octaves) and not with the ‘exact’ midband frequencies (according to e.g. ISO
266-1997 and ANSI S1.6-1984(R2006)). For a list of normalized midband
frequencies, see the table further.

Note: With base-10 system, midband frequencies of 1/3 octave band will
include e.g. 10, 100, 1000, 10000 (ratio of 10). Other midband frequencies
digits will also repeat themselves apart from the location of the decimal
points. For the base-2 system, the 100 Hz (nominal) third octave band will
have a midband frequency of 99.2126 Hz while the 10000 Hz (nominal) third
octave band will have a midband frequency of 10079.37 Hz.

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 Chapter 3 Octave bands

Note on even fractional octaves:

Different formulas are used to locate center frequencies for b=odd (1, 1/3
octave) and b=even (1/2; 1/6, 1/12, 1/24). This means that the reference
frequency (1000 Hz) is a center frequency for b=odd and an edge frequency
for b=uneven (e.g. 1/2, 1/6, 1/12 octave). The purpose of this is to be able to
split an octave band in smaller fractions, covering the same edge frequencies
as the original one: e.g. the 1000 octave band can be split in 2 ½ - octave band
with 1000 lying just at the edge of both.

The definition of the even fractional octaves changed in the period 1995-1997.
Until that period, the center frequencies of even fractional octaves such as 1/2,
1/6, 1/12 and 1/24 octaves were based on the known octave center frequencies
(e.g.: 250Hz, 500Hz, 1000Hz, etc). This approach has the disadvantage that the
sum of two ½ octaves does not add up to an octave level.

In the above mentioned period 1995-1997, ISO 226 removed the definition of
even fractional octaves from the standard and mentions only 1/1 and 1/3
octaves. The IEC standard on the calculation of time based fractional octaves,
adapted the definition to the alternative approach to keep the filter cut-off
frequencies as references, hence allowing even fractional octaves to add up to
the next level of octaves. Consequently, the known octave center frequencies
are no longer valid center frequencies. Data from other sources, CADA-X data
and older Test.Lab data (before 8B) might still be measured according to this
older convention.

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 Chapter 3 Octave bands

Section 3.2 Octave filter shapes

When implemented in the time domain as digital band-pass filter banks on

sampled data, the relative attenuation of the filters is never ‘perfect’ (no
attenuation between the edges and full attenuation outside the edges). The
current IEC 61260:1995 and ANSI S1.11-2004 standards give an upper and a
lower limit for the relative attenuation, depending on the ‘Class’ of analyzer.
These limits allow a shape of the filter response which attenuates before the
edge frequencies and with a finite slope beyond the edge frequencies.

When converting data to octave band in the frequency domain, it is much easier
(and common practice) to implement a (nearly) ideal filter (i.e. a ‘square’ filter
shape): only energy on the frequency lines within the octave band will be
summed. However, in order to match as closely as possible data processed with
time-domain filters with data processed with FFT, it is also possible to use filter
shapes with a ‘smoother’ shape (ANSI Emulation). This requires more
processing, as for each octave band, (weighted) integration over more frequency
lines will be done.

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 Chapter 3 Octave bands

Section 3.2.1 Octave Filtering Options in Test.Lab

Section 3.2.1.1 Time-based

For time-based octave filtering, it is possible to select 3 methods in Test.Lab:
 ANSI-IEC – Class1 – base 10: the recommended setting as it complies with
the latest IEC and ANSI standard and uses base 10 as recommended by the
standard
 ANSI – base 2: for compatibility with previous (before 8A) releases (ANSI
method used in Signature throughput processing)
 IEC – base 2: for compatibility with previous releases (before 8A) (IEC
method used in Signature throughput processing)
This choice influences host-based octave calculations performed with the
ANSI-IEC Octave Filtering add-in (e.g. Signature throughput processing or
RTO in parallel with Fixed sampling acquisition). Front-end based octave
filtering (available in the RTO workbook) will always use an ANSI-IEC –
Class1 – base 2 filter implementation.

Section 3.2.1.2 FFT-based

For FFT-based octave filtering (e.g. done in Octave display of narrowband data,
in Signature Fixed sampling, by the ‘Octave’ function in the Data calculator
etc…), it is possible to select between:
 Ideal – base 10: the recommended setting (only method before 8A)
 Ideal – base 2: for compatibility with e.g. CADA-X data
 ANSI Emulation – base 10: if similarity with time-domain filters is
important (before 8A, only possible in Signature Processing with the ‘ANSI
Emulation’ option set in the setup)
 ANSI Emulation – base 2: for compatibility with e.g. CADA-X data

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 Chapter 4 Acoustic weighting

Chapter 4 Acoustic weighting

In This Chapter
Frequency weighting ..........................................................17

Section 4.1 Frequency weighting

The human ear has nonlinear, frequency dependent characteristics, which means
that the sensation of loudness cannot be perfectly described by the sound
pressure level or its spectrum. To derive an experienced loudness level from the
sound pressure signal, the frequency spectrum of the sound pressure signal is
multiplied by a frequency weighting function. These weighting functions are
based on experimentally determined equal loudness contours which express the
loudness sensation as a function of sound pressure level and frequency. A
number of equal loudness contours are shown in Figure 2-1. The loudness level
is expressed in ‘Phons’. 1 kHz-tones are used as the reference, which means that
for a 1000 Hz tone, the Phon value corresponds to the dB sound pressure level.

Figure 2-1 Equal loudness perception contours

A, B and C - weighting for acoustic signals. A-weighting modifies the

frequency response such that it follows approximately the equal loudness curve
of 40 phons and is applied to signals with a sound pressure level of 40dB. The
A-weighted sound level has been shown to correlate extremely well with
subjective responses. The B and C-weighting follow more or less the 70 and
100 phon contours respectively. These contours can be seen in Figure 2-2. The
resulting value is then denoted by LA, LB,.... with unit dBA, dBB...

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 Chapter 4 Acoustic weighting

Table 2.2 (overleaf) shows the relative response attenuations or amplifications

of the 3 types of filters. In between the listed normal frequencies, these filter
spectra are linearly interpolated on a log-log scale. Figure 2-2 shows the same
information in a graphical form.

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 Chapter 4 Acoustic weighting

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 Index

A
Acoustic impedance (Z) • 8
Acoustic quantities • 5
Acoustic weighting • 23
F
FFT-based • 21
Free field • 7
Frequency weighting • 23
O
Octave bands • 15
Octave filter midband and edge frequencies •
15
Octave filter shapes • 20
Octave Filtering Options in Test.Lab • 21
P
Particle velocity • 7
Particle velocity level Lv • 12
R
Reference conditions • 11
S
Sound (Acoustic) intensity • 6
Sound (Acoustic) intensity level LI • 12
Sound power (P) • 5
Sound power level Lw • 11
Sound pressure • 5
Sound pressure level LP • 13
T
Time-based • 21

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