Wave

Waves can occur whenever a system is disturbed from equilibrium and

when the disturbance can travel or propagate from one region of the
system to another.

Wave can carry energy and momentum.

When a wave motion passes through a medium, particles of the

medium only vibrate simple harmonically about their mean position.
They do not leave their position and move with the disturbance.

In wave motion, the phase of particles of medium keeps on changing.

The velocity of the particle during their vibration is different at
different position.

The velocity of wave motion through a particular medium is constant.

It depends only on the nature of medium not on the frequency,
wavelength or intensity.

Energy is propagated along with the wave motion without any net
transport of the medium.

Types of Waves
1. Mechanical waves
Require medium for their propagation e.g. Waves on string and spring,
waves on water surface, sound waves.

2. Non-mechanical waves
Do not require medium for their propagation e.g, Light, heat (Infrared),
radio waves, γ- rays, X-rays etc

3. Transverse waves
Particles of the medium vibrates in a direction perpendicular to the
direction of propagation of wave.
Transverse waves can be transmitted through
solids, they can be setup on the surface of liquids.
But they can not be transmitted into liquids and
gases.

Transverse waves can be polarised.

It travels in the form of crests (C) and troughs

(T).

4. Longitudinal waves
Particles of a medium vibrate in the direction of wave motion.

These waves can be transmitted through solids, liquids and gases

because for these waves propagation, volume elasticity is necessary.

Longitudinal waves can not be polarised.

It travels in the form of compression (C) and rarefaction (R).

One dimensional waves

Energy is transferred in a single direction only e.g. Wave propagating in
a stretched string.

Two dimensional waves

Energy is transferred in a plane in two mutually perpendicular
directions e.g. Wave propagating on the surface of water.

Three dimensional waves

Energy in transferred in space in all direction e.g. Light and sound
waves propagating in space.

Matter waves
The waves associated with the moving particles are called matter
waves.

Audible or sound waves

Range 20 Hz to 20 KHz. These are generated by vibrating bodies such as
vocal cords, stretched strings etc.

Infrasonic waves
Frequency lie below 20 Hz and wavelengths are greater than 16.6 cm.

Ultrasonic waves
Frequency greater than 20 KHz and wavelength < 1.66 cm.

Shock waves When an object moves with a velocity greater than that
of sound, it is termed as Supersonic.

Amplitude (A)
Maximum displacement of a vibrating particle of medium from it's
mean position is called amplitude.

Wavelength (λ)
It is equal to the distance travelled by the wave during the time in
which any one particle of the medium completes one vibration about
its mean position.
Or distance travelled by the wave in one time period is known as
wavelength.

Or is the distance between the two successive crests or troughs.

Frequency (f)
Frequency of vibration of a particle is defined as the number of
vibrations completed by particle in one second.

It is the number of complete wavelengths traversed by the wave in one

second.

Units of frequency are hertz (Hz) and per second.

Time period (T)
Time period of vibration of particle is defined as the time taken by the
particle to complete one vibration about its mean position.

It is the time taken by the wave to travel a distance equal to one

wavelength

Wave pulse
It is a short wave produced in a medium when the
disturbance created for a short time.

Wave train
A series of wave pulse is called wave train.

Wave velocity (v)

It is the distance travelled by the disturbance in one time period. It only
depends on the properties of the medium and it independent of time
and position.

Velocity of Longitudinal Wave (Speed of Sound)

Velocity of sound in any elastic medium is given by
In solids where Y = Young's modulus of elasticity

In a liquid and gaseous medium

where B = Bulk modulus of elasticity of liquid or gaseous medium.

As solids are most elastic while gases least. So the velocity of sound is
maximum in solids and minimum in gases, hence
vsteel > vwater > vair
5000 m/s > 1500 m/s > 330 m/s

Newton's formula :
He assumed that when sound propagates through air temperature
remains constant. i.e. the process is isothermal. For isothermal process
B = Pressure (P)

For air at NTP : P = 1.01x105 N/m2 and p = 1.29 kg/m3.

However the experimental value of sound in air is 332 m/sec which is

greater than that given by Newton's formula.

Laplace correction
He modified Newton's formula assuming that propagation of sound in
gaseous medium is adiabatic process. For adiabatic process
B = Adiabatic elasticity (E ) = γP
Factors Affecting Velocity of Sound in Gaseous Medium

1. Effect of pressure at constant temperature

Velocity of sound is independent of the pressure of gas, because as
pressure increases, density also increases hence ratio remains
constant.

2. Effect of temperature
With rise in temperature velocity of sound increases.

When the temperature change is small then vt = vo + 0.61t

where vt = Velocity of sound at t°C

v0 = Velocity of sound at 0°C = 332 m/sec
t = Small temperature change

If t = 1°C then vt = (vo + 0.61) m/sec. Hence for 1°C rise, speed of sound
in air increases by 0.61 m/sec.

4. Effect of humidity
With increase in humidity, density of air decreases. So with rise in
humidity velocity of sound increases. Sound travels faster in humid air
(rainy season) than in dry air (summer) at the same temperature
because
ρmoist air < ρdry air => vmoist air > vdry air

Principle of Superposition
The displacement at any time due to any number
of waves meeting simultaneously at a point in a
medium is the vector sum of the individual
displacements due each one of the waves at that
point at the same time.

If y1 , y2 , y3 ………. Are the displacements at a particular time at a

particular position, due to individual waves, then the resultant
displacement.
y = y1 + y2 + y3 + ……

Interference of Sound Waves

When two waves of same frequency, same wavelength, same velocity
(nearly equal amplitude) moves in the same direction, Their
superimposition results in the interference.

Due to interference the resultant intensity of sound at that point is

different from intensities of each wave separately.

Interference is of two type

(i) Constructive interference
(ii) Destructive interference.

In interference energy is neither created nor destroyed but is

redistributed.

Constructive interference
When the waves meets a point with same phase, constructive
interference is obtained at that point (i.e. maximum sound).
Path difference between the waves at the point of observation S = mλ

Destructive interference
When the wave meets a point with opposite phase, destructive
interference is obtained at that point (i.e. minimum sound)
Path difference S = ( m + ½ )λ m=0,1,2,3,4….

Standing Waves or Stationary Waves

When two sets of progressive wave trains of same type (both
longitudinal or both transverse) having the same amplitude and same
time period/frequency/wavelength travelling with same speed along
the same straight line in opposite directions superimpose, a new set of
waves are formed. These are called stationary waves or standing
waves.

In practice, a stationary wave is formed when a wave train is reflected

at a boundary. The incident and reflected waves then interfere to
produce a stationary wave.

Nodes (N)
The points where amplitude is minimum are called nodes.
Distance between two successive nodes is λ/2 .
Nodes are at permanents rest.
At nodes air pressure and density both are high.

Antinodes (A)
The points of maximum amplitudes are called antinodes.
The distance between two successive antinodes is λ/2 .
At nodes air pressure and density both are low.
The distance between a node (N) and adjoining antinode (A) is λ/4.
Note
Any musical sound produced by the simple harmonic oscillations of the
source is called note.

Tone
Every musical sound consists of a number of components of different
frequencies every component is known as a Tone.

Fundamental note and fundamental frequency

The note of lowest frequency produced by an instrument is called
fundamental note. The frequency of this note is called fundamental
frequency.

Harmonics
The frequency which are the integral multiple of the fundamental
frequency are known as harmonics e.g. if n be the fundamental
frequency, then the frequencies n, 2n, 3n .... are termed as first,
second, third .... harmonics.

Overtone
The harmonics other than the first (fundamental note) which are
actually produced by the instrument are called overtones. e.g. the tone
with frequency immediately higher than the fundamental is defined as
first overtone.

Standing Waves on a String/ fundamental frequency and

harmonics
Consider a string of length l, stretched under tension T between two
fixed points.

If the string is plucked and then released, a transverse harmonic wave

propagate along it's length and is reflected at the end.
The incident and reflected waves will superimpose to produce
transverse stationary waves in a string.

Nodes (N) are formed at rigid end and antinodes (A) are formed in
between them.

Number of antinodes = Number of nodes – 1

Velocity of wave (incident or reflected wave) is given by

m = Mass per unit length of the wire And T is the tension

For obtaining n loops (n-segments) in string, it has to be plucked at a

distance L / 2n from one fixed end.

Fundamental mode of vibration

Number of loops n = 1
Plucking at L/2 (from one fixed end)
Fundamental frequency or first harmonic
𝑣
f1= 𝜆1 = 2𝑙𝑣

Second mode of vibration (First over tone or second harmonic)

Number of loops n = 2
Plucking at L/4 (from one fixed end)
l = λ2
Second harmonic or first over tone

𝑣
f2 = 𝜆2 = 𝑣𝑙 = 2f1

Third mode of vibration (Second over tone or third harmonic)

Number of loops n = 3
Plucking at L/6 (from one fixed end)
l = 3 λ3 / 2 => λ3 = 2 l / 3
Second harmonic or first over tone
𝑣
f3 = 𝜆3 = 3 2𝑙𝑣 = 3f1

All even and odd harmonics are present. Ratio of harmonic = 1:2: 3 .....
Ratio of over tones = 2 : 3 : 4 ....
𝑣
General formula for frequency f = n x 2𝑙 where n = 1,2,3, …
correspond to 1st , 2nd, 3rd modes of vibration of the string.

2𝑙
General formula for wavelength λ= 𝑛 where n = 1,2,3, …
correspond to 1st , 2nd, 3rd modes of vibration of the string.

Beats
When two sound waves of slightly different frequencies, travelling in a
medium along the same direction, superimpose on each other, the
intensity of the resultant sound at a particular position rises and falls
regularly with time. This phenomenon of regular variation in intensity
of sound with time at a particular position is called beats.

Beat period
The time interval between two successive beats (i.e. two successive
maxima of sound) is called beat period.

Beat frequency
The number of beats produced per second is called beat frequency.

Doppler's Effect
Whenever there is a relative motion between a source of sound and
the observer (listener), the frequency of sound heard by the observer is
different from the actual frequency of sound emitted by the source.

The frequency observed by the observer is called the apparent

frequency. It may be less than or greater than the actual frequency
emitted by the sound source. The difference depends on the relative
motion between the source and observer.
Doppler's effect was discovered by Doppler, an Australian physicist, in
1845.

1. When source is moving but observer is at rest

Apparent frequency of sound heard by stationary listener due to
source moving towards him at speed Vs is given as
𝑣
f’ = ( 𝑣 −𝑣𝑠 )f f’ > f
Apparent frequency of sound heard by stationary listener due to
source moving away from him at speed 'u' is given as;

f’ = ( 𝑣 +𝑣 𝑣𝑠 ) f f’ < f

2. When observer is moving but source is at rest

Apparent frequency of sound heard by a person moving towards a
stationary source with speed 'u' is given as;

f’ = ( 𝑣 +𝑣𝑣𝑜 ) f f’ > f

Apparent frequency of sound heard by a listener moving away from a

stationary source with speed 'u' is given as;

f’ = ( 𝑣 −𝑣𝑣𝑜 ) f f’ < f

3. When source and observer both are moving

When both source and observer moves towards each other.

f’ = ( 𝑣𝑣+−𝑣𝑠
𝑣𝑜
)f f’ > f

When both source and observer moves away from each other.
 f’ = ( 𝑣𝑣 −+ 𝑣𝑜
𝑣𝑠
)f f’ < f

Applications of Doppler's effect

i) Ships and submarine (sonar devices)
ii) Bats (for traveling)
iii) Radar (for detection)
iv) Determining velocity of a star w.r.t earth
[ When a star is moving away from Earth then wavelength of
light increases and red shift of spectrum is observed. When a
star is moving towards the Earth then wavelength of light
decreases and blue shift of spectrum is observed.]
v) To monitor blood flow in major arteries.

Simple Harmonic Motion

Simple harmonic motion is a special type of periodic motion, in which a
particle moves to and fro repeatedly about a mean position.

In linear S.H.M. a restoring force which is always directed towards the

mean position and whose magnitude at any instant is directly
proportional to the displacement of the particle from the mean
position i.e. Restoring force α Displacement of the particle from mean
position.

F α – x => F = – kx

Where k is known as force constant. Its S.I. unit is Newton/meter.

Some Important Definitions

Time period (T)

It is the least interval of time after which the periodic motion of a body
repeats itself. S.I. unit of time period is second.
Frequency (f)
It is defined as the number of oscillations executed by body per
second. S.I unit of frequency is hertz (Hz).

Angular Frequency (ω)

Angular frequency of a body executing periodic motion is equal to
product of frequency of the body with factor 2π. Its unit is rad/sec.
Angular frequency = 2 π n