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Mechanical waves

Electromagnetic waves

Matter waves

Transverse waves

Longitudinal waves


Travelling or progressive waves

Displacement of particle in SHM

Amplitude of wave in SHM


Initial phase angle

Time period of wave

Frequency of wave

Angular frequency

Angular wave number or propagation constant

Sound waves

Infrasonic waves

Audible waves

Ultrasonic waves

Velocity of travelling wave or wave velocity

Particle velocity in SHM

Particle acceleration in SHM

Speed of a transverse wave on stretched string

Velocity of longitudinal (sound) waves

Newton’s formula for velocity of sound

Laplace’s correction for velocity of sound

Factors affecting velocity of longitudinal (Sound) waves

Effect of pressure on velocity of sound

Effect of temperature on velocity of sound

Effect of density on velocity of sound

Superposition of Waves

Reflection of waves

The boundary effect


Interference of waves

Standing waves and normal modes

Node and antinode

Constructive Interference

Destructive Interference

Modes of a stretched string fixed at both ends

Normal modes

Fundamental note



Frequency of an air column in a closed pipe

Frequency of an air column in a open pipe

End correction of organ pipe

Factors Affecting Frequency of Pipe

Resonance tube


Frequency of beat

Doppler’s effect

Doppler’s effect when source is moving and observer is at rest

Doppler’s effect when source is at rest and observer is moving

Doppler’s effect when source and observer both are moving

Applications of doppler’s effect




A wave is motion of vibratory disturbance in a medium or free space which carries energy from one point to another point without any actual transfer of the particles of medium.

There are three types of waves

  1. Mechanical Waves

    Those waves which require a material medium for their propagation, are called mechanical waves, e.g., sound waves, water waves etc.

  2. Electromagnetic Waves

    Oscillating electrical and magnetic fields perpendicular to each other and also perpendicular to the direction of propagation of energy, are called EM waves. They do not require a material medium for their propagation, e.g., light waves, radio waves etc.

  3. Matter Waves

    These waves are associated with electrons, protons and other fundamental particles.


Transverse waves

A wave, in which the particles of the medium vibrate at right angles to the direction of propagation of wave, is called a transverse wave.

These waves travel in the form of crests and troughs.

A point of maximum positive displacement in a wave is called crest, and a point of maximum negative displacement is called trough.

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Longitudinal waves

A wave, in which the particles of the medium vibrate in the same direction in which wave is propagating, is called a longitudinal wave.

These waves travel in the form of compressions and rarefactions.

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The distance between two nearest points in a wave which are in the same phase of vibration is called the wavelength (λ).

Travelling or Progressive Waves

A wave, transverse or longitudinal, is said to be travelling or progressive if it travels from one point of the medium to another.

Sound Waves

Sound waves are longitudinal mechanical waves that occur in nature. They require medium to travel.

Infrasonic Waves

The sound waves of frequency lies between 0 to 20 Hz are called infrasonic waves.

Audible Waves

The sound waves of frequency lies between 20 Hz to 20000 Hz are called audible waves.

Ultrasonic Waves

The sound waves of frequency greater than 20000 Hz are called ultrasonic waves.

Important points to note


A wave can be represented as

y (x, t) = A sin (kx - ωt + ϕ)

Various terms in the above equation are described as follows,

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The equation can also be represented as,

y (x, t) = C sin(kx - ωt) + D sin(kx - ωt )


 A = C2+ D2



Time Period of Wave

Time taken to complete one vibration is called time period (T).

Frequency of Wave

The number of vibrations completed in one second is called frequency of the wave.

Frequency ν = 1Time period T=ω2π

Its SI unit is hertz.

Angular Frequency

ω =2πT

Its SI unit is rad s-1

Angular wave number or propagation constant

For t = 0 and ϕ = 0.

At this instant,

Since the displacement y is same at x and x + λ

This condition can be satisfied only when, kλ = 2πn, where n = 1, 2, 3, …

For n = 1, we have k = 2πλ, which is called the angular wave number or propagation constant.

Its SI unit is rad m–1

Also, when x = 0 and ϕ = 0,

y(x, t) = A sin (kx  ωt )

yx, t= A sin kλ ωtkλ  xλ

yx, t= A sin 2π tT  xλ

yx, t=A sin 2π tT  xvT

yx, t= A sin 2πT t  xv

yx, t= A sin2πνt  xv

yx, t= A sin2πνvvt  x

yx, t= A sin2πλvt  x

Velocity of Travelling Wave or Wave Velocity

The distance travelled by a wave in one second is called velocity of the wave (v). Relation among velocity, frequency and wavelength of a wave is given by 

v =λT= νλ =ωk

Particle Velocity in SHM

The velocity of the particles executing SHM is called particle velocity.

Particle velocity vy = dydt

We know that,

y(x, t) = A sin (kx - ωt)

Keeping x constant,

V=dydt=Aω cos (kx - ωt)

V=Aω sin (kx - ωt + π/2)


dydx= -k A cos (kx - ωt)


Vdy/dx=A ω cos (kx - ωt)-k A cos (kx - ωt)

= -ωk=2πν2π/λ= -νλ= -v

Hence, we can write,

V = -vdydx

Particle Acceleration in SHM

a=dVdt=- Aω2 sin (kx - ωt)=-ω2y

a= Aω2 sin (kx - ωt + π)

The maximum particle acceleration,


Speed of a Transverse Wave on Stretched String

The speed of transverse waves on a string is determined by two factors,

  1. the linear mass density or mass per unit length, μ,

  2. the tension F.

    Velocity v =Fμ

Velocity of Longitudinal (Sound) Waves

Velocity of longitudinal (sound) wave in any medium is given by

For solids v= Yρ    

For fluids v = Bρ

where,Y is Young’s modulus, B is bulk coefficient of elasticity of the medium and ρ is density of the medium.

Although the densities of liquids and solids are much higher than those of the gases, the speed of sound in them is higher. It is because liquids and solids are less compressible than gases, i.e. have much greater bulk modulus.

Newton’s Formula for Velocity of Sound

In the case of an ideal gas, the relation between pressure P and volume V is given by

PV = NkBT 


N is the number of molecules in volume V,

kB is the Boltzmann constant and

T the temperature of the gas (in Kelvin).

For an isothermal change

VΔP + PΔV = 0


 B = P

Hence for an ideal gas under isothermal conditions,

 v = Pρ

Laplace’s Correction for Velocity of Sound

Laplace pointed out that the pressure variations in the propagation of sound waves are so fast that there is little time for the heat flow to maintain constant temperature. These variations, therefore, are adiabatic and not isothermal. For adiabatic processes the ideal gas satisfies the relation,

PVγ = constant 

 Δ(PVγ) = 0

 P γVγ 1 ΔV + Vγ ΔP = 0


Thus for an ideal gas the adiabatic bulk modulus is given by,

Bad=γ P

Therefore, velocity of longitudinal wave in gas should be,

v = γPρ

Factors Affecting Velocity of Longitudinal (Sound) Waves

  1. Effect of Pressure on Velocity of Sound

    Velocity of longitudinal wave in gas is given by,

    v = γPρ

    But, Pρ remains constant at constant temperature, hence, there is no effect of pressure on velocity of longitudinal wave at constant pressure.

  2. Effect of Temperature on Velocity of Sound

    Velocity of longitudinal wave in gas is given by,

    v = γPρ

    Now from ideal gas law

    PV = nRT nV=PRT 

    And also,



    v = γPPMRT

    v = γRTM   

    Or Velocity of sound in a gas is directly proportional to the square root of its absolute temperature.

  3. Effect of Density on Velocity of Sound

    The velocity of sound in a gas is inversely proportional to the square root of density of the gas.

    v ∝ 1ρ

     v1v2= ρ2ρ1

  4. Effect of Humidity on Velocity of Sound

    The velocity of sound increases with increase in humidity in air.


When two or more waves are simultaneously travelling through a medium, the resultant displacement of each particle of the medium at any instant is equal to vector sum of the displacements produced by the individual waves separately. This principle is called principle of superposition.

If y1(x,t) and y2(x,t) are two waves, then, their resultant wave is given by,

y (x, t) = y1(x, t) + y2(x, t)

The principle implies that the overlapping waves do not, in any way, alter the travel of each other.

Consider two waves

y1 = A sin (kx  ωt)


y2= A sin (kx  ωt + ϕ) 

having phase difference of ϕ

Applying the superposition principle, we get,

y x, t =A sin (kx  ωt) + A sin (kx  ωt + ϕ)

Using the trigonometric relation,

sinα+sinβ=2sinα + β2 cosα  β2 

We have,

y=2Acosϕ2 sinkx  ωt +ϕ2

The resultant wave is also a sinusoidal wave, travelling in the positive direction of x-axis.

The resultant wave differs from the constituent waves in two respects:

(1) Its phase angle is ϕ2 and

(2) Its amplitude is

A(ϕ) = 2A cos ϕ2

If ϕ = 0, i.e. the two waves are in phase,

A(0) = 2A sin (kx  ωt) 

The amplitude of the resultant wave is 2A, which is the largest possible value of A(ϕ).

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The boundary effect

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If the incident wave is

yi(x, t) = A sin (ωt - kx)

then, after reflection at a rigid boundary the reflected wave will be

yr(x, t) = A sin (ωt - kx + π) = - A sin (ωt - kx)

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The reflection at an open boundary takes place without any phase change.

yr(x, t) = A sin (ωt - kx)

If a wave is incident obliquely on the boundary between two different media the transmitted wave is called the refracted wave.

The incident and refracted waves obey Snell’s law of refraction, and the incident and reflected waves obey the usual laws of reflection.


The repetition of sound caused by the reflection of sound waves is called an echo.

Sound persists on ear for 0.1s. The minimum distance from a sound reflecting surface to hear an echo is approximately 17.2 m.

If first echo is heard after t1 s, second echo after t2 s, then third echo will be heard after (t1 + t2) s.

Interference of Sound Waves

Interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. At some points the intensity of the resultant wave is very large while at some other points it is very small or zero.

Standing Waves and Normal Modes

Standing wave, also called stationary wave is combination of two waves moving in opposite directions, each having the same amplitude and frequency. This phenomenon is the result of interference - that is, when waves are superimposed, their energies are either added together or cancelled out.

The points of maximum or minimum amplitude stay at one position.

Node and Antinode

The positions of zero amplitude in a standing wave are called nodes.

The positions corresponding to maximum amplitude in a standing wave are called antinodes.

The distance between two consecutive nodes is λ2 or half a wavelength.

The antinodes are separated by λ2 and are located half way between pairs of nodes.

Constructive Interference

For maximum amplitude = 2A, which occurs for the values of kx that gives,

|sin kx| = 1

kx = n +12π

For n = 0, 1, 2, 3, …

Substituting k = 2πλ in this equation, we get

x =n +12λ2 

For n = 0, 1, 2, 3, …

For two waves of amplitude A and B respectively,

Maximum amplitude = (A + B)

Intensity ∝ (Amplitude)2 ∝ (A + B)2

In general amplitude = A2 + B2+2AB cos ϕ

Destructive Interference

The amplitude is zero for values of kx that give,

sin kx= 0 

 kx = nπ

For n = 0, 1, 2, 3, …

Substituting k = 2πλ in this equation, we get

x = nλ2

For n = 0, 1, 2, 3, …

For destructive interference, phase difference between two waves = π, 3π, 5π

For two waves of amplitude A and B respectively,

Minimum amplitude = (A ~ B) = Difference of component amplitudes.

Intensity ∝ (Amplitude)2 ∝ (A – B)2

Modes of a stretched string fixed at both ends

The stretched string produces certain waves with certain natural frequencies. These frequencies are called normal modes.

Fundamental Note

It is the sound of lowest frequency produced in vibration of a system.


Tones having frequencies greater than the fundamental note are called overtones.


When the frequencies of overtone are integral multiples of the fundamental, then they are known as harmonics.

The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic.

The second harmonic is the oscillation mode with n = 2.

The third harmonic corresponds to n = 3 and so on.

The frequencies associated with these modes are labeled as ν1, ν2, ν3 and so on. The collection of all possible modes is called the harmonic series and n is called the harmonic number.

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For a stretched string of length L, fixed at both ends, the two ends of the string have to be nodes. [due to phase change at the reflecting surface]

If one of the ends is chosen as position x = 0, then the other end is x = L. In order that this end is a node; the length L must satisfy the condition

The frequencies corresponding to these wavelengths are

The frequencies produced by vibrating string fixed at both ends are in the ratio 1 : 2 : 3 : 4, …

Sitar and violin are designed on this principle (both ends tied to rigid supports)

Standing Waves in Organ Pipes

Organ pipes are cylindrical pipes which are used to produce musical (longitudial) sounds. Organ pipes are of two types

  1. Open Organ Pipe

    Cylindrical pipes open at both ends.

  2. Closed Organ Pipe

    Cylindrical pipes open at one end and closed at other end.

Frequency of vibrating air column in a closed pipe

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The open end, x = L, is an antinode, and the closed end is a node.

That is, L = n + ½λ2 

  λ=2Ln + ½

For n = 0, 1, 2, 3, …

The frequency will be, ν = n + ½v2L  





The ratio of frequencies is 1 : 3 : 5 : 7…

Hence they are called odd harmonics.

Frequency of vibrating air column in an open pipe

In the case of a pipe open at both ends, there will be antinodes at both ends, and all harmonics will be generated.

The open end, x = L,

That is, L = n λ2 


For n = 0, 1, 2, 3, …

The frequency will be, ν = n v2L  





The ratio of frequencies is 1 : 2 : 3 : 4…

Hence all harmonics will be produced.

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End Correction in Organ Pipe

Antinode is not obtained at exact open end but slightly above it. The distance between open and antinode is called end correction.

It is denoted by e.

Factors Affecting Frequency of Pipe

  1. Length of air column, ν ∝ 1L

  2. Radius of air column, ν ∝ 1r

  3. Temperature of air column, ν ∝ T

  4. Pressure of air inside air column, ν ∝ p

  5. Density of air, ν ∝ 1ρ

  6. Velocity of sound in air column, ν ∝ v

Resonance Tube

Resonance tube is a closed organ pipe in which length of air column can be changed by changing height of liquid column in it.


The phenomenon of wavering of sound intensity, when two waves of nearly same frequencies and amplitudes, travelling in the same direction, are superimposed on each other, is called beats.

Frequency of beat

Let the time dependent variations of the displacements due to two sound waves at a particular location be

where ω1 > ω2.

We have assumed, for simplicity, that the waves have same amplitude and phase.

According to the superposition principle, the resultant displacement is

Using trigonometric identity

cosα+cos β= 2cosα + β2cosα - β2

We have,

s = 2Acosω1+ω22tcosω1-ω22t

If we write,



ωb =ω1-ω22

We have,

If |ω1 ω2| << ω1 , ω2, ωa >> ωb, the main time dependence arises from cosine function whose angular frequency is ωa.

The quantity in the brackets can be regarded as the amplitude of this function (which is not a constant but, has a small variation of angular frequency ωb). It becomes maximum whenever cos ωbt has the value +1 or –1, which happens twice in each repetition of cosine function. Since ω1, ω2 and ωa are very close, they cannot be differentiated easily.

Thus, the result of superposition of two waves having nearly the same frequencies is a wave with nearly same angular frequency but its amplitude is not constant and the intensity of resultant sound varies with an angular frequency

Now ω = 2πν, therefore, the beat frequency,

Thus we hear a waxing and waning of sound with a frequency equal to the difference between the frequencies of the superposing waves.

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The number of maxima or minima heard in one second, when two waves of almost same frequency are superimposed, is called beat frequency and is equal to the difference of two frequencies.

[The difference of frequencies should not be more than 10. Sound persists on human ear drums for 0.1 second. Hence, beats will not be heard if the frequency difference exceeds 10]

Maximum amplitude = (A1 + A2)

Maximum intensity = (Maximum amplitude)2 = (A1 + A2)2


The phenomenon of apparent change in frequency of source due to relative motion of the source and observer is called Doppler’s effect.

When Source is Moving and Observer is at Rest

Let us take the velocity from the observer towards the source as positive.

Consider a source S moving with velocity vs and an observer who is stationary in a frame in which the medium is also at rest.

Let the speed of a wave of angular frequency ω and time period To, both measured by an observer at rest with respect to the medium, be v.

At time t = 0 the source is at point S1, located at a distance L from the observer, and emits a crest. This reaches the observer at time

 t1 =Lv

At time t = To the source has moved a distance vsTo from S1 and is at point S2, located at a distance (L+ vsTo) from the observer.

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At S2, the source emits a second crest. This reaches the observer at


At time nTo, the source emits its (n+1)th crest and this reaches the observer at time


Hence in a time interval

tn+1- t1= nTo+L+nvsTov-Lv

The observer detects n crests, or

T = nTo+L+nvsTov-Lvn

 ν= νo1+vsv-1 

If vsv, we can ignore higher order terms of  vsv in the binomial exapnsion

ν= νo1-vsv

If the surce is approaching the observer,

 ν= νo1+vsv 

When Source is at Rest and Observer is Moving

When observer is moving with velocity vo towards the source, which is at rest, the source and medium are approaching at speed vo and the speed with which the wave approaches is vo + v.

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The distance between the two consecutive pulses is voTo.

The relative speed of the pulse with respect to the observer = v + vo

Therefore the first pulse will be received by the observer at time

t1 =Lv+vo

2nd pulse will be received after


The (n+1)th pulse will be received at time,


The time interval between the arrival of the first and the (n+1)th crests is

tn+1- t1=nTo-nvoTovo+v   

 T=  To1-vovo+v


ν=1T=1To v+vov 


When Source and Observer Both are Moving

(a) When both object and source are moving in same direction along the direction of propagation of sound, then

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Let L be the distance between O1 and S1 at t = 0, when the source emits the first crest.

Since the observer is moving, the velocity of the wave relative to the observer is v + vo.

Therefore the first crest reaches the observer at time

t1 =Lv+vo

At time t = To, both the observer and the source have moved to their new positions O2 and S2 respectively. The new distance between the observer and the source, O2 S2, would be L + (vs - vo)To.

At S2, the source emits a second crest. This reaches the observer at time.

t2=To+ L+vs-voTovo+v

(n+1)th crest will be received at time,

tn+1=nTo+ L+nvs-voTovo+v

In time interval tn+1 – t1 the observer counts n crests

T= To1+vs-vovo+v=Tov+vsvo+v

 ν= νovo+vv+vs  

(b) If the observer is moving towards the source, vo has a positive (numerical) value whereas if O is moving away from S, vo has a negative value. On the other hand, if S is moving away from O, vs has a positive value whereas if it is moving towards O, vs has a negative value.

The sound emitted by the source travels in all directions. It is that part of sound coming towards the observer which the observer receives and detects.

Therefore the relative velocity of sound with respect to the observer is v + vo in all cases.

Applications of Doppler’s Effect

The measurement of Doppler shift has been used

  1. By police to check over speeding of vehicles.

  2. At airports to guide the aircraft.

  3. To study heart beats and blood flow in different parts of the body.

  4. By astrophysicist to measure the velocities of planets and stars.