CHAPTER FIFTEEN

WAVES

15.1 INTRODUCTION

In the previous Chapter, we studied the motion of objects

oscillating in isolation. What happens in a system, which is

a collection of such objects? A material medium provides

such an example. Here, elastic forces bind the constituents

to each other and, therefore, the motion of one affects that of

the other. If you drop a little pebble in a pond of still water,

the water surface gets disturbed. The disturbance does not

remain confined to one place, but propagates outward along

a circle. If you continue dropping pebbles in the pond, you

see circles rapidly moving outward from the point where the

water surface is disturbed. It gives a feeling as if the water is

moving outward from the point of disturbance. If you put

some cork pieces on the disturbed surface, it is seen that

the cork pieces move up and down but do not move away

from the centre of disturbance. This shows that the water

mass does not flow outward with the circles, but rather a

moving disturbance is created. Similarly, when we speak,

the sound moves outward from us, without any flow of air

from one part of the medium to another. The disturbances

produced in air are much less obvious and only our ears or

a microphone can detect them. These patterns, which move

without the actual physical transfer or flow of matter as a

whole, are called waves. In this Chapter, we will study such

waves.

Waves transport energy and the pattern of disturbance has

information that propagate from one point to another. All our

communications essentially depend on transmission of sig-

nals through waves. Speech means production of sound

waves in air and hearing amounts to their detection. Often,

communication involves different kinds of waves. For exam-

ple, sound waves may be first converted into an electric cur-

rent signal which in turn may generate an electromagnetic

wave that may be transmitted by an optical cable or via a

15.1 Introduction

15.2 Transverse and

longitudinal waves

15.3 Displacement relation in a

progressive wave

15.4 The speed of a travelling

wave

15.5 The principle of

superposition of waves

15.6 Reflection of waves

15.7 Beats

15.8 Doppler effect

Summary

Points to ponder

Exercises

Additional exercises

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PHYSICS368

satellite. Detection of the original signal will usu-

ally involve these steps in reverse order.

Not all waves require a medium for their

propagation. We know that light waves can

travel through vacuum. The light emitted by

stars, which are hundreds of light years away,

reaches us through inter-stellar space, which

is practically a vacuum.

The most familiar type of waves such as waves

on a string, water waves, sound waves, seismic

waves, etc. is the so-called mechanical waves.

These waves require a medium for propagation,

they cannot propagate through vacuum. They

involve oscillations of constituent particles and

depend on the elastic properties of the medium.

The electromagnetic waves that you will learn

in Class XII are a different type of wave.

Electromagnetic waves do not necessarily require

a medium - they can travel through vacuum.

Light, radiowaves, X-rays, are all electromagnetic

waves. In vacuum, all electromagnetic waves

have the same speed c, whose value is :

c = 299, 792, 458 ms

–1

. (15.1)

A third kind of wave is the so-called Matter

waves. They are associated with constituents of

matter : electrons, protons, neutrons, atoms and

molecules. They arise in quantum mechanical

description of nature that you will learn in your

later studies. Though conceptually more abstract

than mechanical or electro-magnetic waves, they

have already found applications in several

devices basic to modern technology; matter

waves associated with electrons are employed

in electron microscopes.

In this chapter we will study mechanical

waves, which require a material medium for

their propagation.

The aesthetic influence of waves on art and

literature is seen from very early times; yet the

first scientific analysis of wave motion dates back

to the seventeenth century. Some of the famous

scientists associated with the physics of wave

motion are Christiaan Huygens (1629-1695),

Robert Hooke and Isaac Newton. The

understanding of physics of waves followed the

physics of oscillations of masses tied to springs

and physics of the simple pendulum. Waves in

elastic media are intimately connected with

harmonic oscillations. (Stretched strings, coiled

springs, air, etc., are examples of elastic media).

We shall illustrate this connection through

simple examples.

Consider a collection of springs connected to

one another as shown in Fig. 15.1. If the spring

at one end is pulled suddenly and released, the

disturbance travels to the other end. What has

happened? The first spring is disturbed from its

equilibrium length. Since the second spring is

connected to the first, it is also stretched or

compressed, and so on. The disturbance moves

from one end to the other; but each spring only

executes small oscillations about its equilibrium

position. As a practical example of this situation,

consider a stationary train at a railway station.

Different bogies of the train are coupled to each

other through a spring coupling. When an

engine is attached at one end, it gives a push to

the bogie next to it; this push is transmitted from

one bogie to another without the entire train

being bodily displaced.

Now let us consider the propagation of sound

waves in air. As the wave passes through air, it

compresses or expands a small region of air. This

causes a change in the density of that region,

say

δρ

, this change induces a change in pressure,

p, in that region. Pressure is force per unit area,

so there is a restoring force proportional to

the disturbance, just like in a spring. In this

case, the quantity similar to extension or

compression of the spring is the change in

density. If a region is compressed, the molecules

in that region are packed together, and they tend

to move out to the adjoining region, thereby

increasing the density or creating compression

in the adjoining region. Consequently, the air

in the first region undergoes rarefaction. If a

region is comparatively rarefied the surrounding

air will rush in making the rarefaction move to

the adjoining region. Thus, the compression or

rarefaction moves from one region to another,

making the propagation of a disturbance

possible in air.

Fig. 15.1 A collection of springs connected to each

other. The end A is pulled suddenly

generating a disturbance, which then

propagates to the other end.

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WAVES 369

In solids, similar arguments can be made. In

a crystalline solid, atoms or group of atoms are

arranged in a periodic lattice. In these, each

atom or group of atoms is in equilibrium, due to

forces from the surrounding atoms. Displacing

one atom, keeping the others fixed, leads to

restoring forces, exactly as in a spring. So we

can think of atoms in a lattice as end points,

with springs between pairs of them.

In the subsequent sections of this chapter

we are going to discuss various characteristic

properties of waves.

15.2 TRANSVERSE AND LONGITUDINAL

WAVES

We have seen that motion of mechanical waves

involves oscillations of constituents of the

medium. If the constituents of the medium

oscillate perpendicular to the direction of wave

propagation, we call the wave a transverse wave.

If they oscillate along the direction of wave

propagation, we call the wave a longitudinal

wave.

Fig.15.2 shows the propagation of a single

pulse along a string, resulting from a single up

and down jerk. If the string is very long compared

position as the pulse or wave passes through

them. The oscillations are normal to the

direction of wave motion along the string, so this

is an example of transverse wave.

We can look at a wave in two ways. We can fix

an instant of time and picture the wave in space.

This will give us the shape of the wave as a

whole in space at a given instant. Another way

is to fix a location i.e. fix our attention on a

particular element of string and see its

oscillatory motion in time.

Fig. 15.4 describes the situation for

longitudinal waves in the most familiar example

of the propagation of sound waves. A long pipe

filled with air has a piston at one end. A single

sudden push forward and pull back of the piston

will generate a pulse of condensations (higher

density) and rarefactions (lower density) in the

medium (air). If the push-pull of the piston is

continuous and periodic (sinusoidal), a

Fig. 15.3 A harmonic (sinusoidal) wave travelling

along a stretched string is an example of a

transverse wave. An element of the string

in the region of the wave oscillates about

its equilibrium position perpendicular to the

direction of wave propagation.

Fig. 15.2 When a pulse travels along the length of a

stretched string (x-direction), the elements

of the string oscillate up and down (y-

direction)

to the size of the pulse, the pulse will damp out

before it reaches the other end and reflection

from that end may be ignored. Fig. 15.3 shows a

similar situation, but this time the external

agent gives a continuous periodic sinusoidal up

and down jerk to one end of the string. The

resulting disturbance on the string is then a

sinusoidal wave. In either case the elements of

the string oscillate about their equilibrium mean

Fig. 15.4 Longitudinal waves (sound) generated in a

pipe filled with air by moving the piston up

and down. A volume element of air oscillates

in the direction parallel to the direction of

wave propagation.

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PHYSICS370

sinusoidal wave will be generated propagating

in air along the length of the pipe. This is clearly

an example of longitudinal waves.

The waves considered above, transverse or

longitudinal, are travelling or progressive waves

since they travel from one part of the medium

to another. The material medium as a whole

does not move, as already noted. A stream, for

example, constitutes motion of water as a whole.

In a water wave, it is the disturbance that moves,

not water as a whole. Likewise a wind (motion

of air as a whole) should not be confused with a

sound wave which is a propagation of

disturbance (in pressure density) in air, without

the motion of air medium as a whole.

In transverse waves, the particle motion is

normal to the direction of propagation of the

wave. Therefore, as the wave propagates, each

element of the medium undergoes a shearing

strain. Transverse waves can, therefore, be

propagated only in those media, which can

sustain shearing stress, such as solids and not

in fluids. Fluids, as well as, solids can sustain

compressive strain; therefore, longitudinal

waves can be propagated in all elastic media.

For example, in medium like steel, both

transverse and longitudinal waves can

propagate, while air can sustain only

longitudinal waves. The waves on the surface

of water are of two kinds: capillary waves and

gravity waves. The former are ripples of fairly

short wavelength—not more than a few

centimetre—and the restoring force that

produces them is the surface tension of water.

Gravity waves have wavelengths typically

ranging from several metres to several hundred

meters. The restoring force that produces these

waves is the pull of gravity, which tends to keep

the water surface at its lowest level. The

oscillations of the particles in these waves are

not confined to the surface only, but extend with

diminishing amplitude to the very bottom. The

particle motion in water waves involves a

complicated motion—they not only move up and

down but also back and forth. The waves in an

ocean are the combination of both longitudinal

and transverse waves.

It is found that, generally, transverse and

longitudinal waves travel with different speed

in the same medium.

u Example 15.1 Given below are some

examples of wave motion. State in each case

if the wave motion is transverse, longitudinal

or a combination of both:

) Motion of a kink in a longitudinal spring

produced by displacing one end of the

spring sideways.

(b) Waves produced in a cylinder

containing a liquid by moving its piston

back and forth.

in water.

(d) Ultrasonic waves in air produced by a

vibrating quartz crystal.

Answer

(a) Transverse and longitudinal

(b) Longitudinal

(d) Longitudinal t

15.3 DISPLACEMENT RELATION IN

A PROGRESSIVE WAVE

For mathematical description of a travelling

wave, we need a function of both position x and

time t. Such a function at every instant should

give the shape of the wave at that instant. Also,

at every given location, it should describe the

motion of the constituent of the medium at that

location. If we wish to describe a sinusoidal

travelling wave (such as the one shown in Fig.

15.3) the corresponding function must also be

sinusoidal. For convenience, we shall take the

wave to be transverse so that if the position of

the constituents of the medium is denoted by x,

the displacement from the equilibrium position

may be denoted by y. A sinusoidal travelling

wave is then described by:

( , ) sin( )

= −ω + φ

y x t a kx t

(15.2)

The term

in the argument of sine function

means equivalently that we are considering a

linear combination of sine and cosine functions:

( , ) sin( ) cos( )

y x t A kx t B kx t

ω ω

= − + −

(15.3)

From Equations (15.2) and (15.3),

2 2

a A B

= +

and

tan

−













To understand why Equation (15.2)

represents a sinusoidal travelling wave, take a

fixed instant, say t = t

. Then, the argument of

the sine function in Equation (15.2) is simply

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WAVES 371

kx + constant. Thus, the shape of the wave (at

any fixed instant) as a function of

is a sine

wave. Similarly, take a fixed location, say x = x

Then, the argument of the sine function in

Equation (15.2) is constant -ωt. The

displacement y, at a fixed location, thus, varies

sinusoidally with time. That is, the constituents

of the medium at different positions execute

simple harmonic motion. Finally, as t increases,

x must increase in the positive direction to keep

kx –

t + φ constant. Thus, Eq. (15.2) represents

a sinusiodal (harmonic) wave travelling along

the positive direction of the x-axis. On the other

hand, a function

( , ) sin( )

= + ω + φ

y x t a kx t

(15.4)

represents a wave travelling in the negative

direction of x-axis. Fig. (15.5) gives the names of

the various physical quantities appearing in Eq.

(15.2) that we now interpret.

Fig. 15.6 shows the plots of Eq. (15.2) for

different values of time differing by equal

intervals of time. In a wave, the crest is the

point of maximum positive displacement, the

trough is the point of maximum negative

displacement. To see how a wave travels, we

can fix attention on a crest and see how it

progresses with time. In the figure, this is

shown by a cross (×) on the crest. In the same

manner, we can see the motion of a particular

constituent of the medium at a fixed location,

say at the origin of the x-axis. This is shown

by a solid dot (•). The plots of Fig. 15.6 show

that with time, the solid dot (•) at the origin

moves periodically, i.e., the particle at the

origin oscillates about its mean position as

the wave progresses. This is true for any other

location also. We also see that during the time

the solid dot (•) has completed one full

oscillation, the crest has moved further by a

certain distance.

Using the plots of Fig. 15.6, we now define

the various quantities of Eq. (15.2).

15.3.1 Amplitude and Phase

In Eq. (15.2), since the sine function varies

between 1 and –1, the displacement y (x,t) varies

between a and –a. We can take a to be a positive

constant, without any loss of generality. Then,

a represents the maximum displacement of the

constituents of the medium from their

equilibrium position. Note that the displacement

y may be positive or negative, but a is positive.

It is called the amplitude of the wave.

The quantity (kx –

t +

) appearing as the

argument of the sine function in Eq. (15.2) is

called the phase of the wave. Given the

amplitude a, the phase determines the

displacement of the wave at any position and

at any instant. Clearly

is the phase at x = 0

and t = 0. Hence,

is called the initial phase

angle. By suitable choice of origin on the x-axis

and the intial time, it is possible to have

= 0.

Thus there is no loss of generality in dropping

, i.e., in taking Eq. (15.2) with

= 0.

Fig. 15.5 The meaning of standard symbols in

Eq. (15.2)

y(x,t) : displacement as a function of

position x and time t

a : amplitude of a wave

: angular frequency of the wave

k : angular wave number

kx–

: initial phase angle (a+x = 0, t = 0)

Fig. 15.6 A harmonic wave progressing along the

positive direction of x-axis at different times.

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PHYSICS372

15.3.2 Wavelength and Angular Wave

Number

The minimum distance between two points

having the same phase is called the wavelength

of the wave, usually denoted by λ. For simplicity,

we can choose points of the same phase to be

crests or troughs. The wavelength is then the

distance between two consecutive crests or

troughs in a wave. Taking

= 0 in Eq. (15.2),

the displacement at t = 0 is given by

( , 0) sin

y x a kx

(15.5)

Since the sine function repeats its value after

every 2π change in angle,

sin sin( ) sinkx kx n k x

= + = +













That is the displacements at points x and at

are the same, where n=1,2,3,... The 1east

distance between points with the same

displacement (at any given instant of time) is

obtained by taking n = 1.

is then given by

(15.6)

k is the angular wave number or propagation

constant; its SI unit is radian per metre or

rad m

−

15.3.3 Period, Angular Frequency and

Frequency

Fig. 15.7 shows again a sinusoidal plot. It

describes not the shape of the wave at a certain

instant but the displacement of an element (at

any fixed location) of the medium as a function

of time. We may for, simplicity, take Eq. (15.2)

with φ = 0 and monitor the motion of the element

say at

. We then get

(0, ) sin( )

y t a t

= −

sin

a t

= −

Now, the period of oscillation of the wave is the

time it takes for an element to complete one full

oscillation. That is

sin sin ( T)

a t a t

ω ω

− = − +

sin( T)

a t

ω ω

= − +

Since sine function repeats after every

T 2

ω π

(15.7)

is called the angular frequency of the wave.

Its SI unit is rad s

–1

. The frequency ν is the

number of oscillations per second. Therefore,

T 2

= =

(15.8)

is usually measured in hertz.

In the discussion above, reference has always

been made to a wave travelling along a string or

a transverse wave. In a longitudinal wave, the

displacement of an element of the medium is

parallel to the direction of propagation of the

wave. In Eq. (15.2), the displacement function

for a longitudinal wave is written as,

s(x, t) = a sin (kx –

t +

) (15.9)

where s(x, t) is the displacement of an element

of the medium in the direction of propagation

of the wave at position x and time t. In Eq. (15.9),

is the displacement amplitude; other

quantities have the same meaning as in case

of a transverse wave except that the

displacement function y (x, t) is to be replaced

by the function s (x, t).

* Here again, ‘radian’ could be dropped and the units could be written merely as m

–1

. Thus, k represents 2

times the number of waves (or the total phase difference) that can be accommodated per unit length, with SI

units m

–1

Fig. 15.7 An element of a string at a fixed location

oscillates in time with amplitude a and

period T, as the wave passes over it.

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WAVES 373

u Example 15.2 A wave travelling along a

string is described by,

y(x, t) = 0.005 sin (80.0 x – 3.0 t),

in which the numerical constants are in

SI units (0.005 m, 80.0 rad m

–1

, and

3.0 rad s

–1

). Calculate (a) the amplitude,

(b) the wavelength, and (c) the period and

frequency of the wave. Also, calculate the

displacement y of the wave at a distance

x = 30.0 cm and time t = 20 s ?

Answer On comparing this displacement

equation with Eq. (15.2),

y (x, t) = a sin (kx –

t),

we find

(a) the amplitude of the wave is 0.005 m = 5 mm.

(b) the angular wave number k and angular

frequency

are

k = 80.0 m

–1

and

= 3.0 s

–1

We, then, relate the wavelength

to k through

Eq. (15.6),

= 2π/k

80.0 m

−

= 7.85 cm

by the relation

T = 2π/

3.0 s

−

= 2.09 s

and frequency, v = 1/T = 0.48 Hz

The displacement y at x = 30.0 cm and

time t = 20 s is given by

y = (0.005 m) sin (80.0 × 0.3 – 3.0 × 20)

= (0.005 m) sin (–36 + 12π)

= (0.005 m) sin (1.699)

= (0.005 m) sin (97

) j 5 mm t

15.4 THE SPEED OF A TRAVELLING WAVE

To determine the speed of propagation of a

travelling wave, we can fix our attention on any

particular point on the wave (characterised by

some value of the phase) and see how that point

moves in time. It is convenient to look at the

motion of the crest of the wave. Fig. 15.8 gives

the shape of the wave at two instants of time,

which differ by a small time internal ∆t. The

entire wave pattern is seen to shift to the right

(positive direction of x-axis) by a distance ∆x. In

particular, the crest shown by a dot (

•

) moves a

distance ∆x in time ∆t. The speed of the wave is

then ∆x/∆t. We can put the dot (

•

) on a point

with any other phase. It will move with the same

speed v (otherwise the wave pattern will not

remain fixed). The motion of a fixed phase point

on the wave is given by

kx –

t = constant (15.10)

Thus, as time t changes, the position x of the

fixed phase point must change so that the phase

remains constant. Thus,

kx –

t = k(x+∆x) –

(t+∆t)

or k

∆

x –

∆t =0

Taking

∆

t vanishingly small, this gives

= =

t k

(15.11)

Relating ω to T and k to λ, we get

2 /

πν λ

λν

π λ

= = =v

(15.12)

Eq. (15.12), a general relation for all

progressive waves, shows that in the time

required for one full oscillation by any

constituent of the medium, the wave pattern

travels a distance equal to the wavelength of the

wave. It should be noted that the speed of a

mechanical wave is determined by the inertial

(linear mass density for strings, mass density

Fig. 15.8 Progression of a harmonic wave from time

t to t + ∆t. where ∆t is a small interval.

The wave pattern as a whole shifts to the

right. The crest of the wave (or a point with

any fixed phase) moves right by the distance

∆x in time ∆t.

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PHYSICS374

in general) and elastic properties (Young’s

modulus for linear media/ shear modulus, bulk

modulus) of the medium. The medium

determines the speed; Eq. (15.12) then relates

wavelength to frequency for the given speed. Of

course, as remarked earlier, the medium can

support both transverse and longitudinal waves,

which will have different speeds in the same

medium. Later in this chapter, we shall obtain

specific expressions for the speed of mechanical

waves in some media.

15.4.1 Speed of a Transverse Wave on

Stretched String

The speed of a mechanical wave is determined

by the restoring force setup in the medium when

it is disturbed and the inertial properties (mass

density) of the medium. The speed is expected to

be directly related to the former and inversely to

the latter. For waves on a string, the restoring

force is provided by the tension T in the string.

The inertial property will in this case be linear

mass density

, which is mass m of the string

divided by its length L. Using Newton’s Laws of

Motion, an exact formula for the wave speed on

a string can be derived, but this derivation is

outside the scope of this book. We shall,

therefore, use dimensional analysis. We already

know that dimensional analysis alone can never

yield the exact formula. The overall

dimensionless constant is always left

undetermined by dimensional analysis.

The dimension of

is [ML

–1

] and that of T is

like force, namely [MLT

–2

]. We need to combine

these dimensions to get the dimension of speed

[LT

–1

]. Simple inspection shows that the

quantity T/µ has the relevant dimension

MLT

L T

−

























2 2

Thus if T and µ are assumed to be the only

relevant physical quantities,

v = C

(15.13)

where C is the undetermined constant of

dimensional analysis. In the exact formula, it

turms out, C=1. The speed of transverse waves

on a stretched string is given by

v =

(15.14)

Note the important point that the speed v

depends only on the properties of the medium T

and

(T is a property of the stretched string

arising due to an external force). It does not

depend on wavelength or frequency of the wave

itself. In higher studies, you will come across

waves whose speed is not independent of

frequency of the wave. Of the two parameters

and

the source of disturbance determines the

frequency of the wave generated. Given the

Propagation of a pulse on a rope

You can easily see the motion of a pulse on a rope. You can also see

its reflection from a rigid boundary and measure its velocity of travel.

You will need a rope of diameter 1 to 3 cm, two hooks and some

weights. You can perform this experiment in your classroom or

laboratory.

Take a long rope or thick string of diameter 1 to 3 cm, and tie it to

hooks on opposite walls in a hall or laboratory. Let one end pass on

a hook and hang some weight (about 1 to 5 kg) to it. The walls may

be about 3 to 5 m apart.

Take a stick or a rod and strike the rope hard at a point near one

end. This creates a pulse on the rope which now travels on it. You

can see it reaching the end and reflecting back from it. You can

check the phase relation between the incident pulse and reflected

pulse. You can easily watch two or three reflections before the pulse

dies out. You can take a stopwatch and find the time for the pulse

to travel the distance between the walls, and thus measure its

velocity. Compare it with that obtained from Eq. (15.14).

This is also what happens with a thin metallic string of a musical instrument. The major difference is

that the velocity on a string is fairly high because of low mass per unit length, as compared to that on a

thick rope. The low velocity on a rope allows us to watch the motion and make measurements beautifully.

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WAVES 375

speed of the wave in the medium and the

frequency Eq. (15.12) then fixes the wavelength

(15.15)

u Example 15.3 A steel wire 0.72 m long has

a mass of 5.0 ×10

–3

kg. If the wire is under

a tension of 60 N, what is the speed of

transverse waves on the wire ?

Answer Mass per unit length of the wire,

m 72.0

kg 100.5

3−

= 6.9 ×10

–3

kg m

–1

Tension, T = 60 N

The speed of wave on the wire is given by

m 93

mkg 109.6

N 60

−

−−

== s

15.4.2 Speed of a Longitudinal Wave

(Speed of Sound)

In a longitudinal wave, the constituents of the

medium oscillate forward and backward in the

direction of propagation of the wave. We have

already seen that the sound waves travel in the

form of compressions and rarefactions of small

volume elements of air. The elastic property that

determines the stress under compressional

strain is the bulk modulus of the medium defined

by (see Chapter 9)

V/V

∆

= −

∆

(15.16)

Here, the change in pressure ∆P produces a

volumetric strain

∆

. B has the same dimension

as pressure and given in SI units in terms of

pascal (Pa). The inertial property relevant for the

propagation of wave is the mass density ρ, with

dimensions [ML

–3

]. Simple inspection reveals

that quantity B/ρ has the relevant dimension:

(15.17)

Thus, if B and

are considered to be the only

relevant physical quantities,

v = C

(15.18)

where, as before, C is the undetermined constant

from dimensional analysis. The exact derivation

shows that C=1. Thus, the general formula for

longitudinal waves in a medium is:

v =

(15.19)

For a linear medium, like a solid bar, the

lateral expansion of the bar is negligible and we

may consider it to be only under longitudinal

strain. In that case, the relevant modulus of

elasticity is Young’s modulus, which has the

same dimension as the Bulk modulus.

Dimensional analysis for this case is the same

as before and yields a relation like Eq. (15.18),

with an undetermined C, which the exact

derivation shows to be unity. Thus, the speed of

longitudinal waves in a solid bar is given by

v =

(15.20)

where Y is the Young’s modulus of the material

of the bar. Table 15.1 gives the speed of sound

in some media.

Table 15.1 Speed of Sound in some Media

Liquids and solids generally have higher speed

of sound than gases. [Note for solids, the speed

being referred to is the speed of longitudinal

waves in the solid]. This happens because they

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are much more difficult to compress than gases

and so have much higher values of bulk modulus.

Now, see Eq. (15.19). Solids and liquids have

higher mass densities (

) than gases. But the

corresponding increase in both the modulus (B)

of solids and liquids is much higher. This is the

reason why the sound waves travel faster in

solids and liquids.

We can estimate the speed of sound in a gas

in the ideal gas approximation. For an ideal gas,

the pressure P, volume V and temperature T are

related by (see Chapter 11).

PV = Nk

T (15.21)

where N is the number of molecules in volume

V, k

is the Boltzmann constant and T the

temperature of the gas (in Kelvin). Therefore, for

an isothermal change it follows from Eq.(15.21)

that

V∆P + P∆V = 0

V/V

∆

−

Hence, substituting in Eq. (15.16), we have

B = P

Therefore, from Eq. (15.19) the speed of a

longitudinal wave in an ideal gas is given by,

v =

(15.22)

This relation was first given by Newton and

is known as Newton’s formula.

u Example 15.4 Estimate the speed of

sound in air at standard temperature and

pressure. The mass of 1 mole of air is

29.0×10

–3

kg.

Answer We know that 1 mole of any gas

occupies 22.4 litres at STP. Therefore, density

of air at STP is:

= (mass of one mole of air)/ (volume of one

mole of air at STP)

3 3

29.0 10 kg

22.4 10 m

−

= 1.29 kg m

–3

According to Newton’s formula for the speed

of sound in a medium, we get for the speed of

sound in air at STP,

= 280 m s

–1

(15.23)

The result shown in Eq.(15.23) is about 15%

smaller as compared to the experimental value

of 331 m s

–1

as given in Table 15.1. Where

did wegowrong ? Ifweexaminethebasic

assumption made by Newton that the pressure

variations in a medium during propagation of

sound are isothermal, we find that this is not

correct. It was pointed out by Laplace 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 (see Section 12.8),

= constant

i.e. ∆(PV

) = 0

or P

–1

∆V + V

∆P = 0

where

is the ratio of two specific heats,

Thus, for an ideal gas the adiabatic bulk

modulus is given by,

V/V

∆

−

γP

The speed of sound is, therefore, from Eq.

(15.19), givenby,

v =

(15.24)

This modification of Newton’s formula is referred

to as the Laplace correction. For air

= 7/5. Now using Eq. (15.24) to estimate the speed

of sound in air at STP, we get a value 331.3 m s

–1

which agrees with the measured speed.

15.5 THE PRINCIPLE OF SUPERPOSITION

OF WAVES

What happens when two wave pulses travelling

in opposite directions cross each other

(Fig. 15.9)? It turns out that wave pulses

continue to retain their identities after they have

crossed. However, during the time they overlap,

the wave pattern is different from either of the

pulses. Figure 15.9 shows the situation when

two pulses of equal and opposite shapes move

towards each other. When the pulses overlap,

the resultant displacement is the algebraic sum

of the displacement due to each pulse. This is

known as the principle of superposition of waves.

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WAVES 377

According to this principle, each pulse moves

as if others are not present. The constituents of

the medium, therefore, suffer displacments due

to both and since the displacements can be

positive and negative, the net displacement is

an algebraic sum of the two. Fig. 15.9 gives

graphs of the wave shape at different times. Note

the dramatic effect in the graph (c); the

displacements due to the two pulses have exactly

cancelled each other and there is zero

displacement throughout.

To put the principle of superposition

mathematically, let y

(x,t) and y

(x,t) be the

displacements due to two wave disturbances in

the medium. If the waves arrive in a region

simultaneously, and therefore, overlap, the net

displacement y

(x,t) is given by

(x, t) = y

(x, t) + y

(x, t) (15.25)

If we have two or more waves moving in the

medium the resultant waveform is the sum of

wave functions of individual waves. That is, if

the wave functions of the moving waves are

= f

(x–vt),

= f

(x–vt),

..........

= f

(x–vt)

thenthewavefunctiondescribingthe

disturbance in the medium is

y = f

(x – vt)+ f

(x – vt)+ ...+ f

(x – vt)

( )

=1i

f x vt

= −

∑

(15.26)

The principle of superposition is basic to the

phenomenon of interference.

For simplicity, consider two harmonic

travelling waves on a stretched string, both with

the same

(angular frequency) and k (wave

number), and, therefore, the same wavelength

. Their wave speed will be identical. Let us

further assume that their amplitudes are equal

and they are both travelling in the positive

direction of x-axis. The waves only differ in their

initial phase. According to Eq. (15.2), the two

waves are described by the functions:

(x, t) = a sin (kx –

t) (15.27)

and y

(x, t) = a sin (kx –

t +

) (15.28)

The net displacement is then, by the principle

of superposition, given by

(x, t) = a sin (kx –

t) + a sin (kx –

t +

)

(15.29)

(

)

(

)

2sin cos

2 2

kx t kx t

ω ω φ

 

− + − +

 

 

 

(15.30)

where we have used the familiar trignometric

identity for

sin sin

A B

. We then have

( )

, 2 cos sin

2 2

y x t a kx t

φ φ

 

= − +

 

 

(15.31)

Eq. (15.31) is also a harmonic travelling wave in

the positive direction of x-axis, with the same

frequency and wavelength. However, its initial

phase angle is

. The significant thing is that

its amplitude is a function of the phase difference

φ between the constituent two waves:

) = 2a cos ½

(15.32)

For φ = 0, when the waves are in phase,

(

)

(

)

, 2 sin

y x t a kx t

= −

(15.33)

i.e., the resultant wave has amplitude 2a, the

largest possible value for A. For

φ π

, the

Fig. 15.9 Two pulses having equal and opposite

displacements moving in opposite

directions. The overlapping pulses add up

to zero displacement in curve (c).

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PHYSICS378

reflected. The phenomenon of echo is an example

of reflection by a rigid boundary. If the boundary

is not completely rigid or is an interface between

two different elastic media, the situation is some

what complicated. A part of the incident wave is

reflected and a part is transmitted into the

second medium. 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.

Fig. 15.11 shows a pulse travelling along a

stretched string and being reflected by the

boundary. Assuming there is no absorption of

energy by the boundary, the reflected wave has

the same shape as the incident pulse but it

suffers a phase change of π or 180

on reflection.

This is because the boundary is rigid and the

disturbance must have zero displacement at all

times at the boundary. By the principle of

superposition, this is possible only if the reflected

and incident waves differ by a phase of π, so that

the resultant displacement is zero. This

reasoning is based on boundary condition on a

rigid wall. We can arrive at the same conclusion

dynamically also. As the pulse arrives at the wall,

it exerts a force on the wall. By Newton’s Third

Law, the wall exerts an equal and opposite force

on the string generating a reflected pulse that

differs by a phase of π.

Fig. 15.11 Reflection of a pulse meeting a rigid boundary.

Fig. 15.10 The resultant of two harmonic waves of

equal amplitude and wavelength

according to the principle of superposition.

The amplitude of the resultant wave

depends on the phase difference

, which

is zero for (a) and

for (b)

waves are completely, out of phase and the

resultant wave has zero displacement

everywhere at all times

(x, t) = 0 (15.34)

Eq. (15.33) refers to the so-called constructive

interference of the two waves

where the amplitudes add up in

the resultant wave. Eq. (15.34)

is the case of destructive

intereference where the

amplitudes subtract out in the

resultant wave. Fig. 15.10

shows these two cases of

interference of waves arising

from the principle of

superposition.

15.6 REFLECTION OF

WAVES

So far we considered waves

propagating in an unbounded

medium. What happens if a

pulse or a wave meets a

boundary? If the boundary is

rigid, the pulse or wave gets

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WAVES 379

If on the other hand, the boundary point is

not rigid but completely free to move (such as in

the case of a string tied to a freely moving ring

on a rod), the reflected pulse has the same phase

and amplitude (assuming no energy dissipation)

as the incident pulse. The net maximum

displacement at the boundary is then twice the

amplitude of each pulse. An example of non- rigid

boundary is the open end of an organ pipe.

To summarise, a travelling wave or pulse

suffers a phase change of π on reflection at a

rigid boundary and no phase change on

reflection at an open boundary. To put this

mathematically, let the incident travelling wave

(

)

(

)

, sin

= −

y x t a kx t

At a rigid boundary, the reflected wave is given

(x, t) = a sin (kx –

t +

= – a sin (kx –

t) (15.35)

At an open boundary, the reflected wave is given

(x, t) = a sin (kx –

t + 0).

= a sin (kx –

t) (15.36)

Clearly, at the rigid boundary,

= + =

y y y

at all times.

15.6.1 Standing Waves and Normal Modes

We considered above reflection at one boundary.

But there are familiar situations (a string fixed

at either end or an air column in a pipe with

either end closed) in which reflection takes place

at two or more boundaries. In a string, for

example, a wave travelling in one direction will

get reflected at one end, which in turn will travel

and get reflected from the other end. This will

go on until there is a steady wave pattern set

up on the string. Such wave patterns are called

standing waves or stationary waves. To see this

mathematically, consider a wave travelling

along the positive direction of x-axis and a

reflected wave of the same amplitude and

wavelength in the negative direction of x-axis.

From Eqs. (15.2) and (15.4), with φ = 0, we get:

(x, t) = a sin (kx –

(x, t) = a sin (kx +

The resultant wave on the string is, according

to the principle of superposition:

(x, t) = y

(x, t) + y

(x, t)

= a [sin (kx –

t) + sin (kx +

t)]

Using the familiar trignometric identity

Sin (A+B) + Sin (A–B) = 2 sin A cosB we get,

(x, t) = 2a sin kx cos

t (15.37)

Note the important difference in the wave

pattern described by Eq. (15.37) from that

described by Eq. (15.2) or Eq. (15.4). The terms

kx and

t appear separately, not in the

combination kx -

t. The amplitude of this wave

is 2a sin kx. Thus, in this wave pattern, the

amplitude varies from point-to-point, but each

element of the string oscillates with the same

angular frequency

or time period. There is no

phase difference between oscillations of different

elements of the wave. The string as a whole

vibrates in phase with differing amplitudes at

different points. The wave pattern is neither

moving to the right nor to the left. Hence, they

are called standing or stationary waves. The

amplitude is fixed at a given location but, as

remarked earlier, it is different at different

locations. The points at which the amplitude is

zero (i.e., where there is no motion at all) are

nodes; the points at which the amplitude is the

largest are called antinodes. Fig. 15.12 shows

a stationary wave pattern resulting from

superposition of two travelling waves in

opposite directions.

The most significant feature of stationary

waves is that the boundary conditions constrain

the possible wavelengths or frequencies of

vibration of the system. The system cannot

oscillate with any arbitrary frequency (contrast

this with a harmonic travelling wave), but is

characterised by a set of natural frequencies or

normal modes of oscillation. Let us determine

these normal modes for a stretched string fixed

at both ends.

First, from Eq. (15.37), the positions of nodes

(where the amplitude is zero) are given by

sin kx = 0 .

which implies

kx = n

; n = 0, 1, 2, 3, ...

Since, k = 2

, we get

x =

; n = 0, 1, 2, 3, ... (15.38)

Clearly, the distance between any two

successive nodes is

In the same way, the

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PHYSICS380

positions of antinodes (where the amplitude is

the largest) are given by the largest value of sin

kx :

sin kx

= 1

which implies

kx = (n + ½)

; n = 0, 1, 2, 3, ...

With k = 2

, we get

x = (n + ½)

; n = 0, 1, 2, 3, ... (15.39)

Again the distance between any two consecutive

antinodes is

. Eq. (15.38) can be applied to

the case of a stretched string of length L fixed

at both ends. Taking one end to be at x = 0, the

boundary conditions are that x = 0 and x = L

are positions of nodes. The x = 0 condition is

already satisfied. The x = L node condition

requires that the length L is related to λ by

L = n

; n = 1, 2, 3, ... (15.40)

Thus, the possible wavelengths of stationary

waves are constrained by the relation

; n = 1, 2, 3, … (15.41)

with corresponding frequencies

v =

, for n = 1, 2, 3, (15.42)

We have thus obtained the natural frequencies

- the normal modes of oscillation of the system.

The lowest possible natural frequency of a

system is called its fundamental mode or the

first harmonic. For the stretched string fixed

at either end it is given by v =

, corresponding

to n = 1 of Eq. (15.42). Here v is the speed of

wave determined by the properties of the

medium. The n = 2 frequency is called the

second harmonic; n = 3 is the third harmonic

Fig. 15.12 Stationary waves arising from superposition of two harmonic waves travelling in opposite directions.

Note that the positions of zero displacement (nodes) remain fixed at all times.

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WAVES 381

and so on. We can label the various

harmonics by the symbol

( n = 1,

2, ...).

Fig. 15.13 shows the first six

harmonics of a stretched string

fixed at either end. A string need not

vibrate in one of these modes only.

Generally, the vibration of a string

will be a superposition of different

modes; some modes may be more

strongly excited and some less.

Musical instruments like sitar or

violin are based on this principle.

Where the string is plucked or

bowed, determines which modes are

more prominent than others.

Let us next consider normal

modes of oscillation of an air column

with one end closed and the other

open. A glass tube partially filled

with water illustrates this system.

The end in contact with water is a

node, while the open end is an

antinode. At the node the pressure

changes are the largest, while the

displacement is minimum (zero). At

the open end - the antinode, it is

just the other way - least pressure

change and maximum amplitude of

displacement. Taking the end in

contact with water to be x = 0, the

node condition (Eq. 15.38) is already

satisfied. If the other end x = L is an

antinode, Eq. (15.39) gives

L =

n +













, for n = 0, 1, 2, 3, …

The possible wavelengths are then restricted by

the relation :

( )

1 2

n + /

, for n = 0, 1, 2, 3,... (15.43)

The normal modes – the natural frequencies –

of the system are

ν =

n +













; n = 0, 1, 2, 3, ... (15.44)

The fundamental frequency corresponds to n = 0,

and is given by

. The higher frequencies

are odd harmonics, i.e., odd multiples of the

fundamental frequency : 3

, 5

, etc.

Fig. 15.14 shows the first six odd harmonics of

air column with one end closed and the other

open. For a pipe open at both ends, each end is

an antinode. It is then easily seen that an open

air column at both ends generates all harmonics

(See Fig. 15.15).

The systems above, strings and air columns,

can also undergo forced oscillations (Chapter

14). If the external frequency is close to one of

the natural frequencies, the system shows

resonance.

Fig. 15.13 The first six harmonics of vibrations of a stretched

string fixed at both ends.

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PHYSICS382

Normal modes of a circular membrane rigidly

clamped to the circumference as in a tabla are

determined by the boundary condition that no

point on the circumference of the membrane

vibrates. Estimation of the frequencies of normal

modes of this system is more complex. This

problem involves wave propagation in two

dimensions. However, the underlying physics is

the same.

u Example 15.5 A pipe, 30.0 cm long, is

open at both ends. Which harmonic mode

of the pipe resonates a 1.1 kHz source? Will

resonance with the same source be

observed if one end of the pipe is closed ?

Take the speed of sound in air as

330 m s

–1

Answer The first harmonic frequency is given

(open pipe)

where L is the length of the pipe. The frequency

of its nth harmonic is:

, for n = 1, 2, 3, ... (open pipe)

First few modes of an open pipe are shown in

Fig. 15.15.

For L = 30.0 cm, v = 330 m s

–1

330 (m s )

0.6 (m )

−

= 550 n s

–1

Clearly, a source of frequency 1.1 kHz will

resonate at v

, i.e. the second harmonic.

Now if one end of the pipe is closed (Fig. 15.15),

it follows from Eq. (14.50) that the fundamental

frequency is

(pipe closed at one end)

and only the odd numbered harmonics are

present :

, ν

, and so on.

For L = 30 cm and v = 330 m s

–1

, the

fundamental frequency of the pipe closed at one

end is 275 Hz and the source frequency

corresponds to its fourth harmonic. Since this

harmonic is not a possible mode, no resonance

will be observed with the source, the moment

one end is closed. t

15.7 BEATS

‘Beats’ is an interesting phenomenon arising

from interference of waves. When two harmonic

sound waves of close (but not equal) frequencies

Fig. 15.14 Normal modes of an air column open at

one end and closed at the other end. Only

the odd harmonics are seen to be possible.

Fundamental

or third fifth

first harmonic harmonic harmonic

seventh ninth eleventh

harmonic harmonic harmonic

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WAVES 383

are heard at the same time, we hear a sound of

similar frequency (the average of two close

frequencies), but we hear something else also.

We hear audibly distinct waxing and waning of

the intensity of the sound, with a frequency

equal to the difference in the two close

frequencies. Artists use this phenomenon often

while tuning their instruments with each other.

They go on tuning until their sensitive ears do

not detect any beats.

To see this mathematically, let us consider

two harmonic sound waves of nearly equal

angular frequency ω

and ω

and fix the location

to be x = 0 for convenience. Eq. (15.2) with a

suitable choice of phase (φ = π/2 for each) and,

assuming equal amplitudes, gives

= a

cos

t and s

= a

cos

t (15.45)

Here we have replaced the symbol y by s,

since we are referring to longitudinal not

transverse displacement. Let ω

be the (slightly)

greater of the two frequencies. The resultant

displacement is, by the principle of

superposition,

s = s

+ s

= a (cos

t + cos

Using the familiar trignometric identity for

cos A + cosB, we get

(

)

(

)

1 2 1 2

2 cos cos

2 2

t t

ω ω ω ω

− +

(15.46)

which may be written as :

s = [2 a cos

t] cos

t (15.47)

If |

–

| <<

, th

where

(

)

1 2

ω ω

−

and

(

)

1 2

ω ω

Now if we assume |

–

| <<

which means

, we can interpret Eq. (15.47) as follows.

The resultant wave is oscillating with the average

angular frequency ω

; however its amplitude is

not constant in time, unlike a pure harmonic

wave. The amplitude is the largest when the

term cos

t takes its limit +1 or –1. In other

words, the intensity of the resultant wave waxes

and wanes with a frequency which is 2

–

Fig. 15.15 Standing waves in an open pipe, first four

harmonics are depicted.

Musical Pillars

Temples often have

some pillars

portraying human

figures playing

musical instru-

ments, but seldom

do these pillars

themselves produce

music. At the

Nellaiappar temple

in Tamil Nadu,

gentle taps on a

cluster of pillars carved out of a single piece

of rock produce the basic notes of Indian

classical music, viz. Sa, Re, Ga, Ma, Pa, Dha,

Ni, Sa. Vibrations of these pillars depend on

elasticity of the stone used, its density and

shape.

Musical pillars are categorised into three

types: The first is called the Shruti Pillar,

as it can produce the basic notes — the

“swaras”. The second type is the Gana

Thoongal, which generates the basic tunes

that make up the “ragas”. The third variety

is the Laya Thoongal pillars that produce

“taal” (beats) when tapped. The pillars at the

Nellaiappar temple are a combination of the

Shruti and Laya types.

Archaeologists date the Nelliappar

temple to the 7th century and claim it was

built by successive rulers of the Pandyan

dynasty.

The musical pillars of Nelliappar and

several other temples in southern India like

those at Hampi (picture), Kanyakumari, and

Thiruvananthapuram are unique to the

country and have no parallel in any other

part of the world.

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PHYSICS384

decreases as it recedes away. When we

approach a stationary source of sound with high

speed, the pitch of the sound heard appears to

be higher than that of the source. As the

observer recedes away from the source, the

observed pitch (or frequency) becomes lower

than that of the source. This motion-related

frequency change is called Doppler effect. The

Austrian physicist Johann Christian Doppler

first proposed the effect in 1842. Buys Ballot in

Holland tested it experimentally in 1845.

Doppler effect is a wave phenomenon, it holds

not only for sound waves but also for

electromagnetic waves. However, here we shall

consider only sound waves.

We shall analyse changes in frequency under

three different situations: (1) observer is

Reflection of sound in an open

pipe

When a high

pressure pulse of

air travelling down

an open pipe

reaches the other

end, its momentum

drags the air out

into the open, where

pressure falls

rapidly to the

atmospheric

pressure. As a

result the air following after it in the tube is

pushed out. The low pressure at the end of

the tube draws air from further up the tube.

The air gets drawn towards the open end

forcing the low pressure region to move

upwards. As a result a pulse of high pressure

air travelling down the tube turns into a

pulse of low pressure air travelling up the

tube. We say a pressure wave has been

reflected at the open end with a change in

phase of 180

. Standing waves in an open

pipe organ like the flute is a result of this

phenomenon.

Compare this with what happens when

a pulse of high pressure air arrives at a

closed end: it collides and as a result pushes

the air back in the opposite direction. Here,

we say that the pressure wave is reflected,

with no change in phase.

Fig. 15.16 Superposition of two harmonic waves, one

of frequency 11 Hz (a), and the other of

frequency 9Hz (b), giving rise to beats of

frequency 2 Hz, as shown in (c).

. Since

= 2

πν

, the beat frequency ν

beat

, is

given by

beat

= ν

– ν

(15.48)

Fig. 15.16 illustrates the phenomenon of

beats for two harmonic waves of frequencies 11

Hz and 9 Hz. The amplitude of the resultant wave

shows beats at a frequency of 2 Hz.

u Example 15.6 Two sitar strings A and B

playing the note ‘Dha’ are slightly out of

tune and produce beats of frequency 5 Hz.

The tension of the string B is slightly

increased and the beat frequency is found

to decrease to 3 Hz. What is the original

frequency of B if the frequency of A is

427 Hz ?

Answer Increase in the tension of a string

increases its frequency. If the original frequency

of B (

) were greater than that of A (

), further

increase in

should have resulted in an

increase in the beat frequency. But the beat

frequency is found to decrease. This shows that

. Since

–

= 5 Hz, and

= 427 Hz, we

get

= 422 Hz. t

15.8 DOPPLER EFFECT

It is an everyday experience that the pitch (or

frequency) of the whistle of a fast moving train

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WAVES 385

stationary but the source is moving, (2) observer

is moving but the source is stationary, and (3)

both the observer and the source are moving.

The situations (1) and (2) differ from each other

because of the absence or presence of relative

motion between the observer and the medium.

Most waves require a medium for their

propagation; however, electromagnetic waves do

not require any medium for propagation. If there

is no medium present, the Doppler shifts are

same irrespective of whether the source moves

or the observer moves, since there is no way of

distinction between the two situations.

15.8.1 Source Moving ; Observer Stationary

Let us choose the convention to take the

direction from the observer to the source as

the positive direction of velocity. Consider a

source S moving with velocity v

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

and period T

both

measured

by an observer at rest with respect to

the medium, be v. We assume that the observer

has a detector that counts every time a wave

crest reaches it. As shown in

Fig. 15.17, at time t = 0 the source is at point S

located at a distance L from the observer, and

emits a crest. This reaches the observer at time

= L/v. At time t = T

the source has moved a

distance v

and is at point S

, located at a

distance (L + v

) from the observer. At S

, the

source emits a second crest. This reaches the

observer at

(

)

s 0

2 0

L T

t T

= +

At time n T

, the source emits its (n+1)

crest

and this reaches the observer at time

(

)

s 0

+1 0

L n T

t n T

= +

Hence, in a time interval

L nv T

( )

−













the observer’s detector counts n crests and the

observer records the period of the wave as T

given by

T nT

L n T

s 0

= +

( )

−













s 0

v T

 

 

 

(15.49)

Equation (15.49) may be rewritten in terms

of the frequency v

that would be measured if

the source and observer were stationary, and

the frequency v observed when the source is

moving, as

v =

−

 

 

 

(15.50)

If v

is small compared with the wave speed v,

taking binomial expansion to terms in first order

in v

/v and neglecting higher power, Eq. (15.50)

may be approximated, giving

v =

1 –

 

 

 

(15.51)

For a source approaching the observer, we

replace v

by – v

to get

v =

 

 

 

(15.52)

The observer thus measures a lower frequency

when the source recedes from him than he does

when it is at rest. He measures a higher

frequency when the source approaches him.

15.8.2 Observer Moving; Source

Stationary

Now to derive the Doppler shift when the

observer is moving with velocity v

towards the

source and the source is at rest, we have to

proceed in a different manner. We work in the

Fig. 15.17 Doppler effect (change in frequency of

wave) detected when the source is moving

and the observer is at rest in the medium.

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PHYSICS386

reference frame of the moving observer. In this

reference frame the source and medium are

approaching at speed v

and the speed with

which the wave approaches is v

+ v. Following

a similar procedure as in the previous case, we

find that the time interval between the arrival

of the first and the (n+1) th crests is

0 0

+1 1 0

nv T

t t n T

v v

− = −

The observer thus, measures the period of the

wave to be













–1

= +













–

giving

v =

1 +













(15.53)

is small, the Doppler shift is almost same

whether it is the observer or the source moving

since Eq. (15.53) and the approximate relation

Eq. (15.51 ) are the same.

15.8.3 Both Source and Observer Moving

We will now derive a general expression for

Doppler shift when both the source and the

observer are moving. As before, let us take the

direction from the observer to the source as the

positive direction. Let the source and the

observer be moving with velocities v

and v

respectively as shown in Fig.15.18. Suppose at

time t = 0, the observer is at O

and the source

is at S

, O

being to the left of S

. The source

emits a wave of velocity v, of frequency v and

period T

all measured by an observer at rest

with respect to the medium. Let L be the

distance between O

and S

at t = 0, when the

source emits the first crest. Now, since the

observer is moving, the velocity of the wave

relative to the observer is v+v

Therefore, the

first crest reaches the observer at time t

= L/

(v+v

). At time t = T

, both the observer and the

source have moved to their new positions O

and

respectively. The new distance between the

observer and the source, O

, would be

L+(v

–v

) T

]. At S

, the source emits a

second crest.

Application of Doppler effect

The change in frequency caused by a moving object

due to Doppler effect is used to measure their

velocities in diverse areas such as military,

medical science, astrophysics, etc. It is also used

by police to check over-speeding of vehicles.

A sound wave or electromagnetic wave of

known frequency is sent towards a moving object.

Some part of the wave is reflected from the object

and its frequency is detected by the monitoring

station. This change in frequency is called Doppler

shift.

It is used at airports to guide aircraft, and in

the military to detect enemy aircraft.

Astrophysicists use it to measure the velocities

of stars.

Doctors use it to study heart beats and blood

flow in different parts of the body. Here they use

ulltrasonic waves, and in common practice, it is

called sonography. Ultrasonic waves enter the

body of the person, some of them are reflected

back, and give information about motion of blood

and pulsation of heart valves, as well as pulsation

of the heart of the foetus. In the case of heart,

the picture generated is called echocardiogram.

Fig. 15.18 Doppler effect when both the source and

observer are moving with different

velocities.

This reaches the observer at time.

= T

+ [L + (v

– v

)] /(v + v

)

At time nT

the source emits its (n+1) th crest

and this reaches the observer at time

n+1

= nT

+ [L + n (v

– v

)] /(v + v

)

Hence, in a time interval t

n+1

–t

, i.e.,

+ [L + n (v

– v

)] /(v + v

) – L /(v + v

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WAVES 387

the observer counts n crests and the observer

records the period of the wave as equal to T given by

0 0

s o s

v v v v

T T T

v v v v

   

= + =

   

+ +

   

(15.54)

The frequency v observed by the observer is

given by

(15.55)

Consider a passenger sitting in a train moving

on a straight track. Suppose she hears a whistle

sounded by the driver of the train. What

frequency will she measure or hear? Here both

the observer and the source are moving with

the same velocity, so there will be no shift in

frequency and the passenger will note the

natural frequency. But an observer outside who

is stationary with respect to the track will note

a higher frequency if the train is approaching

him and a lower frequency when it recedes

from him.

Note that we have defined the direction from

the observer to the source as the positive

direction. Therefore, if the observer is moving

towards the source, v

has a positive (numerical)

value whereas if O is moving away from S, v

has a negative value. On the other hand, if S is

moving away from O, v

has a positive value

whereas if it is moving towards O, v

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+v

in all cases.

u Example 15.7 A rocket is moving at a

speed of 200 m s

–1

towards a stationary

target. While moving, it emits a wave of

frequency 1000 Hz. Some of the sound

reaching the target gets reflected back to the

rocket as an echo. Calculate (1) the

frequency of the sound as detected by the

target and (2) the frequency of the echo as

detected by the rocket.

Answer (1) The observer is at rest and the

source is moving with a speed of 200 m s

–1

. Since

this is comparable with the velocity of sound,

330 m s

–1

, we must use Eq. (15.50) and not the

approximate Eq. (15.51). Since the source is

approaching a stationary target, v

= 0, and v

must be replaced by –v

Thus, we have

−













−=

v = 1000 Hz × [1 – 200 m s

–1

/330 m s

–1

]

–1

j 2540 Hz

(2) The target is now the source (because it is

the source of echo) and the rocket’s detector is

now the detector or observer (because it detects

echo). Thus, v

= 0 and v

has a positive value.

The frequency of the sound emitted by the source

(the target) is v, the frequency intercepted by

the target and not v

. Therefore, the frequency

as registered by the rocket is

v′ =

v v

 

 

 

j 4080 Hz t

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PHYSICS388

SUMMARY

1. Mechanical waves can exist in material media and are governed by Newton’s Laws.

2. Transverse waves are waves in which the particles of the medium oscillate perpendicular

to the direction of wave propagation.

3. Longitudinal waves are waves in which the particles of the medium oscillate along the

direction of wave propagation.

4. Progressive wave is a wave that moves from one point of medium to another.

5. The displacement in a sinusoidal wave propagating in the positive x direction is given

y (x, t) = a sin (kx –

t +

)

where a is the amplitude of the wave, k is the angular wave number,

is the angular

frequency, (kx –

t +

) is the phase, and

is the phase constant or phase angle.

6. Wavelength

of a progressive wave is the distance between two consecutive points of

the same phase at a given time. In a stationary wave, it is twice the distance between

two consecutive nodes or antinodes.

7. Period T of oscillation of a wave is defined as the time any element of the medium

takes to move through one complete oscillation. It is related to the angular frequency

through the relation

T =

8. Frequency v of a wave is defined as 1/T and is related to angular frequency by

9. Speed of a progressive wave is given by

k T

ω λ

λν

= = =

10. The speed of a transverse wave on a stretched string is set by the properties of the

string. The speed on a string with tension T and linear mass density

v =

11. Sound waves are longitudinal mechanical waves that can travel through solids, liquids,

or gases. The speed v of sound wave in a fluid having bulk modulus B and density

The speed of longitudinal waves in a metallic bar is

For gases, since B = γP, the speed of sound is

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WAVES 389

12. When two or more waves traverse simultaneously in the same medium, the

displacement of any element of the medium is the algebraic sum of the displacements

due to each wave. This is known as the principle of superposition of waves

( )

y f x vt

= −

∑

13. Two sinusoidal waves on the same string exhibit interference, adding or cancelling

according to the principle of superposition. If the two are travelling in the same

direction and have the same amplitude a and frequency but differ in phase by a phase

constant

, the result is a single wave with the same frequency

(x, t) =

a kxcos sin

φ ω φ













− +













= 0 or an integral multiple of 2

, the waves are exactly in phase and the interference

is constructive; if

, they are exactly out of phase and the interference is destructive.

14. A travelling wave, at a rigid boundary or a closed end, is reflected with a phase reversal

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

For an incident wave

(x, t) = a sin (kx –

t )

the reflected wave at a rigid boundary is

(x, t) = – a sin (kx +

t )

For reflection at an open boundary

(x,t ) = a sin (kx +

15. The interference of two identical waves moving in opposite directions produces standing

waves. For a string with fixed ends, the standing wave is given by

(x, t) = [2a sin kx ] cos

Standing waves are characterised by fixed locations of zero displacement called nodes

and fixed locations of maximum displacements called antinodes. The separation between

two consecutive nodes or antinodes is

/2.

A stretched string of length L fixed at both the ends vibrates with frequencies given by

n v

n = 1, 2, 3, ...

The set of frequencies given by the above relation are called the normal modes of

oscillation of the system. 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 and so on.

A pipe of length L with one end closed and other end open (such as air columns)

vibrates with frequencies given by

( )

n ½

= +

, n = 0, 1, 2, 3, ...

The set of frequencies represented by the above relation are the normal modes of

oscillation of such a system. The lowest frequency given by v/4L is the fundamental

mode or the first harmonic.

16. A string of length L fixed at both ends or an air column closed at one end and open at

the other end or open at both the ends, vibrates with certain frequencies called their

normal modes. Each of these frequencies is a resonant frequency of the system.

17. Beats arise when two waves having slightly different frequencies,

and

comparable amplitudes, are superposed. The beat frequency is

beat

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PHYSICS390

18. The Doppler effect is a change in the observed frequency of a wave when the source (S)

or the observer (O) or both move(s) relative to the medium. For sound the observed

frequency

is given in terms of the source frequency

v = v

v v

 

 

 

here v is the speed of sound through the medium, v

is the velocity of observer relative

to the medium, and v

is the source velocity relative to the medium. In using this

formula, velocities in the direction OS should be treated as positive and those opposite

to it should be taken to be negative.

POINTS TO PONDER

1. A wave is not motion of matter as a whole in a medium. A wind is different from the

sound wave in air. The former involves motion of air from one place to the other. The

latter involves compressions and rarefactions of layers of air.

2. In a wave, energy and not the matter is transferred from one point to the other.

3. In a mechanical wave, energy transfer takes place because of the coupling through

elastic forces between neighbouring oscillating parts of the medium.

4. Transverse waves can propagate only in medium with shear modulus of elasticity,

Longitudinal waves need bulk modulus of elasticity and are therefore, possible in all

media, solids, liquids and gases.

5. In a harmonic progressive wave of a given frequency, all particles have the same

amplitude but different phases at a given instant of time. In a stationary wave, all

particles between two nodes have the same phase at a given instant but have different

amplitudes.

6. Relative to an observer at rest in a medium the speed of a mechanical wave in that

medium (v) depends only on elastic and other properties (such as mass density) of

the medium. It does not depend on the velocity of the source.

7. For an observer moving with velocity v

relative to the medium, the speed of a wave is

obviously different from v and is given by v ± v

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WAVES 391

EXERCISES

15.1 A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched

string is 20.0 m. If the transverse jerk is struck at one end of the string, how long

does the disturbance take to reach the other end?

15.2 A stone dropped from the top of a tower of height 300 m splashes into the water of

a pond near the base of the tower. When is the splash heard at the top given that

the speed of sound in air is 340 m s

–1

? (g = 9.8 m s

–2

)

15.3 A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the

tension in the wire so that speed of a transverse wave on the wire equals the speed

of sound in dry air at 20

C = 343 m s

–1

15.4 Use the formula

to explain why the speed of sound in air

(a) is independent of pressure,

(b) increases with temperature,

15.5 You have learnt that a travelling wave in one dimension is represented by a function

y = f (x, t) where x and t must appear in the combination x – v t or x + v t, i.e.

y = f (x ± v t). Is the converse true? Examine if the following functions for y can

possibly represent a travelling wave :

(a) (x – vt )

(b) log [(x + vt)/x

]

15.6 A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a

water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted

sound? Speed of sound in air is 340 m s

–1

and in water 1486 m s

–1

15.7 A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the

wavelength of sound in the tissue in which the speed of sound is 1.7 km s

–1

? The

operating frequency of the scanner is 4.2 MHz.

15.8 A transverse harmonic wave on a string is described by

y(x, t) = 3.0 sin (36 t + 0.018 x +

/4)

where x and y are in cm and t in s. The positive direction of x is from left to right.

(a) Is this a travelling wave or a stationary wave ?

If it is travelling, what are the speed and direction of its propagation ?

(b) What are its amplitude and frequency ?

(d) What is the least distance between two successive crests in the wave ?

15.9 For the wave described in Exercise 15.8, plot the displacement (y) versus (t) graphs

for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does

the oscillatory motion in travelling wave differ from one point to another: amplitude,

frequency or phase ?

15.10 For the travelling harmonic wave

y(x, t) = 2.0 cos 2

(10t – 0.0080 x + 0.35)

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PHYSICS392

where x and y are in cm and t in s. Calculate the phase difference between oscillatory

motion of two points separated by a distance of

(a) 4 m,

(b) 0.5 m,

(c)

/2,

(d) 3

15.11 The transverse displacement of a string (clamped at its both ends) is given by

y(x, t) = 0.06 sin













cos (120

where x and y are in m and t in s. The length of the string is 1.5 m and its mass is

3.0 ×10

–2

kg.

Answer the following :

(a) Does the function represent a travelling wave or a stationary wave?

(b) Interpret the wave as a superposition of two waves travelling in opposite

directions. What is the wavelength, frequency, and speed of each wave ?

15.12 (i) For the wave on a string described in Exercise 15.11, do all the points on the

string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain

your answers. (ii) What is the amplitude of a point 0.375 m away from one end?

15.13 Given below are some functions of x and t to represent the displacement (transverse

or longitudinal) of an elastic wave. State which of these represent (i) a travelling

wave, (ii) a stationary wave or (iii) none at all:

(a) y = 2 cos (3x) sin (10t)

(b)

y x vt = −2

(d) y = cos x sin t + cos 2x sin 2t

15.14 A wire stretched between two rigid supports vibrates in its fundamental mode with

a frequency of 45 Hz. The mass of the wire is 3.5 × 10

–2

kg and its linear mass density

is 4.0× 10

–2

kg m

–1

. What is (a) the speed of a transverse wave on the string, and

(b) the tension in the string?

15.15 A metre-long tube open at one end, with a movable piston at the other end, shows

resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when

the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the

temperature of the experiment. The edge effects may be neglected.

15.16 A steel rod 100 cm long is clamped at its middle. The fundamental frequency of

longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of

sound in steel?

15.17 A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is

resonantly excited by a 430 Hz source ? Will the same source be in resonance with

the pipe if both ends are open? (speed of sound in air is 340 m s

–1

15.18 Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce

beats of frequency 6 Hz. The tension in the string A is slightly reduced and the

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WAVES 393

beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz,

what is the frequency of B?

15.19 Explain why (or how):

(a) in a sound wave, a displacement node is a pressure antinode and vice versa,

(b) bats can ascertain distances, directions, nature, and sizes of the obstacles

without any “eyes”,

distinguish between the two notes,

(d) solids can support both longitudinal and transverse waves, but only

longitudinal waves can propagate in gases, and

(e) the shape of a pulse gets distorted during propagation in a dispersive medium.

15.20 A train, standing at the outer signal of a railway station blows a whistle of frequency

400 Hz in still air. (i) What is the frequency of the whistle for a platform observer

when the train (a) approaches the platform with a speed of 10 m s

–1

, (b) recedes

from the platform with a speed of 10 m s

–1

? (ii) What is the speed of sound in each

case ? The speed of sound in still air can be taken as 340 m s

–1

15.21 A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still

air. The wind starts blowing in the direction from the yard to the station with a

speed of 10 m s

–1

. What are the frequency, wavelength, and speed of sound for an

observer standing on the station’s platform? Is the situation exactly identical to

the case when the air is still and the observer runs towards the yard at a speed of

10 m s

–1

? The speed of sound in still air can be taken as 340 m s

–1

Additional Exercises

15.22 A travelling harmonic wave on a string is described by

y(x, t) = 7.5 sin (0.0050x +12t +

/4)

(a)what are the displacement and velocity of oscillation of a point at

x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?

(b)Locate the points of the string which have the same transverse displacements

and velocity as the x = 1 cm point at t = 2 s, 5 s and 11 s.

15.23 A narrow sound pulse (for example, a short pip by a whistle) is sent across a

medium. (a) Does the pulse have a definite (i) frequency, (ii) wavelength, (iii) speed

of propagation? (b) If the pulse rate is 1 after every 20 s, (that is the whistle is

blown for a split of second after every 20 s), is the frequency of the note produced

by the whistle equal to 1/20 or 0.05 Hz ?

15.24 One end of a long string of linear mass density 8.0 × 10

–3

kg m

–1

is connected to an

electrically driven tuning fork of frequency 256 Hz. The other end passes over a

pulley and is tied to a pan containing a mass of 90 kg. The pulley end absorbs all

the incoming energy so that reflected waves at this end have negligible amplitude.

At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement

(y = 0) and is moving along positive y-direction. The amplitude of the wave is 5.0

cm. Write down the transverse displacement y as function of x and t that describes

the wave on the string.

15.25 A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy

submarine moves towards the SONAR with a speed of 360 km h

–1

. What is the

frequency of sound reflected by the submarine ? Take the speed of sound in water

to be 1450 m s

–1

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15.26 Earthquakes generate sound waves inside the earth. Unlike a gas, the earth can

experience both transverse (S) and longitudinal (P) sound waves. Typically the speed

of S wave is about 4.0 km s

–1

, and that of P wave is 8.0 km s

–1

. A seismograph

records P and S waves from an earthquake. The first P wave arrives 4 min before the

first S wave. Assuming the waves travel in straight line, at what distance does the

earthquake occur ?

15.27 A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the

sound emission frequency of the bat is 40 kHz. During one fast swoop directly

toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air.

What frequency does the bat hear reflected off the wall ?

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