NCERT Textbook - Waves - Class 11 PDF Download

 Page 1


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
2020-21
Page 2


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
2020-21
PHYSICS 368
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 d?, this change induces a change in pressure,
dp, 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.
2020-21
Page 3


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
2020-21
PHYSICS 368
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 d?, this change induces a change in pressure,
dp, 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.
2020-21
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.
2020-21
Page 4


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
2020-21
PHYSICS 368
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 d?, this change induces a change in pressure,
dp, 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.
2020-21
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.
2020-21
PHYSICS 370
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 u u u 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:
(a) 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.
(c) Waves produced by a motorboat sailing
in water.
(d) Ultrasonic waves in air produced by a
vibrating quartz crystal.
Answer
(a) Transverse and longitudinal
(b) Longitudinal
(c) Transverse and 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( ) = - ? + f y x t a kx t (15.2)
The term f 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 ? ? = - + - (15.3)
From Equations (15.2) and (15.3),
2 2
a A B = +
 and  
1
tan f
-
=
B
A
?
?
?
?
?
?
To understand why Equation (15.2)
represents a sinusoidal travelling wave, take a
fixed instant,  say t = t
0
. Then, the argument of
the sine function in Equation (15.2) is simply
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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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PHYSICS 368
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 d?, this change induces a change in pressure,
dp, 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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PHYSICS 370
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 u u u 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:
(a) 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.
(c) Waves produced by a motorboat sailing
in water.
(d) Ultrasonic waves in air produced by a
vibrating quartz crystal.
Answer
(a) Transverse and longitudinal
(b) Longitudinal
(c) Transverse and 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( ) = - ? + f y x t a kx t (15.2)
The term f 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 ? ? = - + - (15.3)
From Equations (15.2) and (15.3),
2 2
a A B = +
 and  
1
tan f
-
=
B
A
?
?
?
?
?
?
To understand why Equation (15.2)
represents a sinusoidal travelling wave, take a
fixed instant,  say t = t
0
. 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 x is a sine
wave. Similarly, take a fixed location, say x = x
0
.
Then, the argument of the sine function in
Equation (15.2) is constant -?t. 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 + f  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( ) = +? +f 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 + f) 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 f is the phase at x = 0
and t = 0. Hence, f is called the initial phase
angle. By suitable choice of origin on the x-axis
and the intial time, it is possible to have f = 0.
Thus there is no loss of generality in dropping
f, i.e., in taking Eq. (15.2) with f = 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–?t+f : 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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