CHAPTER-14: WAVES

Marks: 10
Wave: The disturbance set up in the medium is called wave.
Wave motion: The spreading of a disturbance in a medium with transfer of energy and momentum
without actual movement of particles is called wave motion.
Waves around us: The world is full of waves and we live together with waves of different kinds.
They are sound waves, light waves, microwaves, radio waves, seismic waves, matter waves.

Note:
1. Matter waves: The waves associated with particles in motion are called matter waves.
Ex: Matter waves are associated with fast moving electrons.
Mechanical waves: Waves which require a material medium for their propagation are called
mechanical waves. Ex: Sound waves, seismic waves.
Note: For the propagation of mechanical waves, the medium should have elastic property and
inertia.

Non-mechanical waves or Electromagnetic waves:

Waves which donot require a material medium for their propagation are called non-
mechanical waves.
Ex: Light waves, Radio waves, microwaves.
Note: Electromagnetic waves are produced due to the disturbance in electric and magnetic fields.

Transverse waves: The waves in which the oscillations of a particles are perpendicular to the
direction of wave propagation are called transverse waves. Ex: Electromagnetic waves, waves on a
string, waves on a surface of water.

In a transverse waves crests and troughs are formed

Crest C

Mean positions

Trough T

Longitudinal waves: The waves in which the oscillations of the particles are parallel to the
direction of wave propagation are called longitudinal waves. Ex: Sound waves, seismic waves.

C=Compressions
R=Rarefactions

C R C R C
In a longitudinal waves compressions and rarefactions are formed in the medium. At
compressions density of medium is maximum At rarefaction density of medium is minimum

KIRAN N 1
Note:
1. Mechanical waves can be longitudinal (Ex: sound) or transverse (waves on a string).
Electromagnetic waves are always transverse.
2. Transverse wave cannot propagate inside fluid. Because a fluid cannot sustain a shearing
stress.
3. The longitudinal wave can be propagated in a gas or a liquid.
4. Both longitudinal and transverse waves can be propagated in solid.

Progressive wave:
A wave in which disturbance propagates continuously is called progressive wave.

Expression for displacement of progressive wave (Progressive wave equation):

It is given by y=asin(Kxt)
Where, y=displacement
a=amplitude of a wave
K=angular wavenumber or propagation constant
=angular velocity, (Kxt)=Phase angle
x=change in position along x-axis, t=time
The above equation represents a sinusoidal wave travelling along positive direction of x-axis.
Note:
1. Progressive wave travelling along negative x-direction, then y=asin(Kx+t).
2. If  is the initial angle at x=0 and t=0 then y=a sin(Kxt+)

Graphical representation of progressive wave: The displacement of progressive wave varies

according to the equation y=a sin(Kx-t) as shown


Y
a t
T=0
T T(2)
()
2


Amplitude of a wave (a): The maximum displacement of a particle of a medium from its mean
position is called amplitude.
Phase: The state of vibration of a particle in a medium is called phase.
In above equation, (Kx-t) is the phase or phase angle. It is measured in radian (rad)

Wavelength (): The distance between two consecutive points which are in same phase is called
wavelength. It is measured in metre.

Note:
1. For transverse wave, wavelength is the distance between two consecutive crests or troughs.
2. For longitudinal wave, wavelength is the distance between two consecutive compression or
rarefactions.

KIRAN N 2
Angular wave number or propagation constant (K): It is the number of waves present in unit
length.
2
It can be written as K 

S.I. unit is rad/m or m-1

Period of wave (T): It is the time taken to complete one wave.

Frequency (f or ) : It is the number of waves produced in one second. S.I. unit is S-1 or Hz (hertz).

Derive an expression for speed of progressive wave (Travelling wave):

Consider a progressive wave travelling along positive x-axis

=wavelength, T=time period, f=frequency.
v=velocity of wave, =angular frequency, K=propagation constant
Let, x=distance travelled by a wave in t time,
x
v
t
Phase of a wave at a point A is (Kxt)
Phase of a wave at a point B is [K(x+x)(t+t)]
Since x and t are very very small.
 Phase at B= Phase at A
K(x+x)(t+t)=Kxt
Kx+Kxtt=Kxt
Kx=t
x 

t K

 v
K
2 2
But  and K 
T 
2
v T
2

 1
v but  f
T T
v = f This is the expression for speed of progressive wave.

KIRAN N 3
Expression for speed of transverse wave on a stretched string:
T
v Where, v=sped of transverse wave

T=Tension on the string
= Mass per unit length of string

Derive an expression for Newton’s formula and explain Laplace correction (Derive an
expression for speed of longitudinal wave or sound wave):
The sound waves travel in the form of compression and rarefaction in a medium. Therefore
there is a change in volume and pressure.
Let, P=pressure, V=volume, =density, v=speed of sound,
P= change in pressure, V=change in volume.
But bulk modulus of a medium is given by
 
P  Stress P 
B B   
V  strain V 
V  V 
V P
 B
V
But speed of sound in any medium is
B
v --------------- (1)

When sound travels through air medium, volume and pressure change at constant
temperature. This is called isothermal process.
According to isothermal process PV=constant
Differentiate, PV+VP=0
PV=-VP
VP P
P  B
V V
V
Then , B=P equation (1) becomes
P
V ---------------(2)

This relation was first given by Newton and is known as Newton’s formula
By using above formula, speed of sound at STP is sound that v=280ms-1.
But experimental value is 330 ms-1.
Therefore above formula requires correction, Laplace made a correction
Laplace correction: According to Laplace, volume and pressure of air changes under adiabatic
process
According to adiabatic process PV =constant
Differentiate (PV)=0
PV-1V+VP=0
PV-1V=V
V  P P
P=  
V V V
 1

V
P=B
 B=P

KIRAN N 4
 Where,  = specific heat ratio
 Equation (1) becomes
P
V

This is called Newton’s Laplace formula.
For air =1.41, by using above formula, speed of sound at STP is found that V=331.3 ms-1 which
agrees with the measured speed.

Principle of super position: It states that “When two wave pulses overlap, the resultant
displacement is the algebraic sum of the displacements of two wave pulses”
Ex:

Intensity of Wave: It is the energy transported by the wave per second per unit area perpendicular
to direction of propagation. S.I. Unit is JS-1m-2.

Loudness: The amount sensation produced in the ear by the sound is called loudness.
S.I. unit is sone.

Stationary wave or standing waves:

When two identical progressive waves travelling in opposite direction are superposed
stationary waves are formed. They are also called standing waves.
A /2 A

N=Node
N A=Antinode
loop N N

/2

Node: The point at which the amplitude is zero is called node, at nodes the particles are at rest.
Antinode: The point at which the amplitude is maximum is called antinode.


Note: Length of one loop =
2

Derive an expression for equation of stationary wave:

Let y1=a sin(kx-t) and y2= a sin (kx+t) are the two identical progressive waves travelling
in opposite direction to form stationary waves.
According to principle of superposition,
y = y1+y2 where, y=resultant displacement

KIRAN N 5
 y = asin(kx-t)+a sin(kx+t)
y=a[sin(kx-t)+sin(kx+t)]
  kx  t  kx  t   kx  t  kx  t 
y= a 2 sin    cos 
  2   2 
 kx   2t 
y  2a sin  2  cos 
 2   2 
y=2a sinkx cost
This is the expression for stationary wave.

Note:
1. 2a sinkx=amplitude of stationary wave.
2. The lowest possible natural frequency of a system is called its fundamental mode or first
harmonic.
3. The frequency of a system above the fundamental mode is called overtone.
Obtain the expression for first three harmonics (Normal modes) of stretched string
(Stationary waves produced in a stretched string):

x=0 x=L x=0 x=L

N N N N
loop loop N
L L
Second harmonic

Consider a stretched string of length ‘L’. The stationary wave equation, y=2a sinkx cost
at nodes, sinkx=0
 Kx=n, where n=0, 1, 2, 3…….
2
but K=

2
 x  n

n
 x
2
If x=L node is formed

 Ln
2
v
W.K.T. v=f,   v=Velocity of sound, f=frequency
f
nv nv nv
L or f  or f n 
2f 2L 2L
v
If n=1, f1  , This is first harmonic (fundamental mode)
2L
2v
If n=2, f2 
2L
v
f2  , This is 2nd harmonic
L
3v
If n=3, f3  this is 3rd harmonic
2L
 In stretched string, f1:f2:f3…….=1:2:3:…………

KIRAN N 6
Obtain the expression for first three harmonics produced in closed pipe [Stationary waves in
closed pipe]:
Consider an air column in a pipe. One end is closed and other end is open. At open end antinode is
formed.
 sinkx=1
 1
 kx=  n   Where n=0, 1, 2, ……..
 2
2 A N
but K 

2  n 1
x   
  2 
 3
 1 L L
x  n   4 4
 2 2
A
x=L, antinode is formed
 1
L  n  
 2 2 N
N
v
but,   v=velocity, f=frequency
f
 1 v
L  n  
 2  2f
 1 v  1 v
f  n   or fn  n  
 2  2L  2  2L
For first harmonic fundamental frequency, n=0
V
 f1 
4L
 1 v 3v
For 2nd harmonic, n=1 f 2  1   
 2  2L 4L
 1 v 5v
For 3rd harmonic, n=2, f3   2   
 2  2L 4L
 f1:f2:f3 ………….. = 1:3:5:……………
In closed pipe only odd harmonics are present

Note: For open pipe (i.e. both ends are open)

v
1st harmonic, f 1 
2L
2v
2nd harmonic, f 2 
2L
3v
3rd harmonic, f 3  etc
2L
 f1:f2:f3……………= 1:2:3………….
nv
i.e. all harmonics are present in open pipe  for open pipe f n 
2L

What are beats?

The waxing and waning of intensity of sound due to superpose of sound waves of nearly
equal frequency are called beats.

KIRAN N 7
Theory of beats: Consider two sound waves of nearly equal frequency but equal amplitude.
i.e. y1=a cos1t and y2 =acos2t, where 1=2f1, 2=2f2
after, superposition, y=y1+y2
y=a cos1t+a cos2t
y=a[cos1t+cos2t]
 1 t   2 t    t  2 t 
y= 2a cos   cos 1 
 2   2 
1  2   2
put  a & 1  b
2 2
 y=2acosat . cosbt
y=2acos(bt) cos(at)
According to above equation, the amplitude of resultant sound wave is not constant.
Therefore the intensity of sound waxes and wanes with an angular frequency.
1   2
b 
2
Where,  b  2f b fb= beat frequency, 1=2f1, 2=2f2
 2fb=2f1-2f2  fb=f1-f2
In general beat frequency is given by fb=f1f2

***********

KIRAN N 8