UNIT VIII: Oscillation and wave 8660658206

Chapter; 15 WAVES Duration- 10 hours;

1. All our communication essentially depends on transmission of signal through wave.
2. Sending and receiving of light and sound through wave.
3. Electron move around the nucleus along the wave motion. (called matter wave i.e. the wave
associated with moving material particles are matter wave).
When particle in a medium starts to vibrate the neighboring particles also starts to vibrate, as a result a
disturbance will be setup in the medium periodically. This sort of disturbance which is transmitted in a
medium without the bulk movement of particles of the medium and it carry & transfer the energy.
Examples; When stone thrown into a pond of water, circular ripples are formed on the surface of the
water and these ripples appeared to move outward and reach the shore of pond from the centre of
disturbance due to continues dropping of stone. If some piece of cork is placed on the disturbed surface,
it seen that simply up and down periodically but not move away from the centre of the disturbance. This
shows that, the mass of the water does not flow from the centre of the circle but rather a moving
disturbance is created. Finally total media are set into vibration periodically and obey the elastic
properties.
Similarly when speak a sound move outward from us, without any flow of air from one part of medium
to another but disturbance produced in air.
WAVES; A wave is a sort of disturbance which is transmitted in a medium without the bulk
movement of particles of the medium. OR The disturbance setup in elastic media due to
continuous periodic vibration of particle is called wave.
Example; light waves, sound waves, Radio waves, Gamma waves, micro waves, waves on
water surface, waves on stretched string, etc.
Wave motion: The propagation of disturbance from a region to another region in a medium is
called wave motion. During wave motion, Wave can carry and transfer the energy and
momentum from one to another.
Method of transferring energy from one place to another place: 1) Wave motion 2) Particle
motion
1. Energy transferred by a wave motion: The energy carried by a wave with out actual motion
of particles.
Ex: light waves, gamma waves, sound waves, micro waves etc……,
2. Energy transferred by a particle motion: The energy carried by a particle, when two
particles colloid with each other with actual motion.
Ex: Bullet fired from a gun
Note: During wave motion energy transferred in space but not in matter between two points.
Type of waves: Depending on appearance, waves can be classified into three type,
1. Based on media 2. Based on vibration 3. Based on propagation
1. Based on media: On the based on media waves can be classified into two types.
1) Mechanical waves or Elastic waves
2) Non-mechanical waves or Electromagnetic waves
Distinguish between Mechanical and Non mechanical waves:
Mechanical wave Non mechanical wave
1 The wave which requires material for The wave which does not require material
their propagation is called mechanical for their propagation is called non-
waves. mechanical waves.
2 Particles of the medium oscillate. Electric and magnetic field oscillate.
3 It can be longitudinal (or) transverse in It can be always transverse in nature.
nature.
4 For the propagation of these waves the They can travel in vacuum as well as in
media must be in elastic properly and matter.
inertia

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 UNIT VIII: Oscillation and wave 8660658206
5 these waves travels at relatively lower these waves travels at relatively higher
speed in a medium speed in a medium
6 Ex: Sound waves, seismic waves, waves Ex: light waves, u-v rays, I.R rays, radio
generated on stretched string etc…… waves etc…….
2. Mode of vibration: There are two type of wave for mode of vibration.
1. Longitudinal wave
2. Transverse wave
Distinguish between Longitudinal wave and Transverse wave
Longitudinal wave Transverse wave
1 The wave in which the particle of the media The wave in which the particle of the
vibrates along the direction of propagation is media vibrate perpendicular to direction
called longitudinal wave. Ex: sound wave, of the propagation is called transverse
seismic wave, wave due earth quek etc…… wave. Ex: Light wave, visible wave,
radio wave etc……
2 The angle between vibration and their The angle between vibration and
propagation is 00 propagation is 900
3 These waves travel in alternate compression These waves travels in alternate crests
& rarefaction. and troughs.
4

5 They are always mechanical wave. They are either mechanical (or) non
Mechanical wave.
6 The pressure and density varies as the wave The pressure and density do not vary as
propagates. Hence it is called pressure wave the wave propagates. it is not pressure
wave
7 They can travel in solids, liquids and gases. They can travel in solids and on the
Surface of liquids
8 They do not exhibit polarization. They exhibit polarization.
Velocity of a longitudinal wave in a gas is Velocity of a transverse wave on a
B T
given by v  where B &  is the bulk stretched string is given by v 
 
modulus and  is the density of media Where T is the tension &  is linear
density of string.
3. Based on mode of propagation: There are three type of wave for mode of
propagation.
1) One dimensional wave: The wave which travels along straight line
Ex: The waves on stretched string.
2) Two dimensional waves: The wave which travels along plane. Ex: water waves.
3) Three dimensional waves: The waves which travels along space. Ex: light waves.

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 UNIT VIII: Oscillation and wave 8660658206
Progressive wave or travelling wave: The waves which travel
continuously in same media in same direction with
constant amplitude is called Progressive wave.
When Progressive wave propagate in the media phase of
vibrating particle changes from particle to particle
resultant all particles in the media begin to vibrate with
constant amplitude and their execute SHM. Progressive waves
may exhibit both longitudinal and transverse waves.
Progressive waves graphically represented as shown in figure.
Characteristics of progressive wave:
The progressive waves travels along the same direction with
constant amplitude.
1. It is setup by a continuously vibrating particle. No
particle in the media is at rest.
2. All particle vibrate with same amplitude and same frequency and execute SHM.
3. It can change the phase continuously.
4. It carries energy, transfer the energy and momentum.
5. It is not localized.
6. It can exhibit reflection, refraction, diffraction etc…….
Displacement relation in the progressive waves:
Consider a SHM wave travel with velocity v along a positive x-axis.
Let particle begin to vibrate from ‘O’ be
point at origin of disturbance at time, t=0.
The displacement of particle at any instant
time t is y=a sin  t
Where a  amplitude
  angular frequency =2
Let ‘P’ be another point of particle at
distance ‘x’ from ‘O’. it will vibrating x/v
second after the vibration of particle O. because time taken by disturbance to reach the point P
from O is x/v.
The displacement of the particle at P compare to O at instant time is
y=a sin    t 
x
v 
x
y=a sin  
 t 
 v 

y=a sin kx  t  where k=
is called propagation constant.
v
The displacement of the progressive wave at any phase (  ) is y=Asin kx  t  
This is a progressive wave equation.
Note: The displacement of the progressive wave at any instant along a negative x-axis is
y=a sin kx  t 
Amplitude and Phase;
Wave amplitude (a): the maximum displacement of the particle from its mean position.
Phase (ϕ): The phase of the particle indicated state of vibration of the particle. The phase of
wave a time t varies from point to point and at position x varies from time to time.
ie ϕ (x,t)=(kx-t)

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 UNIT VIII: Oscillation and wave 8660658206
Wavelength and angular wave number;
Wave length ;() The distance between two successive
vibrating particles of the medium which are in the
same phase is called wave length. OR Distance
between any two crests or trough is called wave length
of transverse wave. OR Distance between any two
compressions or rarefaction is wave length of
longitudinal wave.
Angular wave number or propagation constant (k); The number of angular wave present in
unit length is called angular wave number.
W.K.T. y=a sin kx  t  
if t=0 & ϕ=0 then y=a sin kx ………………..(1)
from periodic sine function sin kx =sin(kx|+2nπ)
 y= sin(kx|+2nπ)
 2n 
y  sin k x |   ……………..(2)
 k 
Comparing equation (1) &(2),
 2n 
x   x|  
 k 
2n
x  x|  this is difference between n successive particle.
k
for least two particle ie n=1
2 2
  or k  angular wave number or propagation constant.
k 
SI unit is radian per metre or rad m-1
Alternative method; W.K.T. y=a sin kx  t 
 2f 2 2
where k=    k=
v f  
Period, angular frequency and frequency;
Wave period (T): Time taken to complete one wave.
Wave frequency (f): Number of complete wave in one
second.
Angular frequency ();the ratio of angular wave to
period of wave.
W.K.T. y=a sin kx  t  but x=0 then y=a sin  t 
y=  a sin t ………. (1)
from periodic sine function is  a sin t  a sin (t  T)
y  a sin(t  T) ……… (2)
Comparing equation (1) & (2), T=2π
1 

T 2
2
 SI unit is rad/s
T
this is an equation of angular frequency.
1 
But f  f
T 2
Speed of a travelling wave or progressive wave; it is defined as Distance travelled by a wave
in one second is called speed of a travelling wave.

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 UNIT VIII: Oscillation and wave 8660658206
Consider a progressive wave, let x be the
distance covered by crest in a small time t
displacement
W.K.T velocity 
time taken
x
v
t
But y=a sin kx  t   ……… (1)
As time increases with increasing displacement
as shown above figure,
For periodic sine function, amplitude & sine function are constant for every alternative crest
and ϕ is also constant during the motion crest
The equation (1) becomes,
(kx  t)=constant
x
Differentiate w.r.t time k 0
t
x
k 
t
x 

t k
 2f
 v v
k 2

v=f this is an speed travelling wave.
wave lenght
Alternative method; v =
time taken

v
T
1
But f   v=f
T
Note: When a wave travels from one media to another media frequency remains constant but
velocity wavelength changes. Because frequency depends on source, velocity and wavelength
depends on media.
Speed of a transverse wave on a stretched string
T
Speed of a transverse wave on a stretched string is dimensionally written as v 

Where T is the tension of string &  is linear density of string.
T=mg= Force on the string and  is the mass per unit length of string.
Newton-Laplace formula for speed of sound wave or Speed of a longitudinal wave;
When longitudinal wave propagate through elastic media, the pressure & density changes.
During compression pressure & density are increases and they decreased during the
rarefaction.
B
Speed of a longitudinal wave in a gas is given by Newton is, v 

it is dimensionally correct.
Where B &  are the bulk modulus and the density of medium.
Newton assumed that, when sound wave propagates through a gas is changes isothermally.
For isothermal relation, PV= constant

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 UNIT VIII: Oscillation and wave 8660658206
Differentiate partially w.r.t P & V
PdV+VdP=0
VdP dp
P  ……… (1)
dV dV
V
dp
W.K.T bulk modulus B   ……. …………(2)
dV
V
From equation (1) and (2) we get  B=P
P
Newton’s formula for velocity of sound is v 

For air, at STP P=1.013x105 N/m2 and  = 1.29kg/m3
1.013x10 5
v  280m / s
1.29
But, the experimental value of velocity of sound in air at STP is found to be 331m/s. Thus, the
value of velocity of sound in air obtain by Newton’s formula does not agree with the
experimental value. and this was correction by Laplace.
According to Laplace the condition, when sound wave propagates through a gas is changes
adiabatically not isothermally.
Cp
For adiabatic relation, PV=constant.  = Where  is specific heat ratio,
Cv
For air  =1.4
Cp&Cv are specific heat at constant pressure& volume respectively.
Differentiate partially w.r.t P&V
P  V-1dV+ VdP=0
P  V-1dV=  VdP
V  dP
P  
V 1dV
dP dp
P   but B  
V dV
dV V
 B=P
P
Newton-Laplace formula for velocity of sound is v 

1.4x1.013x 105
v  331m / s
1.29
This is in close agreement with the experimental value.
Factors affecting velocity of sound in gas;
1. Effect of change in pressure;
According in to Boyle’s law, at constant temperature PV=constant or Pm/=constant
Where m is mass of the given gas of density 
P
Velocity of sound is v  =constant

Hence velocity longitudinal wave does not depends on pressure of gas.
2. Effect of change in temperature;
According to Charle’s law at constant pressure V/T=constant V=m/

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 UNIT VIII: Oscillation and wave 8660658206
  1/V1/T
 1/T
velocity sound is v  PT  v T
Note ; the velocity of sound increased by 0.61 m/s by increasing temperature of 10c.
3. Effect of humidity: The presence of water vapor in a atmosphere is known as moisture or
P
humidity. Velocity of sound is v  =constant

v1 /  Density of the moisture is less than density of the dry air.
4. Effect of wind; Velocity of sound is increases along direction of the wind and decreases in
its opposite direction.
Intensity of wave :Amount of energy present in the unit area in one second perpendicular to
direction of propagation is called intensity of wave. SI unit of its w/m2
ie, I= 22f2a2v where, f= frequency of wave, a= amplitude of wave
v= velocity of wave = density of medium
Pulse; it is single disturbance that move through a medium from one point to the next point.
Example; A single water wave from a splash, a sonic boom from on aircraft breaking the sound
barrier & an electromagnetic pulse from a nuclear explosion etc,.
Relation between phase difference and path difference.We know that equation of progressive
wave y=a sin(kx-t+)Let  is the phase of the particle at a distance x from the origin along the
positive x direction. The equation of phase is (kx-t) =

O x1
x2 Q

From above the figure phase changes periodically with distance x. At a given time (t 1=t2=t), let
1 and 2 be the phase of two particles at a distance x1 and x2 from the origin to a point P and Q
respectively.
Let y1,y2 displacement of two progressive waves and y is resultant displacement
1=kx1-t1 and 2=kx2-t2
=2 - 1=(kx2-t2)-(kx1-t1) =kx2-kx1
=k(x2-x1) =kx
2 2
= x  Phase difference =  path difference
 
Principle of super position; It states that “when two wave pulse overlap, the resultant
displacement is the algebraic sum of displacement of two wave pulses”.
Explanation; when two or more wave travels along same media in same or opposite direction
which are overlap to each other then their resultant displacement is equal to algebraic sum of
individual wave displacement and the resultant wave obeys the SHM”
Let y1,y2 displacement of two progressive waves and y is resultant displacement y=y1+y2
Application of super position principle (uses):
1. Stationary wave 2. Beats and 3.Interference

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 UNIT VIII: Oscillation and wave 8660658206
Derive an equation for resultant displacement & amplitude of two wave in super
imposing
Consider two progressive waves having velocity (v), angular frequency () & wave length 
are same amplitude which are traveling along positive x-axis on stretched string, but their
initial phase is different. The displacement of two progressive wave y1= a sin (kx-t) & y2= a
sin (kx-t+)
Where  is phase difference between two waves,
According in to super position principle y=y1+y2 where y is resultant displacement.
y=a sin (kx-t) + a sin (kx-t+)
kx  t  kx  t     kx  t  kx  t   
y=2a sin   cos  
 2   2 
   A  B  A B
y=2a sin  kx  t   cos    {since sin A+sin B= 2 sin   cos }
 2  2  2   2 
 
y=2a cos   sin  kx  t   ………………. (1)
2  2
By comparing equation (1) with individual wave equations,
We get A=2a cosϕ/2 This is equation of resultant amplitude of two waves.
The equation (1) becomes

y=A sin  kx  t   This is an equation of resultant displacement of two waves.
 2
Thus resultant displacement is also sinusoidal wave.
Constructive interference; The super position of two waves is in phase, the resultant amplitude
is maximum. As shown in figure (a). ie ϕ=0 A=2a is maximum.
Destructive interference; The super position of two waves is out of phase, the resultant
amplitude is minimu. As shown in figure (b). ie ϕ=180 A=0 is minimum.

Reflection of wave; It is defined as return of wave in same media when wave pulse incident
on two different densities of media. They are two type of reflection of wave,
1. Hard reflection; The wave which reflected from fixed end (or rigid boundary) with change
of 1800 phase is called hard reflection. The phase
reversed reflected wave from fixed end is as shown in
figure.
Example; The phenomenon of echo
Let y1& y2 be the displacement of incident and
reflected wave which are travels in same media but
0
opposite direction with changes of the phase 180 due
to fixed boundary.
According in to principle of super position,
y=y1+y2
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 UNIT VIII: Oscillation and wave 8660658206
ie, y= a sin (kx  ωt)+a sin(kx  ωt+π)
y= a sin (kx  ωt)  a sin(kx  ωt) { sin (+180)=  sin}
y=0 Thus,the resultant displacement is zero.
2. Soft reflection; The wave which reflected from free end (or non- rigid boundary) with out
change of phase is called soft reflection. The reflected
Free end
wave from free end is same as shown in figure.
Example; Wave produce inside open end of organ pipe.
Let y1& y2 be the displacement of incident and
reflected wave which are travels in same media but
opposite direction with out change of the phase due to
free boundary.
According in to principle of super position, y=y1+y2
ie, y= a sin (kx  ωt)+a sin(kx  ωt)
y= 2a sin (kx  ωt)
Thus, the resultant displacement is twice of individual displacement.
Note; wave reflected from air boundary the phase changes with 1800 other than air boundary
no changes phase.
standing waves and Normal modes;
Standing waves or stationary waves :
The wave which form due to super position of two identical progressive wave (both having
same velocity, amplitude, frequency, & Wavelength) travelling in the same media but in
opposite direction is called stationary waves.
Note;
1. Longitudinal stationary wave is produce inside the organ pipe.
2. Transverse stationary wave is produce in a stretched string between fixed points.
4. It can create node (N), Antinodes (A) and loops.
5. Stationary wave represented by graphically as
follows.
Node (N): The point at which the amplitude is zero is
called node, at nodes the particles are at rest.
Antinode(A) : The point at which the amplitude is maximum is called antinode. At antinodes
the particles are vibrate maximum.
Loop: the space between successive any two node.
a. The distance between node and antinodes=/4
b. The distance between two successive Nodes or Antinodes =/2
c. The length of an loop between successive two Nodes =/2
Differences between Stationary wave and progressive wave;
Progressive wave Stationary wave
1 The wave which travels continuous along The which form due to super position
the same direction with constant amplitude of two identical progressive wave
travelling in the same media but in
opposite direction.
2 It can travels from one to another medium It does not travels from one point to
another
3 It can carries the energy It can not carries the energy
4 No one particles are rest in the medium The particles at node are permanently
at rest.

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 UNIT VIII: Oscillation and wave 8660658206
5 The wave travel continuously with certain The wave does not move. It remains
velocity called wave velocity localized.
6 Amplitude of vibration is the same for every The amplitude of vibration varies from
particle of the medium along the wave. zero at node & maximum at antinode.
7 Phase changes continuously Same phase
8 The wave equation is of the form y = a The wave equation is of the form.
sin(kx - wt) y=2a sinkx cost
Characteristics of stationary wave;
1. Super position of two identical progressive travels in same direction
2. It does not carry & transfer the energy.
3. The particle at node is permanently at rest.
4. The particle at Antinodes vibrates maximum amplitude.
5. Amplitude of the particle varies from zero to maximum.
6. A particle in the loop vibrates with same phase.
7. The Equation of stationary wave is y = 2a sin kx cos wt.
Equation of stationary wave:
Consider two identical progressive waves which are travelling in opposite direction to
form stationary waves.
Let y1=a sin (kx-t) is a progressive travelling along positive direction
and y2= a sin (kx+t) is a progressive travelling along negative direction
According to principle of superposition,
y=y1+y2 where, y=resultant displacement
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
y=A cost This is the expression for stationary wave.
Where A=2a sinkx is represent the amplitude of standing wave.
Note;
1. For node amplitude is zero.  2a sinkx =0
From periodic sine function kx=n where n=0, 1, 2, 3…….
2. For antinode amplitude is maximum.  2a sinkx =1
 1
From periodic sine function kx=  n   Where n=0, 1, 2, ……..
 2
Normal modes of oscillation; It is set of natural frequencies which vibrate the system of
particle with harmonic travelling wave.
Fundamental mode of vibration; The lowest vibration of the system of the particle to form
least number node & antinode.
Fundamental mode or first harmonic; The lowest possible natural frequency of a system
Fundamental frequency; The lowest possible frequency of a fundamental mode of vibration.
Harmonics: The integral multiple of fundamental frequency are called harmonics
Overtones; The frequency of a system above the fundamental mode.
Example; 1  fundamental frequency  I harmonics
2  fundamental frequency  II harmonics or I overtones
3  fundamental frequency  III harmonics or II overtones

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 UNIT VIII: Oscillation and wave 8660658206
n  fundamental frequency  n harmonics is called normal modes.
Stationary waves produced in a stretched string;
Consider stretched string fixed at both ends of length L when it is plucked to vibrate with
velocity ‘v’, frequency ‘f’ and to form stationary with node & antinode. The number of node
more then antinode as shown in the figures.
W.K.T stationary wave equation, y=2a sinkx cost and its amplitude A=2a sinkx
At nodes, sinkx=0  kx=n where n=0, 1, 2, 3…….
2 2
but K=  x  n
 
n
x 
2

If x=L nodes are formed


 Ln
2
v
W.K.T. v=f, 
f
nv nv nv
L or f  or f n  this is an equation of normal mode.
2f 2L 2L
If n=1, then two nodes & one antinode are formed as shown above the figure (a).
v
 f1  , This is fundamental mode (first harmonic) …………….. (1)
2L
If n=2, then three nodes & two antinodes are formed as shown above the figure (b)
2v
 f2 
2L
v
f2  , This is 2nd mode or 2nd harmonic
L
or f 2  2f1 This is 1st overtone …………….. (2)
If n=3, then four nodes & three antinodes are formed as shown above the figure (c)
3v
 f3  this is 3nd mode or 3rd harmonic
2L
or f 3  3f1 This is 2nd overtone …………….. (3)
From equation (1),(2)& (3) becomes, f1:f2:f3…….=1:2:3:…………
Hence, all harmonic are present in stretched string.
Air column or organ pipe: An air enclosed in a pipe is called air column. at closed end node
is formed and at open end antinode is formed. They are two organ pipe.
Open pipe; the organ pipe which open at both end. The number of antinode greater than node
Closed pipe; the organ pipe which open at one end & closed at another end. The number of
node equal to antinode
Stationary waves in closed pipe:
Consider a closed pipe of length L when air column is vibrated with velocity ‘v’, frequency ‘f’
and stationary waves are formed. Node is formed at closed end & antinode at open end as
shown in the figures.
W.K.T stationary wave equation, y=2a sinkx cost and its amplitude A=2a sinkx
At antinodes, sinkx=1
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 UNIT VIII: Oscillation and wave 8660658206
 kx=  n  
1
 2
Where n=0, 1, 2, ……..
2
but k 

2  n 1 
x   
  2 
 1
x  n  
 2 2
x=L, antinode is formed
 1
L  n  
 2 2
v
W.K.T. v=f, 
f
 1 v
L  n  
 2  2f
 1 v  1 v
f  n   or f n   n   this is an equation of normal mode.
 2  2L  2  2L
If n=0, then one node and one antinode are formed as shown above the figure (a).
v
 f1  This is fundamental mode (first harmonic) …………….. (1)
4L
If n=1, then two node and two antinode are formed as shown above the figure (a).
 1 v 3v
f 2  1    This is 2nd mode or 2nd harmonic
 2  2L 4L
f 2  3f1 This is 1st overtone …………….. (2)
If n=2, then three node and three antinode are formed as shown above the figure (a).
 1 v 5v
f3   2    This is 3rd mode or 3rd harmonic
 2  2 L 4 L
f3  5f3 This is 2nd overtone …………….. (3)
From equation (1),(2)& (3) becomes, f1:f2:f3…….=1:3:5:7:…………
Hence, all odd harmonic are present in closed pipe.
Stationary waves in open pipe: Consider an open pipe of length L when air column is vibrated
with velocity ‘v’, frequency ‘f’ and stationary waves are formed. antinode is formed at both open
end & antinode are greater than node as shown in the figures.
W.K.T stationary wave equation, y=2a sinkx cost and its amplitude A=2a sinkx
At nodes, sinkx=0  kx=n where n=0, 1, 2, 3…….
2 2
but k=  x  n
 
n
 x
2

v
W.K.T. v=f, 
f
nv nv nv
L or f  or f n  this is an equation of normal mode.
2f 2L 2L
If n=1, then two antinodes & one node are formed as shown above the figure (a).
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 UNIT VIII: Oscillation and wave 8660658206
v
 f1  , This is fundamental mode (first harmonic) …………….. (1)
2L
If n=2, then three antinodes & two nodes are formed as shown above the figure (b)
2v
 f2 
2L
v
f2  , This is 2nd mode or 2nd harmonic
L
or f 2  2f1 This is 1st overtone …………….. (2)
If n=3, then four antinodes & three nodes are formed as shown above the figure (c)
3v
 f3  this is 3nd mode or 3rd harmonic
2L
or f 3  3f1 This is 2nd overtone …………….. (3)
From equation (1),(2)& (3) becomes, f1:f2:f3…….=1:2:3:…………
Hence, all harmonic are present in open pipe.
Beats; The waxing (rise) and waning (fall) of intensity of sound due to superpose of sound
waves of nearly equal frequency are called beats.
Example; Sticking of bicycle bell to produce the sound of rise & fall.
Define beat period; Time interval between any two successive waxing and waning of sound.
Define beat frequency (fb ); Number of beats heard in one second. S.I unit is hertz (Hz)
Beat frequency is equal to difference in frequency between two vibrating body.
ie fb=f1~ f2
Uses of beats;
1. The beats are used to find the unknown frequency of the given tuning fork.
2. Beats are used for tuning the musical instrument.
3. By using beats determine harmful gas inside the mines.
Theory of beats: Consider two sound waves of same amplitude and nearly equal frequency but
equal amplitude which are travels in a same media.
W.K.T. y1=a cos1t and y2 =acos2t, where 1=2f1, 2=2f2
According in to super position, 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 
 t  2 t 
y= A cos  1 
 2 
  t  2 t 
A  2a cos 1  is the amplitude of the resultant wave and it is is not constant.
 2 
Therefore the intensity of sound waxes and wanes with an angular frequency.
The intensity of resultant sound is mximum when Ais maximum.
1  2 1  2
put  a &  b
2 2
 y=2acosat . cosbt
1  2
b  Where, b  2f b fb= beat frequency, 1=2f1, 2=2f2
2
 2fb=2f1-2f2  fb=f1-f2
In general beat frequency is given by fb=f1f2
Doppler Effect: The apparent change in frequency of sound heard by a observer/listener due to
relative motion between the source of sound and observer/ listener is called Doppler effect.

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 UNIT VIII: Oscillation and wave 8660658206
Example; The apparent frequency of a train increases as it approaches an observer on the
platform and decreases when the train passes the observer.
Real or actual frequency: The frequency of a sound produced by a source is called real
frequency
Apparent frequency: The frequency of sound perceived by observer is called apparent
frequency.
Derive an expression for apparent frequency:
1. When source move is moving away from the stationary observer with respect to media.
When source moves moving away from a stationary observer (v0=0), Relative velocity between
observer and sound = v+v0 =v. and observer detects the crest in every interval of time (t) is as
in figure.
Let vs=velocity of source (S) v0=velocity of observer (O),
v= velocity of sound,f=real frequency,
f0=apparent frequency, T= period of waves,
T0 = period of instant =real wavelength,
0= apparent wavelength.
At a time t=0, The source is at S1 place and at a
distance L from O. the time taken to 1st crest detect
by the observer is t1=L/v
At a time t=T0, The source is at S2 place and at a
distance (L+vsT0) from O. the time taken to 2nd crest
 L  v s T0 
detect by the observer is t2= T0   
 v 
At a time t=2T0 , The source is at S3 place and at a distance (L+2vsT0) from O. the time taken to
 L  2v s T0 
3rd crest detect by the observer is t3=2 T0   
 v 
Similarly, At time t=nT0 , The source is at Sn place and at a distance (L+nvsT0) from O. the time
 L  nv s T0 
taken to (n+1)th crest detect by the observer is tn+1=n T0   
 v 
The n number crest detect by the observer in time interval (tn+1  t1) = nT0 
 L  nvs T0   L 
 v v 
There fore, the total period (T) taken by the n number of wave recorded by observer is,

nT  nT0 
L  nvs T0   L 
 v v 
vT
T  T0  s 0
v
 v 
T  T0 1  s 
 v
1
 v 
But f=1/T, f  f 0 1  s  ……………………. (1)
 v
Apply binomial expansion and neglect the higher power of term because v > vs, the equation
 v 
(1) becomes f  f 0 1  s 
 v
 v 
f 0  f   This is an expression for apparent frequency.
 v  v s 

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 UNIT VIII: Oscillation and wave 8660658206
Note; When source move is moving towards the stationary observer with respect to media. The
 v 
apparent frequency f 0  f  
 v  vs 
2. When observer is moving towards the stationary source with respect to media.
When observer is moving towards the stationary source (vS=0), Relative velocity between
observer and sound = v+v0 and observer detects the crest in every interval of time (t) is as in
figure.
Let vs=velocity of source (S) v0=velocity of
observer (O),
v= velocity of sound, f=real frequency,
f0=apparent frequency, T= period of waves,
T0 = period of instant =real wavelength,
0= apparent wavelength.
At a time t=0, The observer is at O1 place and at a
distance L from S. the time taken to 1st crest detect
L
by the observer is t1 
v  v0
At a time t=T0 , The observer is at O2 place and at a distance (L  v0T0) from S. the time taken to
 L  v 0 T0 
2nd crest detect by the observer is t2= T0   
 v  v0 
At a time t=2T0 , The observer is at O3 place and at a distance (L  2v0T0) from S. the time taken
 L  2v 0 T0 
to 3rd crest detect by the observer is t3=2 T0   
 v  v0 
Similarly, At time t=nT0, The source is at On place and at a distance (L  nvsT0) from S. the time
 L  v s T0 
taken to (n+1)th crest detect by the observer is tn+1=n T0   
 v  v0 

The n number crest detect by observer in time interval (tn+1  t1)= nT0 
 L  nvs T0   L 

 v  v0 v  v0 
There fore, the total period (T) taken by the n number of wave recorded by observer is,

nT  nT0 
L  nv0T0   L 

 v  v0 v  v0 
vT
T  T0  0 0
v  v0
 v0 
T  T0 1  
 v  v0 
 v 
T  T0  
 v  v0 
 v 
f 0  f  
 v  v0 
Note; When observer moves away from the source at rest with respect to media. The apparent
 v 
frequency f 0  f  
 v  v0 

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 UNIT VIII: Oscillation and wave 8660658206
3. When source& observer are moving opposite to each other with respect to media.
When source & observer are moving same direction, Relative velocity between observer and
sound Relative velocity between observer and sound = v+v0 and observer detects the crest in
every interval of time (t) is as in figure.
Let vs=velocity of source (S) v0=velocity of observer (O),
v= velocity of sound, f=real frequency,
f0=apparent frequency, T= period of waves,
T0 = period of instant =real
wavelength,
0= apparent wavelength.
At a time t=0, The source S1 from observer is at O1
st
place and at a distance L. the time taken to 1 crest
L
detect by the observer is t1 
v  v0
At a time t=T0, the observer at O2 and source at
S2. The distance between the observer and
source is given by [L+(VS-Vo)To].
the time taken to 2nd crest detect by the
 L  ( vs  v 0 )T0 
observer is t2= T0   
 v  v0 
Similarly, At time t=nT0, the observer at On and source at Sn. The distance between the
observer and source is given by [L+n(VS-Vo)To].taken to (n+1)th crest detect by the observer
is
 L  n ( vs  v0 )T0 
tn+1=n T0   
 v  v0 
The n number crest detect by observer in time interval (tn+1  t1)=
 L  n(vs  v0 )T0   L 
nT0  
 v  v0 v  v0 
There fore, the total period (T) taken by the n number of wave recorded by observer is,

nT  nT0 
L  n(vs  v0 )T0   L 

 v  v0 v  v0 
( v  v0 )T0
T  T0  s
v  v0
 ( v  v0 ) 
T  T0 1  S 
 v  v 0 

 ( v  vs ) 
T  T0  
 v  v0 
 ( v  vs ) 
f 0  f  
 v  v0 
One mark question and answer.
1. Name the properties of a medium which are responsible for the propagation of a
mechanical wave? Ans; Elastic and Inertial properties of medium
2. Define amplitude of a wave. Ans; Amplitude is the maximum displacement of the particle
on either side of the equilibrium position during wave propagation

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 UNIT VIII: Oscillation and wave 8660658206
3. Define phase of a vibrating particle? Ans; The phase of a vibrating particle at a given
instant of time is the state of vibration of a particle at that instant of time with reference to
its equilibrium position.
4. Define propagation constant (or) angular wave number. Ans; It is the number of waves
that can be accommodated per unit length.
5. What is sound? Ans; Sound is a form of energy that produces a sensation of hearing.
6. How is sound produced? Ans; Vibrating bodies surrounded by a material medium produces
sound.
7. Why do we see the flash of lightening before we hear the thunder? Ans; Because speed of
light is much greater that the speed of sound.
8. How much energy is transported by a stationary wave? Ans; zero
9. What are normal modes of oscillation in a stationary wave? Ans; In a stationary wave, the
possible frequencies of oscillation of the system are characterized by a set of natural
frequencies called as normal modes of oscillation.
10. Is Doppler effect observed for sound waves only? Ans; No
11. Name the quantity associated with a wave that remains unchanged when a wave travel from
one medium to another? Ans; frequency of wave
12. Name the quantities associated with a wave, that changes when a wave travels from one
medium to another. Ans; wave length & velocity
13. Which harmonics are absent in a closed organ pipe? Ans; even harmonics
14. What is the increase in the speed of sound in air when the temperature of the air rises by 10
C? Ans: The speed of sound increases approximately by 0.61 m/s per degree centigrade
rise in temperature
15. Why a transverse mechanical wave cannot travel in gases? Ans: Sheave modulus of
velocity is absent in gases medium, which is necessary for transvers wave
16. The fundamental frequency of a closed pipe is 80Hz. What is the frequency of first
overtone. :
Ans: Frequency of I overtone in closed pipe = 3 (fundamental frequency) = 380=240Hz
17. Calculate the wavelength of a wave whose angular wave number is 10 radian m-1? Ans: 
2 2
=   0.2m
k 10
18. The distance between a node & an next antinode in a stationary wave pattern is 0.08m.
What is the wavelength of the wave? Ans: The distance between node & an next antinode
 
is  = 0.32m
4 4
19. With what velocity does an electro magnetic wave travel in vacuum. Ans: 3 x108m/s
One mark question.
1. Does, all the waves requires a material medium for their propagation?
2. Does, wave carry energy?
3. What are matter waves?
4. Define period of a wave.
5. Define frequency of a wave.
6. Define wavelength of a wave.
7. Define wave velocity.
8. How propagation is constant related to wavelength of a wave?
9. Name the factors which determine the speed of a propagation of an electromagnetic wave?
10. What is a stationary wave?
11. What is the meaning of the fundamental mode (or) first harmonic of oscillation in a
stationary wave?
12. What are harmonics in a stationary wave?

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 UNIT VIII: Oscillation and wave 8660658206
13. What are overtones in a stationary wave?
14. What are beats?
15. What is beat period?
16. What is Doppler effect?
17. Give the formula for speed of transverse wave on a stretched string.
18. How does the velocity of sound in air vary with temperature?
19. How does the velocity of sound in air vary with pressure?
20. Give the dimensional formula for propagation constant.
21. How is the frequency of an air column in an open pipe related with the temperature of air?
22. A sound wave has a velocity of 330 ms-1 at one atmospheric pressure. What will be its
velocity at 4 atmospheric pressure?
23. What happens to the frequency of the wave when it travels from water to air?
24. Give the relation between time period and frequency of a wave.
25. What is the distance between two consecutive antinodes in a stationary wave of wavelength
2m?
26. How does speed of a transverse wave on a stretched string vary with its tension?
Two Mark questions:
1. What are mechanical waves? Give two examples.
2. What are non-mechanical waves? Give two examples.
3. What are longitudinal waves? Give two examples
4. What are Transverse waves? Give two examples.
5. Obtain the relation connecting v, f and  where symbols have their usual meaning.
4 and 5 marks questions:
1. Give the differences between mechanical and non mechanical (electromagnetic) waves.
2. Give the differences between longitudinal and transverse waves.
3. Write Newton’s formula for speed of sound in a gas. Discuss Laplace correction & arrive at
the formula modified by him.
4. Mention the characteristics of a progressive mechanical wave.
5. Mention the characteristics of a stationary wave.
6. Give the differences between progressive and stationary waves.
7. What are beats? Give the theory of beats.
8. Discuss different modes of vibration (first three harmonics) produced in a open pipe.
9. Discuss different modes of vibration (first three harmonics) produced in a closed pipe.
10. Discuss different modes of vibration on a stretched string.
11. What is Doppler effect? Derive an expression for the apparent frequency when a source
moves towards a stationary listener.
12. What is Doppler effect? Derive an expression for the apparent frequency when a listener
moves towards a stationary source.
13. What is Doppler effect? Derive an expression for the apparent frequency when the source
and listener are moving in the same direction.
Two marks problem
1. Calculate the period of a wave of wavelength 0.005 m which travels with a speed of
50 cm.s-1. Given  = 0.005 m ; V=50 cm s-1=50 x 10-2 ms-1

We have v=
T
 0.005
T=   0.01 s.
v 50x10 2
2. The frequency of a tuning fork is 256 Hz and sound travels a distance of 25m while
the fork executes 20 vibrations. Calculate the wavelength and velocity of the sound
wave.

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 UNIT VIII: Oscillation and wave 8660658206
Wave length = 25  1.25m
20
v=f v=256 x 1.25  v=320 ms-1
3. Velocity of sound in air is 340 ms-1. Two sound waves of frequency 1 KHz each
interfere to produce stationary wave. What is the distance between two successive
nodes?
Given v=340 ms-1; f=1 KHz =1000 Hz.
v 340
We have = =
 0.34m
f 1000
 0.34
Distance between two successive node =   0.17 m
2 2
4. When is the fundamental frequency of the sound emitted by a closed pipe is same as
that emitted by an open pipe? Let Lo & Lc be the length of open & closed pipe
respectively
Given (ffundamental) =(ffundmental)
Open pipe closed pipe
v v Lc 1 Lo
   Lc 
2L 0 4L 0 Lo 2 2
5. A closed pipe & open pipe have same frequency for the first overtone. What is the
ratio of their lengths?
Let Lo & Lc be the length of open and closed pipe respectively
Given & (ffirst overtone) Closed pipe =(ffirst overtone) open pipe
3V 2V Lc 3
 
4L 0 L o Lo 4
6. For what wavelength of waves, does a closed pipe of length 30 cm emit the first
overtone?
The frequency of the first overtone in a closed pipe is given by
3v 3v v
f=  
4e 4x30 40
f
But v=f f= =40 cm
40
7. The second overtone of closed pipe of length 1 m is in unison with the third overtone
of an open pipe. What is the length of the open pipe?
Given (fII overtone) Closed pipe = (fIII overtone) open pipe
5v 4v 8 8
 Lo  Lc Lo  (1)  1.6m
4L c 2L o 5 5
8. The velocity of a sound wave decreases from 330 ms-1 to 220 ms-1 on passing from one
medium to another. If the wavelength in the first medium is 3m. What is the wavelength
in the second medium? We have V=f
v1 f1
 
v 2 f 2
Where V1 & V2 are velocity of sound in first and second medium respectively and 1&
2 are the corresponding wavelength.
v2  220 
 2= 1    x3  2= 2m
v1  330 
V
9. Can sound waves of wavelength 33mm be heard in air? Justify.frequency (f)=

330
f=  10 KHz
33x10 3

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 UNIT VIII: Oscillation and wave 8660658206
Since this wave belong to audible range of sound they can be heard.
10. At what temperature will the velocity of sound becomes 1.25 times that at 270 C?
We have V  T  t  273
Given Vtoc=1.25 V270c
Vt 0C t  273
 
V270C 27  273
1.25V270C t  273

V270C 300
t=195.750C
11. A musical note produces 2 beats per second. When sounded with a tuning fork of
frequency 340Hz & 6 beats per second when sounded with a tuning fork of frequency
344 Hz. Find the frequency of the musical note?
We have fb=f1f2
Given I case f1= 340 Hz & fb = 2 beats per second
possible values of f2=342 Hz (or)338 Hz
possible values of fb = 342 Hz (or) 338 Hz.
II case f1 = 344 Hz & fb= 6 beats per second
possible values of fb = 338 Hz (or) 350 Hz
frequency of the musical note is f2= 338 Hz.
12. Calculate the velocity of sound at -300 C and 30oC given the velocity of sound at 00 C
is 330 ms-1.
W.K.T v T
v 300 c  30  273

v 00 c 0  273
v 300 c  311.3 m / s

30  273
v 300 c

v 00 c 0  273
v300 c  347.7m / s
13. With what velocity should a sound source travel towards a stationary observer so that
the apparent frequency may be double of the actual frequency.
 v   v  v
f  f 0   2f 0  f 0    vs 
 v  vs ,  v  vs  2
14. A bat emits ultrasonic sound of frequency 1000KHz in air. If sound meets a water
surface, what is the wavelength of a) reflected sound b) transmitted sound? (Given
speed of sound in air is 340 ms-1 & in water 1486 ms-1?
v air 340
 reflected sound  r    r  3.4x104 m
f 1000x103
v 1486
 transmited sound  water t   t  1.486x103 m
f 1000x103
15. The sitar strings A & B playing the note ‘Ga’ are slightly out of tune & produce beats of
frequency 6 Hz. The tension in the string A is slightly reduced & the beat frequency is
found to reduce to 3 Hz. If the original frequency of A is 324 Hz. What is the frequency
of B?
We have fb = fA - fB
I case fB = 6 Hz, fA = 324 Hz
possible values of fB = 318 Hz (or) 330 Hz

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 UNIT VIII: Oscillation and wave 8660658206
II case when tension in the string A is reduced, its frequency (fA) also decreases,
the new beat frequency is given to be 3 Hz. This is possible only if fB = 318 Hz.
16. A sinusoidal wave propagating through air has a frequency of 200 Hz. If the wave speed
is 300 ms-1, how far apart are the two points (path difference) with a phase of difference
of 60o c Given f = 200 Hz ; v = 300 ms-1,Phase difference = 600
v 300
We have   1.5m
f 200

Path difference = phase difference
2
1.5 
path difference = x  Path difference = 0.25m
2 3
Five marks problem.
1. A stone dropped from the top of tower of height 300 m high splashes into the water of a
pond near the base of a tower. When is the splash heard at the top given that the speed of
sound in air is 340 ms-1? (given=9.8ms-2)
Let time taken by stone to reach the surface of water dropped from top of the tower of
height 300 m be t1
we have x = vt+1/2at1 2
-300 = 0 (t1) +1/2(-9.8)t12
t1 = 7.82 s
Let the time taken by sound to reach the person on the top of tower be t2.
t2 =300/340=0.88s
The splash of sound is heard after a time (t) equal t1 + t2.
i.e., t = t1 + t2
= 7.82 + 0.88  T = 8.7 s
2. A transverse harmonic wave on a string is described by
y(x, t) = 3 sin (36 t + 0.018x +/4)
where x & y are in cm and t is in s. i) Is the wave traveling (or) stationary.
ii) What is the direction of its propogation
iii) What is its frequency?
iv) What is its initial phase?
v) What is the distance between two consecutive crests in the wave?
Given equation is y =(x, t) = 3 sin (36t + 0.018x +/4 ) ………..(1)
This equation is of form y (x, t) = a sin (w t + kx + f) …………… (2)
i) It is a traveling wave
ii) It travels from right to left (i.e, along the negative x direction)
iii) Comparing equation (1) & (2) we get
w = 36
2f = 36
f=36/  f = 5.73 Hz
iv) Initial phase =/4 radian
v) Distance between two consecutive crests is its wavelength (l)
Comparing (1) & (2) we have
k =2/= 0.018
= 2/0.018
= 349 cm  = 3.49 m
3. The transverse displacement of a string (clamped at both ends) is given by y(x,t) = 0.06 sin
(2/3) x cos(120 t). where x, y are m & t is in s. The length of the string is 1.5 m & its
mass is 3 x 10-2 kg.
i) Does the function represent a traveling wave (or) a stationary wave?
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 UNIT VIII: Oscillation and wave 8660658206
ii) Interpret the wave as a superposition of two waves traveling in opposite directions
what is the wavelength, frequency and speed of each wave?
iii) Determine the tension in the string.
Given y(x,t) = 0.06 sin (2/3) x cos(120 t)…………………………(1)
and y(x, t) = 2a sin kx cos w t ………………………………………….(2)
i) It is a stationary wave.
ii) Comparing equation (1) and (2); we get k=2/=2/3.
= 3m
= 2f = 120
f = 60 Hz
v = f = 60 × 3 = 180 ms-1  v = 180 ms-1
iii) Given m = 3 × 10-2 kg, L= 1.5 m
m 3x10 2
linear density     2x10 2 kg / m
L 1.5
We have T
v

 T = v2  T = (180)2x2x10-2 = 648 N
4. A progressive wave is described by the equation y = 1.2 sin  2t  x  where x & y are
 5 4
in m and t is in s. Determine the amplitude, wavelength, time period & speed of the wave?
 2t x 
Given equation is y = 1.2 sin   
 5 4
 2t x 
y = 1.2 sin    .............................. (1)
 5 4 
This equation is of the form y = a sin (w t – kx) ………… (2)
Comparing equation (1) & (2)
2 
We get, a = 1.2 m k= 
 4
  8m
2 2
= 
T 5
T=5s

v  1.6 m / s
T
5. A closed pipe of length 0.42 m and an open pipe both contain air at 350 C. The frequency
of the first overtone of the closed pipe is equal to the fundamental frequency of the open
pipe. Calculate the length of the open pipe and the velocity of sound in air at 0oC. Given
that the closed pipe is in unison in the fundamental mode with a tuning fork of frequency
210 Hz.
Let Lc & Lo be the length of closed pipe and open pipe respectively
Given l c = 0.42m t = 35oC
(frequency of first overtone Closed pipe) = (fundamental frequency open pipe
3v 350 c v 350 c
ie  210Hz ………………………(1)
4Lc 2L 0
Fundamental frequency of closed pipe = 210 Hz
v 350 c
ie  210Hz ……………..………………(2)
4L c

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 UNIT VIII: Oscillation and wave 8660658206
4L c 4(0.42)
from (1) we get, L0= L0 =
6 6
From (2) we get v 350 c  210x 4L c = 210 x4 x 0× 42
= 352.8 ms-1
We have v a T
v 00 c 0  273 273
   v 00 c  xv 0
v 350 c 35  273 308 35 c
273
 v 00 c  x 352.8 v0 = 332.15 ms-1
308
6. y = 1.4 sin  (300t – x) represents a progressive wave where x, y are in m & t is in s.
Calculate the wave velocity & the phase difference between oscillatory motion of two
points separated by a distance of 0.25 m.
Given equation is y = 1.4 sin  (300t – x)
This can be rewritten as y = 1.4 sin (300 t - x) ………(1)
This equation is of the form y = a sin(t-
We have v=f …………………………………………… (3)
Comparing equation (1) & (2)
We get, =2f=300
f= 150 Hz
2
k 

=2m
from (3) v = 150 x2
v = 300 ms-1
2
phase difference= Path difference

2
phase difference= (0.25)

2
phase difference= (0.25)
2

phase difference= radian.
4

Questions for OSILLATION;

One and two mark Questions :
1. What happens to the time period of a simple pendulum, when it is taken to moon?
2. What is a periodic motion? Give an example.
3. What is an oscillatory motion? Give an example.
4. What is the mean or equilibrium position of an oscillating body.
5. Write the relation between time period and frequency of oscillation.
6. Define amplitude, time period and frequency of oscillations.
7. Define a phase of particle in oscillatory motion.
8. What are free oscillations?
9. What are damped Oscillations? Give an example.
10. What are forced oscillations?
11. What is a resonance?
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 UNIT VIII: Oscillation and wave 8660658206
12. A particle takes 20s to make 10 oscillations. Calculate time period and frequency.
13. The Oscillations of a particle is given by y = 10 sinωt where displacement Y is in metre.
Find the amplitude.
14. Which of the following relationships between the acceleration a and the displacement x of a
particle involve simple harmonic motion?
a) a= 0.7 x b) a= -200x2 c) a=-10x d) a=100x3
Problem;
1. The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m.
If the piston moves with simple harmonic motion with an angular frequency of 200
rad/min, what is its maximum speed?
2. The acceleration due to gravity on the surface of moon is 1.7 m s-2. What is the time period
of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5
s? (g on the surface of earth is 9.8 ms-2)
Four and Five marks questions;
1. For a body executing simple harmonic motion, explain (i) amplitude, (ii) period, (iii)
frequency
and (iv) phase.
2. Define simple harmonic motion. D
3. A simple pendulum of length L and having a bob of mass M is suspended in a car. The car
is moving on a circular track of radius R with a uniform speed v. If the pendulum makes
small oscillations in a radial direction about its equilibrium position, what will be its time
period?
4. Arrive at the expression for time period of oscillation of a mass attached to a vertical
spring.
5. Show that in simple harmonic motion, the acceleration is directly proportional to its
displacement at the given instant.

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