Waves

WAVES SYNOPSIS
CLASSIFICATION OF WAVES
• Mechanical Waves: The waves that requires material medium for their propagation are called
Mechanical Waves.
Ex: Sound waves, waves on strings and springs.
• Charecteristics of medium to propgate waves: Medium should posses
1) Interia 2) Elasticity 3) Low resistance
• Non-mechanical Waves: The waves that do not require material medium for their propagation are
called Non-Mechanical waves. These are the waves that can transfer energy even in vacuum. They
propagate due to variation of electric and magnetic fields.
Ex.: Light waves, X-rays,  -rays, radio waves.
On the Basis of Energy Propagation
• Progressive Wave: A wave that propagates energy from a point in an infinite medium and never
returns to that point is called a progressive wave.
• In these waves all the particles of the medium execute SHM with same amplitude and same
frequency.
• Changes in pressure and densities occur at all points of the medium.
• All the particles of the medium cross the mean position once in one time period.
• Average energy over one time period is equal to the sum of K.E. and P.E.
• Waves may be one dimensional, two dimensional, three dimensional according as they propagate energy
Ex.:
1) Transverse waves along a string are one dimensional
2) Ripples on water surface are two dimensional
3) Sound waves proceeding radially from a point source are three dimensional.
• Stationary Wave: When two identical progressive waves traveling in opposite directions superpose,
stationary waves are formed.
On the Basis of Vibration of Particles
• Transverse Wave: In a Transverse wave the particles of the medium vibrate perpendicular to the
direction of propagation of the wave. Ex : Waves in Strings.
Light waves are transverse electro-magnetic non-mechanical waves.
• It travels in the form of crests and troughs.
• These waves can propagate only in solids and on the surface of water.
• These waves can be represented by
Y  a Sin  t  kx  Z  a Sin  t  kx  Z  a Sin t  ky Y  a Sin   t  kz 
• These waves can be polarised.
• Longitudinal Wave: In a Longitudinal wave the particles of the medium vibrate in the direction of
propagation of the wave.
These are also known as pressure or compression waves. Ex : Sound Waves in air
• Travel in the form of compressions and rarefactions.
• They can be produced in solids, liquids and gases.
• Pressure and density vary. Sound waves are longitudinal mechanical waves.
• These waves can be represented by x  a sin(t  kx) .
• These waves cannot be polarised.
• A mechanical wave may be transverse or longitudinal. A non mechanical wave is always transverse.

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PROGRESSIVE WAVE & PULSE EQUATION
when a wave passes thorugh a point , y = f(t) represents displacement of particle as a function of
x
time t at that point .If t is replaced by  t   , then the same function represents wave pulse
 v
equation . Here ‘- ‘should be taken when the pulse is travelling along positive X-axis . While ‘+’ sign
should be taken when it travells along negative X- axis.
• In general y = f(x, t) represents progressive wave pulse equations .
• y = f(vt  x)represents progressive wave pulse
• In case of harmonic waves (sound wave ), the function is either ‘sin’or ‘cos’.
Equation of Progressive Harmonic Wave
(Or) Equation of Sound Wave
y  A sin(t  kx)
+ sign for a wave travelling along –ve X–direction
– sign for a wave travelling along +ve X direction where,
y is displacement of the particle after a time t, from mean position.
x is displacement of the wave.
A is Amplitude.
 is angular frequency or angular velocity   2 / T  2n
k is propagation constant k  2 / 
• Number of waves acomodated in unit distance is called wave number . It is equal to 1 /  .
• Change in phase per unit advancement incase of harmonic wave is called propagation constant k.
2 
k   , where V is wave velocity..
 V
• The sound wave is also represented as
     t x 
y  a sink  t  x  y  a sink(Vt  x) y  a sin2   
 k    T  
• The equation of sound wave can also be taken as y  a sin(t  kx   ) depending on the intial phase
at the origin .
d2 y 2
2 d y
• Differential equation of wave propagation is  V  0 . The any solution of this equation
dt 2 dx 2
[i.e, y = f(x,t)] represents wave pulse equation. The solution may be in the form of
y  a( x  Vt ) 2 , y  a x  Vt

and y  a log( x  Vt ) ,
but these are not useful to represent wave pulse.The function should be continuous and finite .
Shape Curve
For a given time ‘t’, y  x graph gives the shape of pulse on string.
• The function y = f(x) represents displacement of a particle at a distance x from the origin of wave.
If x is replaced by ( x  Vt ) , then the above function represent wave pulse .
• The shape curve is plotted according to the function y  f ( x  Vt ) .
• We can determined the motion of various particles of medium at given time ‘t’ using shape curve.

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Phase of Harmonic Wave
Phase gives the state of the vibrating particle at any instant of time as regards to its position and
direction of motion.
• Phase is the angular displacement from its mean position.
 = (  t  kx)
• If phase is constant then the shape of wave remains constant.
• Path difference is the difference of distances transferred by two vibrating particles.
• Phase difference between two points on a wave = 2 /  (Path difference)
  ( 2 /  )x
• Time difference is the difference of times taken by the vibrating particle in completing one vibration.
2
   time difference
T
T
Time difference =  path difference.

• If two sources emit waves of frequencies n1 and n2 simultaneously, then the phase difference between
these two waves after a time t is given by
  (2  1 )t  2 (n2  n1 )t
• a) Wave Velocity : It is the displacement covered by the wave per unit time.
Wave velocity, V  n   / T   / k
Wave velocity is constant as it doesnot depend on time.
b) Particle Velocity: When a wave propagates through a medium, the velocity of the particle in
SHM is called particle velocity.
V   A 2  y 2 or V  a cos t

• At y = 0, V is maximum, Vmax   A
• At y   A , V is minimum, Vmin = 0
• Particle velocity varies with time.
• In case of wave travelling along positive X-axis, particle velocity at a given position and time is
equal to negative of the product of wave velocity and slope of shape curve at that point i.e.
 dy 
Vparticle  V wave  
 dx 
• In case of wave travelling along negaitive X-axis, particle velocity at a given position and time is
equal to the product of wave velocity and slope of shape curve at that point i.e.
 dy 
Vparticle  V wave  
 dx 
SUPERPOSITION OF WAVES
• Principle of superposition of waves:
When two or more waves propagating in the medium superimpose over a common particle of the
medium, then the resultant displacement of the particle is the vector sum of the displacement of
the particle due to each wave as if other waves are absent.
    
y  y1  y2  y3  .......  yn
The resultant amplitude
    
a  a1  a2  a 3  .....  an

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• This law is applicable for all types of waves like light waves, sound waves etc. but not applicable to
large amplitude waves like shock waves, laser waves etc.
Phenomena arising due to superposition of waves.
a) Interference b) stationary waves c) Beats d) Lissajous figures
• In superposition phenomena, energy is conserved and redistributed.
• Superposition is possible only in waves of similar nature.
• When a mechanical wave or an electromagnetic wave is reflected from a rigid surface, a phase
difference of  is introduced in the reflected wave. y1  a sin(t  kx) is the wave propagating
along +ve X-direction
The reflected wave is y 2  a sin[(t  kx )   ]
• When a wave is reflected from free end there is no phase change.
y1  a sin(t  kx) propagating along +X- direction y 2  a sin(t  kx) reflected wave along
–X-direction.
• When a wave refracts (transmits) there is no phase change
Refraction of Rigid Surface
T 
• Change in phase is  ; change in time is . Change in path is . Compression returns as compression
2 2
and rarefaction as rarefaction. But crest returns as trough and trough returns as crest.
Refraction from Free End
• Change in phase = 0, change in time = 0, change in path = 0. Compression and rarefaction return as
rarefaction and compression respectively. But crest and trough return as crest and trough respectively.
• Sound produced in air is not heard by a diver inside water since most of the sound energy is
reflected at the surface of water.
Stationary or Standing Waves
• When two identical progressive waves moving in opposite directions superpose, stationary waves
are formed.
• There is no transfer of energy.
• Nodes are the points where the displacement and velocity are zero. At the nodes, the pressure
changes are maximum.
• Antinodes are the points where the displacement and velocity are maximum. At antinodes pressure
changes are zero.
• Distance between two successive nodes or antinodes is  / 2 .
• Distance between a node and an immediate antinode is  / 4.
• In a loop, all the particles vibrate with same frequency, time period. However, their amplitudes
vary ranging from zero to maximum.
• All particles within a loop are in same phase or phase difference between any two particles within
a loop is zero.
• The Phase difference between any two particles in successive loops is  radian or 1800.
• Amplitude is a function of displacement whereas phase is a function of time.
• Both longitudinal and transverse waves exhibit stationary wave phenomena.
• In strings (under tension) if reflected wave exists, the waves are transverse stationary, while in organ
pipes, waves are longitudinal stationary.
• Strain is maximum at nodes and minimum at antinodes.
• Particles do not execute SHM at nodes where as they execute SHM at antinodes.

• Nodes are obtained at x   2n  1 where n = 0, 1, 2 ....
4
n
• Antinodes are obtained at x  where n = 0, 1, 2,.....
2
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• In longitudinal stationary waves, nodes are points of maximum pressure (minimum displacement)
while antinodes of minimum pressure (maximum displacement).
• Total energy confined in a segment (elastic P.E + K.E) always remains the same.
Equation of Stationary Wave
• If the wave is reflected at free boundary
Y = 2A Cos kx Sin  t (where 2 A Cos kx is the amplitude) or
If the wave is reflected at fixed boundary Y = 2A sin kx Cos  t
(where 2 A Sin kx is the amplitude)
• In stationary waves, amplitude is the function of displacement
Interference of Two Waves,
Moving in Same Direction
• If the displacements of two waves reaching at a point are y1 and y2, then
y1 = a1 sin  t .....(1)
y2 = a2sin(  t +  ) .....(2)
• The equation for the resultant displacement in the region of superposition is
y = A sin(  t +  ). Where  is resultant phase constant.
Resultant amplitude A  a12  a 22  2 a1 a2 cos
• The intensity is directly proportional to the square of the amplitude. Hence the resultant intensity
I is given by I = KA2.
• Resultant intensity I  I1  I 2  2 I1 I 2 cos 
• The resultant intensity at any point depends upon the phase difference  between two waves at
that point.
Condition for maximum intensity:
• Phase difference :
  2n , where n = 0,1,2,3,4....
 Path difference  n
I max  I 1  I 2  2 I 1 I 2
Condition for minimum intensity:
• Phase difference :
  ( 2n  1) , where n  1,2 ,3,4.....

 Path difference   2 n  1 2
• I min  I1  I 2  2 I1 I 2
2 2
I max  I 1  I 2   a1  a2 
•   
I min  I 1  I 2   a1  a2 
• The phenomenon of interference is based on conservation of energy.
• When a standing wave is formed due to reflection, then standing wave ratio (SWR)
Amax A1  A2
 
Amin A1  A2 where
A1  incident amplitude,
A2  reflected amplitude
For 100% reflection, SWR  
For no reflection, SWR = 1

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VIBRATION OF STRETCHED STRINGS
• Sonometer: Sonometer consists of a stretched wire whose length can be changed and the tension
in the wire can also be changed. When the sonometer wire is made to vibrate, transverse stationary
waves are produced in it.
• Integral multiples of fundamental frequency are called Harmonics.
• The frequencies that are produced by in instrument successively are called over tones
• When sonometer wire vibrates in 1 loop then its frequency of vibration is called 1st harmonic. (n1)
(or) Fundamental note
V  T / m and 1  2l
1 T
n1  V /  ,  n1  2l m
Where l is the length of the string.
• When the sonometer wire vibrates in 2 loops then its frequency of vibration is called 2nd harmonic
( n2 ) or 1st overtone 2  2l / 2 .
1 T
n2  2    2n1
 2l  m
• When the sonometer wire vibrates in 3 loops, then its frequency of vibration is called
3rd harmonic ( n3 ) or 2nd overtone. 3  2l / 3
1 T
n3  3   3n1
 2l  m
• When sonometer wire is vibrating in ‘x’ loops then its frequency of vibration is called xth harmonic
or (x – 1)th overtone.  x  2l / x
1 T
n x  x   xn1
 2l  m
• The ratio of the frequencies of the various modes of vibration is 1 : 2 : 3 : 4 : …….
• The difference in the frequencies of successive overtones is equal to the fundamental frequency.
1 T
n2  n1  n3  n2  n4  n3  n1   
 2l  m
• Fundamental frequency
1 T 1 T 1 T  1  T 1 T
n1        2
  
 2l  m  2l  A  2l  r   2lr   ld 
Where d is the diameter of the wire.
• Laws of transverse vibrations of stretched strings:
• Law of tension: n  T
n1 T
 1
n2 T2
1
• Law of length: n 
l
n1 l2

n2 l1
1
• Law of linear density : n 
m
n1 m2

n2 m1
These laws can be verified using a sonometer.
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Graphs

n n n

n T T

Mg l 1/l
• A metal wire of length ‘ l ’ and cross- section A is stretched to get elongation ‘e’ and fixed between two
YAe( l  e)
rigid supports. When this wire is plucked, then the velocity of transverse waves is V 
Ml
where Y = Young’s modulus, M= mass of wire
1 YAe( l  e )
Now its fundamental frequency is n 
2(l  e) Ml
• l If a wire held at the two ends by rigid supports is just taut at t°C then the velocity of the transverse

wave at t°Cis given by V  T YA t Y  t

 
m A 
where Y is the Young’s modulus and  is the co-efficient of linear expansion of the material of the
wire.  t  t 1  t 2 (t 1  t 2 )
• A metal wire under tension T fixed between two supports is heated through temperature range t .
1 T  YAt
Now the wire is excited, then its fundamental frequency is n 
2l m
Where Y= young’s modulus, A = cross- sectional area ,  = linear coefficient of expansion and
m = linear mass density.
When the above wire is cooled through temperature t , then fundamental frequency is
1 T  YAt
n
2l m
• A wire of uniform cross–section is fixed at one end and is attached to a load M passing over a pulley
at the other end.
Mg
Velocity of the transverse wave, V =
m
• If the load is submerged in a liquid of density dL then the velocity of wave is,
d
( Mg  vd L g ) Mg (1  dL )
V1= = S
m m

where d S is the density of the material of the load suspended

V1 dS  dL

V dS
• If n and n1 are the fundamental frequencies of the stretched string with the load in air and when
 d 
Mg  1  L 
n1 T  dS    dS  dL 
submerged in liquid then,  1 
n T2 Mg dS

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• Velocity of wave in vertical strings is different at different points:

V3

V 2 V 3 > V2 > V1

V1

• In case of vibration of composite string (string made up by joining two strings of different lengths,
cross sections and densities) having same tension throughout , the joint is a node.The lowest common
fundamental frequency of string will be f  P1 f1  P2 f 2 , Where f1, f2 are the fundamental frequencies
of indvidual strings and P1, P2 are number of loops in the two strings at common frequecncy.
The higher harmonics of composite string are the integral multiples of f.
• When a longitudinal wave is set up in a metal wire , then longitudinal strain at distance x from one
 dy 
end is   .
 dx  at x
 dy 
Now longtiudinal or tensile stress at that point is Tensile stress  Y   
 dx  at x
Where Y = young’s modulus of wire
• When a transverse wave is set up in a metal wire under tension, shape of the string changes along
 dy 
its length. The vaule   represents angle of shear or deformation strain
 dx  at x
Now tangential or shearing stress at that point is
 dy 
Ten gential stress      Where  = rigidity modulus
 dx at x
Pressure Wave
The longitudinal wave propagation in gases is also described in terms of pressure variations. Thus
longitudinal waves in gases are also called pressure waves .
• When the displasement wave is y  a sin(t  kx) ,then the equation of Pressure wave will be ,
P  P0 cos(t  kx) ,where P  change in pressure of gas at a point (x, t)
P0  Pressure amplitude
 Maximum pressure change

• Pressure amplitude P0  Bak , where B = Bulk modulus, a = displacement amplitude and
k = propagation constant. Here B  V 2
where  = density of gas , V= velocity of wave.

• The phase difference between pressure wave and displacement wave is .
2
• In case of standing wave , displacement node is pressure antinode. Whlie displacement antinode is
pressure node
• Standing waves in organ pipes can also be described mathematically using pressure waves.

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SOUND
Hearing of sound is characterised by following three parameters.
Loudness (Refers to Intensity):
It is the sensation received by ear due to intensity of sound
Greater the amplitude of vibration, greater will be intensity ( IA 2 ) and so louder will be sound.
The loudness being the sensation, depends on the sensitivity of listener’s ear. Loudness of a sound
of a given intensity may be different for different listeners.
SI unit of loudness is bell (B).
1 bell (B)= 10 decibell (dB)
I
• The loudness of sound of intensity I is l  10 log 10 ,
I0

where I 0  10 12 watt/m 2 (threshold off audibility)

Pitch (Refers to Frequency):
The shrillness or harshness of sound is known as pitch. Pitch depends on frequency. Higher the
frequency, higher will be the pitch and shriller will be the sound.
Quality or Timber (Refers to Harmonics):
It is the sensation received by ear due to waveform. Quality of a sound depends on number of
overtones. i.e, harmonic present.
• Energy Of Wave:
• The average KE per unit volume during the wave propagation is  2 a 2 n2 .
Here
  Density , a = Amplitude, n = Frequency
2 2 2
• The average PE per unit volume during the wave propagation is  a n
• Total energy per unit volume or energy density is the sum of average kinetic energy , averge potential
energy densities.
1
U  2 2 n 2 a 2  
 2 a 2
2
• Total energy ‘E’= energy density × volume
1
E  2 a 2  volume
2
• The rate of energy transmission is called power of source .
E 1  volume 
P   2 a 2  
t 2  time 
1
P   2 a 2 V  A
2
Where V = wave velocity, A= area normal to energy transmission
• Intensity Of Wave:
The power transmission per unit area is called intensity I.
P 1
I    2 a 2 V  2 2 n 2 a 2 V
A 2
 I = Energy density × wave velocity
P0
• Intensity of longtudial wave in terms of pressure amplitude P0 is I  ,
2a
where, a = Displacement amplitude and  = Angular frequency

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• Human ear responds to sound intensities over a wide range from 10-12 W/m2 to 1 W/m2.
• In a spherical wave ( i.e. wave starting from a point source),
P 1
I , r = distance from the source I 
4r 2 r2
2 1
As I  a  amplitude a 
r
P
• In a cylindrical wave (i.e. wave starting from a linear source ), I  ,
2rl
where l = length source and r = distance from the source
1 1
. I   a 
r r
VELOCITY OF SOUND
E
• The equation for velocity of sound through a medium is given by V 
d
where E = modulus of elasticity; d = density
• As modulus of elasticity is more for solids and less for gases, so
Vsolids  Vliquids  Vgases
Y
• In case of solids V  ,
d
where Y is Young's modulus,
K
• In case of fluids (liquids and gases) V 
d
where K is the Bulk modulus
Velocity of Sound in Gases
• Newton’s formula :
Newton assumed that the propagation of sound in a gas takes place under isothermal conditions.
• Isothermal Bulk modulus , K = P
P
V 
d
1.013  10 5
• At S.T.P. V   280 ms 1
1.29
Which is less than experimental value (332 m/s)
• Laplace’s correction :
Laplace assumed that the propagation of sound in a gas takes place under adiabatic conditions.
• Adiabatic Bulk modulus, K  P
P
V 
d
• For air,  = 1.4.
Therefore, V  280 1.4  330 ms 1
which agrees with the experimentally calculated value.
• From ideal gas equation,
pm m P RT
PV  nRT   RT  
d M d M

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rRT
V 
M
• Velocity of sound in a gas is directly proportional to the square root of the absolute temperature
1
Vt T  t  273  2
 t 
    Vt  Vo 1  
Vo To  273   546 
• When temperature rises by 1°C then velocity of sound increases by 0.61 m/s
• The velocity of sound increases with increase in humidity, sound travels in moist air than in dry air
at the same temperature, because density of humidity air is less than that of dry air.
• The velocity of sound of constant temperature in a gas does not depend upon the pressure of air.
Types of Vibrations
• Natural Vibrations :
When a body is set to vibrate and left free to itself, it vibrates with a frequency called its natural
frequency and the oscillations of such a body are called free oscillations or natural vibrations.
• The natural frequency of a body depends upon its elastic constants, its geometrical shape (or
dimensions) and mode of vibration.
• Ex : The natural frequency of a simple pendulum is given by
1 g
n
2 L
Where L is the length of the pendulum and g is the acceleration due to gravity.
• Forced Vibrations :
• If a body is set to vibrate under the action of an external periodic force, the body vibrates with a
frequency of the external periodic force. Such a frequency is called forced frequency and such
oscillations are called forced oscillations.
• The amplitude of vibration increases.
• Ex : Every musical instrument is provided with a hollow sounding box in which air particles
vibrate with forced frequency and produce intensity of sound.
Resonance
• If the natural frequency of a vibrating body is exactly equal to its external forced frequency, the
body vibrates with a maximum amplitude. This phenomenon is called resonance.
• Resonance is only a special case of forced vibrations.
ORGAN PIPES
• Organpipe:
An organpipe is a cylindrical tube of uniform cross section in which a gas is trapped as a column.
Open Pipe
If both ends of a pipe are open and a system of air is directed against an edge, standing longitudinal
waves can be set up in the tube. The open end is a displacement antinode.
• Due to finite momentum, air molecules undergo certain displacement in the upward direction
hence antinode takes place just above the open end but not exactly at the end of the pipe.
• Due to pressure variations, reflection of longitudinal wave takes place at open end and hence
longitudinal stationary waves are formed in open tube.
In an organ-pipe, closed end is always a node while free end an antinode.

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 Waves
• a) For fundamental mode of vibrations,

L  1 ;  1  2 L
2
A 2 A 3 A
4 4 N
N 3
A
1 2 2 N
N A
2 2 3
A
2 N
2 N 3
4 4 A
A A
(a) (b) (c)
V  1 f1 ; V  2 Lf1 ....(1)
• b) For the second harmonic or first overtone,
L  2
V  2 f 2 V  Lf 2 .....(2)
• c) For the third harmonic or second overtone,
3 2
L  3  3  L
2 3
2
V  3 f 3 V  Lf 3 .....(3)
3
• From (1), (2) and (3) we get, f1 : f 2 : f 3 .....  1: 2 : 3 :......
i.e. for a cylindrical tube, open at both ends, the harmonics excitable in the tube are all integral
multiples of its fundamental.
2L
•  In the general case,   n , where n  1, 2,.....
V nV
• Frequency   , where n  1, 2,.....
 2l

Closed Pipe
If one end of a pipe is closed the reflected wave is 180 out of phase with the wave. Thus the
displacement of the small volume elements at the closed end must always be zero. Hence the
closed end must be a displacement node.

A A 3
A
2 4 N
4
N 3
1 A
L 2
2
N
2
3
N
2 A
2
N N
(a) (b) (c)

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 Waves
• Figure (a) represents the fundamental mode of vibration:
1
L 1  4L
4
If f1 is the fundamental frequency, then the velocity of sound waves is given as
V  1 f1 V  4Lf1 .....(1)
• Figure (b) represents third harmonic or first overtone:
 4
L  5  2 ,  2  L
4 3
4
V  2 f 2 , V  Lf 2 .....(2)
3
• Figure (c) represents fifth harmonic or second overtone:
 4
L  3  3 ,  3  L
4 5
4
V  3 f3 , V  Lf 3 .....(3)
5
From (1), (2) and (3) we get,
f1 : f 2 : f3 .....  1: 3: 5 :......
4l
• In the general case,   , where n  0,1, 2,.....
 2n  1
• Velocity of sound = V
 2n  1V
• Frequency  ,
4l
where n  0,1, 2,.....
Note : The antinode formed at the open end is not exactly at that point but shifted away from pipe
by distance 0.6R (where R is the radius of cross-section of pipe). This is called end correction (e).
• The end correction (e) depends on the internal radius of the pipe.
e = 0.6 R=0.3d
Where R is radius of the pipe and d is diameter of the pipe.
• When the end correction is considered, then the fundamental frequency of open pipe
V
n
2  l  2e 
V
n
2  l  1.2 R 
V
• The fundamental frequency of closed pipe n 
4(l  e )
V
• n
4(l  0.6 R)
Velocity of Sound
(Resonance Column Apparatus)

• If l1 and l2 are the first and second resonating lengths then l1  e 
4
3 
l2  e    l2  l1
4 2
• 1)   2  l2  l1 
• 2) V  n  2n  l2  l1 
V
3) n  l  e  3  l1  e 
•
2  l2  l1  ; 2
l2  3l1
• 4) e 
2
VELOCITY INSTITUTE OF PHYSICS 13
 Waves
BEATS
• It is the phenomenon of periodic change in the intensity of sound when two waves of slightly
different frequencies superpose with each other.
• Maximum Intensity of sound is produced in the beats when constructive Interference takes place.
• Minimum Intensity of sound is produced in the beats when destructive Interference takes place.
• If A1 , A2 are amplitudes of two sound waves that interfere to produce beats then the ratio of
2
I max  A1  A2 
maximum and minimum intensity of sound is,  
I min  A1  A2 

• The variation in the intensity of sound between successive maxima or minima is called one beat.
• The number of beats per second is called beat frequency. If n1 and n2 are the frequencies of the two
o
sound waves that interfere to produce beats then
Beat frequency = n1 ~ n2
• The time period of one beat (or) the time interval between two successive maxima or minima is
1
n1 ~ n2
1
• The time interval between a minimum and the immediate maximum is 2n ~ n 
1 2

• Maximum number of beats that can be heard by a human being is 10 per second.
If more than 10 beats produced then no. of beats produced are same but no. of beats heard are zero
n1  n2
• Frequency of variation of amplitude 
2
n1  n2
• Frequency of resultant wave 
2
In beats, Amplitude is function of time.
Analytical Treatment of Beats
• Equations of waves producing beats are given as y1  a sin 1t and y2  a sin 2t let 1  2
• Resultant wave equation is
   2   1  2 
y  y1  y2  2a cos  1  t sin  t
 2   2 

   2 
y  A  t  cos  1 t
 2 

 1  2 
Here A  t   2a sin t
2

n1  n2
• Amplitude is function of time. Frequency of variation of amplitude 
2
n1  n 2
• Frequency of resultant wave
2

Uses of Beats
• To determine unknown frequency of a tuning fork with the help of a standard tuning fork.
• To tune the stretched string of a musical instrument to a particular frequency.
• To detect the presence of dangerous gases in mines.

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 Waves
DOPPLER’S EFFECT
• It is the phenomenon of apparent change in the frequency of sound which is heard by a listener
due to relative motion between listener and source of sound.
In the following cases, apparent frequency is greater than original frequency.
• Listener approaches stationary source of sound.
• Source of sound approaches stationary listener.
• Both listener and source of sound approach each other.
In the following cases apparent frequency is less than actual frequency.
• Listener moves away from stationary source of sound.
• Source of sound moves away from stationary listener.
• When the sound source moves towards the observer then the waves contract i.e. the wavelength
decreases and consequently frequency increases.
• When the source moves away from the observer, then the waves spread i.e. the wavelength increases
and consequently frequency decreases.
• Both, listener and source of sound move away from each other.
• Let V be the velocity of sound in still air and V w is the velocity of wind.
• If wind is blowing from source of sound to listener then, velocity of sound in air  V  V w
• If wind is blowing from listener to source of sound then velocity of sound in air  V  V w
• Doppler effect in sound is asymmetric.
• Doppler effect in light is symmetric.
• Generally, Doppler effect is well applicable when velocity of listener and velocity of source of sound
are much less than velocity of sound in air.
( V0  V  V s )

• To measure the velocity of automobiles by traffic police.

• This effect is used in accurate navigation of air craft and is basis of target bombing techniques.
• To measure velocities of stars relative to earth.
• To measure speed of rotation of the sun.
• To detect double stars and rotation of saturn rings.
• Expressions for apparent frequency ( n1 ) in terms of actual frequency (n) :
V - Velocity of sound in stationary medium
V0 - Velocity of Listener
VS - Velocity of source of sound
 V  Vo 
n1    n
 V  Vs 
• Listener and source at rest. n1  n
 V  Vo 
• Listener moving towards stationary source. n1   n
 V 
 V  Vo 
• Listener moving away from stationary source. n1   n
 V 
 V 
• Source moving towards listener at rest. n1   n
 V  Vs 
 V 
• Source moving away from listener at rest. n1    n
 V  Vs 
 V V 
o
• Source moving towards listener and listener moving away from source. n1   V  V  n
 s 

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 Waves

o
V V 
• Source moving away from listener and listener moving towards source. n1   V  V  n
 s 
 V  Vo 
• Source and listener moving towards each other. n1   n
 V  Vs 
 V  Vo 
• Source and listener moving away from each other. n1   n
 V  Vs 

VS VO

• 2 1
S O
 V  V0 cos 1 
n1   n
 V  V s cos 2 
• Doppler’s effect in vector form is written as
 V  V0 .r 
n1   n S O
 V  V .r 
 s 
r̂  unit vector along the line joining source and observer
V  Velocity of sound in the medium. Its direction is always taken from source to observer..
• Shift in the position of spectral lines of a star towards greater wavelength side (Red shift) indicates
that universe is expanding.
• If the source of sound is moving towards a wall and the observer is standing between the source
and the wall, no beats are heard by the observer.

• When the source and the observer both are moving with same velocity in same direction.
• When Vs and Vo are perpendicular to line joining source and observer..
• Doppler’s effect in sound does not take place in transverse direction.
• When the medium only is moving then Doppler effect is not observed.
• When the distance between the source and the observer is constant then also Doppler effect is not
observed.
• When source moves and observer is at rest at origin as shown in figure then

VS cos 
P VS Q VS R 
S VS
 

VS cos 

O
 V 
1) When source is at P nP   n 2) When source reaches Q then n Q  n
 V  Vs cos 
 V 
3) When source reaches R then nR   V  V cos n
 s 

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 Waves
• If source is at origin and observer moves with constant velocity V0 as shown

VO cos 
P VO Q VO R 
O VO
 

V O cos 

then

nP  
 V  V0 cos   n
1) When observer at P, 
 V 
2) When observer reaches Q, nQ  n

 V  V0 cos  n
3) When observer at R, nR   
 V 
• When observer is at center and source is rotating along circular path (or) source is at
center and observer is rotating along circular path then there is no change between actual
and apparant frequencies. n1  n
• If the observer is standing outside the circular track and source is rotating along the circle then

VS
VS R

S Q O
P VS

 V 
nP  nmax   n; nQ  n
 V  VS 
 V 
nR  nmin   n
 V  VS 
• If the source is at rest outside the circular track and observer is rotating along the circle
then
VO
 V  V0  VO R
nP  nmax   n ,
 V 
nQ  n ,
Q S
 V  V0  O P VO
nR  nmin   n
 V 

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 Waves
• Consider a source of sound of frequency ‘n’ Hz is moving rapidly towards a wall
with a velocity of V S m/s.
Case (i): If the observer is at rest in between source and wall as shown

Source Observer Image Of Source

Wall

 V   V 
ndirect   n;
 nreflected   n
 No.of beats  0(ndir  nref )
 V  VS   V  VS 

Case (ii): If the source is in between observer and wall

Vs Vs

Observer Source Image Of Source

Wall

 V   V  2 VS n
nd   n ; nr   n No. of beats =
 V  V S   V  V S  V
Case (iii): If the observer and source are moving together with velocity x towards a wall

x
x
Source Image Of Source

V x
nd  n; nr   n
V x
2 xn
Number of beats = 
V x
• When source is in motion and observer is at rest, apparent wavelength of the wave is given by
 V  Vs 
1    ;
 V 
‘+’ for source moving away observer
' ' for source moving towards observer

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 Waves
ECHO
• The sound reflected by an obstacle which is heard by a listener is called an echo.
• Persistence of hearing is the minimum interval of time between two sound notes to distinguish
them.
Persistence of hearing is 0.1s
• A person is at a distance ‘d’ from a reflected surface (a wall, mountain etc). The person sounds a
Vt
horn and hears its echo at the end of a time ‘t’. If V is the velocity of sound in air then. d 
2
V  0.1 V
To hear a clear echo, the minimum distance of the obstacle, dmin  
2 20
If V = 330 ms-1 then dmin = 16.5m
If V = 340 ms-1 then dmin = 17 m
• A person in an open car approaching a vertical wall with a velocity VC , sounds a horn when the car is
at a distance ‘d’ from the wall and listens the echo after a time ‘t’. If V is the velocity of sound in air
then,

 V  VC 
d t
 2 
• If the car is moving away from the mountain then,

 V  VC 
d t
 2 
• A person standing between two parallel cliffs fires a gun and listens to the 1 st and 2 nd
echoes after time t 1 and t 2 second respectively,then
3rd echo  t1  t 2

4 th echo  t1  t 2  t1  2t1  t 2

5 th echo  t1  t 2  t1  t 2  2t1  2t 2
• In above case if d is the distance between the two mountains then

t t 
d  V 1 2 
 2 
• l Echo is used to find the depth of the sea , height of the aeroplane. If ‘d’ is the depth of the sea and
L is the distance between transmitter T and receiver R of sound and t is the time interval between
transmission and reception of the wave, then

d  (V 2 t 2  L2 ) / 4

T L R

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 Waves
CLASSWORK
CLASSIFICATION OF WAVES & DISPLACEMENT 5. A wave has a frequency of 120Hz. Two points
RELATION IN A PROGRESSIVE WAVE
at a distance of 9m apart have a phase difference
of 1080°. The velocity of the wave is
1. The equation y  4 sin  [ 200 t  ( x / 25 )]
represents a transverse wave that travels a) 340m/s b) 300m/s
in a stretched wire, where x, y are in cm
and t in second. Its wavelength and velocity c) 330m/s d) 360m/s
are
6. A plane progressive wave has frequency
a) 0.5 m, 25 ms–1 b) 0.5 m, 50 ms–1 25 Hz and amplitude 2.5 × 10–5 m and initial
phase zero propagates along the negative
c) 1 m, 50 ms–1 d) 1 m, 25 ms–1
x-direction with a velocity of 300 m/s. The
phase difference between the oscillations at
2. A transverse wave is described by
two points 6m apart along the line of
the equation Y  Y0 sin 2 ( ft  x /  ) . The
propagation is
maximum particle velocity is equal to four
times the wave velocity if

a)  b)
2
a)    Y0 / 4 b)    Y0 / 2


c)    Y0 d)   2 Y0 c) 2 d)
4
3. A transverse wave along a string is given
7. If Young’s modulus of the material of a rod is
  Y and density is  then time taken by sound
by y  2sin  2  3t  x    , where x and y are
 4
wave to travel l length from bottom is
in cm and ‘t’ is in second. The acceleration
of a particle located at x = 4 cm at t = 1 s is
 Y
a) l b) l
a) 36 22 cm / s2 b) 362 cm / s 2 Y 

c) 36 2 2 cm / s 2 d) 36 2 cm / s 2
1 Y 1 
c) d)
4. Two waves are represented by the l  l Y
following equations y1  5sin 2  10t  0.1x 
and y2  10 sin 2 10t  0.1x  . The ratio of 8. The equation of a wave is represented as
I2 y  20 cos  ( 50t  x) . Its wavelength is
intensities will be
I1

a) 5 cm b) 2 cm
a) 1 b) 2
c) 50 cm d) 20 cm
c) 4 d) 16

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 Waves
9. A transverse wave is given
by 13. A transverse wave is travelling along a
string from left to right. The figure below
 t x
y  A sin 2    . The maximum particle represents the shape of the string at a given
T  
instant. At this instat, among the following,
velocity is equal to 4 times the wave velocity choose the wrong statement
when
y
a)   2A b)   A / 2
C
B D
c)   A d)   A / 4 A x
E
F G H
10. Two waves represented by the following
equations are travelling in the same medium

y1  5 sin 2 (75t  0.25x) a) Points D, E, F have upward positive velocity

y 2  10 sin 2 (150t  0.50 x) b) Points A, B and H have downward negative

The intensity ratio I1/I2 of the two waves is c) Point C and G have zero velocity

a) 1 : 2 b) 1 : 4 d) points A and E have minimum velocity

c) 1 : 8 d) 1 : 16
SPEED OF A TRAVELLING WAVE
11. The equation of a propagating wave is
14. The pressure of air increases by 100 mm of
y  25 sin(20t  5 x) where y is the
Hg and the temperature decreases by 1oC.
displacement. Which of the following The change in the speed of sound in air is
statements is not true?
a) 61 ms-1 b) 61 mms-1
a) The amplitude of the wave is 25 units

b) The wave is propagating in posi tive c) 61 cms-1 d) 0.61 cms-1

x-direction
15. The temperature at which the speed of
c) The velocity of the wave is 4 units sound will be same in oxygen as the speed
in nitrogen at 15oC is (Densities are in the
d) The maximum velocity of the particles is
ratio 16 : 14)
500 units.

12. The displacement y of a particle in a medium a) 561oC b) 56.1oC

can be expressed as
c) 5.61oC d) 5.061oC

y  10 6 sin(100t  20 x  )m , 16. The ratio of speed of sound in Nitrogen gas
4 to that in Helium gas at 300 K is ( assume
where t is in second and x in metre. The both gases to be ideal)
speed of the wave is
a) 2 : 7 b) 1: 7
a) 2000 m/s b) 5 m/s

c) 20 m/s d) 5  m/s c) 3:5 d) 6 :5

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 Waves
17. The temperature at which the speed of 22. A metallic rod of length 2.0 m is rigidly
sound in air becomes double of its value at clamped at its middle point. A longitudinal
0°C is wave is set up in the rod in such a way that
there will be 3 nodes on either side of the
a) 273 K b) 546 K clamed point. Then the wavelength of the
longitudinal wave so produced is
c) 1092 K d) Zero K
a) 2/7 m b) 4/7 m
18. Velocities of sound measured in hydrogen
and oxygen gas at a given temperature will
c) 2/5 m d) 4/5 m
be in the ratio

a) 1 : 4 b) 4 : 1 23. A wave y  a sin  t  kx  on a string meets

with another wave producing a node at
c) 2 : 1 d) 1 : 1
x  0 . Then the equation of the unknown
19. The velocity of sound is V, in air. If the wave is
density of air is increased to 4 times, the
new velocity of sound will be a) y  a sin  t  kx  b) y  a sin  t  kx 

vS vS c) y  a cos  t  kx  d) y  a cos  t  kx 
a) b)
2 12
24. The path difference between the two waves
3 2
c) 12 vS d) vS
2  2x   2x 
y1  a1 sin  t   & y2  a2 cos  t   
     
20. Two strings A and B, made of same material,
are stretched by same tension. The radius is
of string A is double of the radius B a
transverse wave travels on A with v A and
on B with speed vB. The ratio vA / vB is    
a)  b) 2    2 
2  
a) 1/2 b) 2

c) 1/4 d) 4    2
c) 2    2  d) 
  
PRINCIPLE OF SUPERPOSITION OF WAVES
25. Three waves of equal frequency having
amplitudes 10mm, 4mm and 7mm arrive at
 
21. For a standing wave, y  10sin  x  cos  84 t  m a given point with successive phase
 18 
difference  /2. The amplitude of the
the distance (in metre) between a node and
resulting wave in mm is given by
the next antinode is
a) 7 b) 6
a) 8.5 b) 9.0
c) 5 d) 4
c) 18 d) 19

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 Waves
26. Standing waves are produced by the 30. A sonometer consists of two wires of lengths
superpostion of two waves 1.5 m and 1m made up of different materials
y1  a sin t  kx  , y2  a sin  t  kx  . The whose densities are 5 g/cc, 8 g/cc and their
respective radii are in the ratio 4 : 3. The
amplitude of a particle at distance x is ratio of tensions in those two wires if their
fundamental frequencies are equal is
a) 2a sin kx b) 2a cos kx
a) 5 : 3 b) 5 : 2
c) a sin kx d) a cos kx c) 2 : 5 d) 3 : 5
31. The velocity of a transverse wave in a
REFLECTION OF WAVES & VIBRATIONS OF
stretched w i re i s 100 m s-1. If the length of
STRETCHED STRINGS
wire is doubled and tension in the string is
also doubled, the final velocity of the
27. Standing waves are produced in 10 m long
transverse wave in the wire is
stretched wire. If the wire vibrates in
5 segments and wave velocity is 20m/s , then a) 100 ms-1 b) 141.4 ms-1
the freqency is c) 200 ms-1 d) 282.8 ms-1
32. If the length of a stretched string is shortened
a) 2 Hz b) 4 Hz
by 40% and the tension is increased by 44%,
then the ratio of the final and initial
c) 5 Hz d) 10Hz
fundamental frequencies is

 x  a) 2 : 1 b) 3 : 2
28. The equation y  5 sin  cos( 450t )
 25  c) 3 : 4 d) 1 : 3
represents the stationary wave setup in a
33. In a sonometer a stretched wire is observed
vibrating sonometer wire, where x,y are in
to vibrate with a frequency of 30 Hz in the
cm and t in second. The distances of second
fundamental mode, when the distance
and third nodes from one end are (in cm).
between bridges is 0.6 m. If the string has a
linear mass of 0.05 kg m–1. The velocity of
a) 50, 75 b) 25, 50
propagation of transverse wave in the string
and the tension in the string are
c) 15, 50 d) 20, 50
a) 36m/s, 64.8N b) 46m/s, 64.8N
29. A transverse wave propagating on a
c) 54m/s, 64.8N d) 36m/s, 54.8N
stretched string of linear density 3 × 10–4 kg
m –1 is represented by equation 34. In an experiment it was found that string
vibrates in n loops when a mass M is placed
y  0.2 sin(1.5x  60t ) where x is in metre
on the pan. The mass that should be placed
and t is in seconds. The tension in the string on the pan to make it vibrate in 2 n loops
in N. with same frequency (neglect the mass of
pan) is
a) 0.24 b) 0.48
a) 2M b) M/4
c) 1.20 d) 1.80 c) 4M d) M/2

VELOCITY INSTITUTE OF PHYSICS 23

 Waves
35. Transverse waves are generated in two 39. A closed organ pipe is vibrating in first
uniform wires A and B of the same material overtone and is in resonance with another
by attaching their free ends to a vibrating open organ pipe vibrating in third harmonic.
source of frequency 200 Hz. The area of cross The ratio of lengths of the pipes respectively
section of A is half that of B while tension on is
A is twice than on B. The ratio of wavelengths
of transverse waves in A and B is. a) 1 : 2 b) 4 : 1
a) 1 : 2 b) 2 :1 c) 8 : 3 d) 3 : 8
c) 1 : 2 d) 2 : 1
40. An open pipe and a closed pipe have same
36. A string is stretched between fixed points
length. The ratio of p th overtones of two
separated by 75.0 cm. It is observed to have
pipes is
resonant frequencies of 420 Hz and 315 Hz.
There are no other resonant frequencies
1
between these two. Then, the lowest a) p b) p
resonant frequency for this string is
a) 105 Hz b) 1.05 Hz
2( p  1) 2p  1
c) 1005 Hz d) 10.5 Hz c) 2 p  1 d) 2( p  1)
ORGAN PIPES
37. The air column in a closed end organ pipe is 41. An organ pipe P1 closed at one end vibrating
vibrating in second overtone. The frequency in its first overtone and another pipe P2 open
of vibration is 440 Hz. If the speed of sound at both ends vibrating in third overtone are
in air is 330 ms–1, the length of the air column in resonance with a given tuning fork. The
is ratio of the length P1 to that of P2 is

15 16 a) 8/3 b) 3/8
a) m b) m
16 15
c) 1/2 d) 1/3

3 4
c) m d) m BEATS
4 3
38. A cylindrical resonance tube open at both 42. The natural frequency of a tuning fork P is
ends has a fundamental frequency ‘f ’ in air. 432 Hz. 3 beats/s are produced when tuning
If half of the length is dipped vertically in fork P and another tuning fork Q are
water, the fundamental frequency of the air sounded together. If P is loaded with wax,
column will be the number of beats increases to 5 beats/ s.
The frequency of Q is
f
a) b) f
2 a) 429 Hz b) 435 Hz

3f c) 437 Hz d) 427 Hz
c) d) 2f
2

VELOCITY INSTITUTE OF PHYSICS 24

 Waves
43. Two organ (open) pipes of lengths 50 cm and 48. Tuning fork A gives 3 beats/sec with an
51 cm produce 6 beats/s. Then the speed of oscillator reading 250 Hz. When oscillator
sound is nearly
reading is 256Hz the no of beats per second
a) 300 ms–1 b) 306 ms–1 with A remains same. The actual frequency
c) 303 ms–1 d) 350 ms–1 of A is

44. A source x of unknown frequency produces a) 259 Hz b) 257 Hz

8 beats with a source of 250 Hz and 12 beats
with a source of 270 Hz. The frequency of c) 255 Hz d) 253 Hz
source x is 49. A sonometer has 25 forks. Each produces
a) 258 Hz b) 242 Hz 4 beats with the next one. If the maximum
c) 262 Hz d) 282 Hz frequency is 288 Hz, which is the frequency
of last fork. The lowest frequency is
45. A tuning fork of known frequency 256 Hz
makes 5 beats per second with the vibrating a) 72 Hz b) 96 Hz
string of a piano. The beat frequency
decreases to 2 beats per second when the c) 128 Hz d) 192 Hz
tension in the piano string is slightly
50. A tuning fork produces 6 beats/sec with
increased. The frequency of the piano string
before increasing the tension was sonometer wire when its tensions are either
169N or 196N. The freqency of that fork is
a) (256+b) Hz b) (256-b) Hz
a) 162 Hz b) 190 Hz
c) (256-5) Hz d) (256+5) Hz
46. In an experiment it was found that when a c) 200 Hz d) 80 Hz
sonometer in its fundamental mode of
51. In an open pipe when air column is 20 cm
vibration and a tuning fork gave 5 beats
when length of wire is 1.05 metre or it is in resonance with tuning fork A. When
1 metre. The velocity of transverse waves length is increased by 2cm then the air
in sonometer wire when its length is 1m column is in resonance with fork B. When
A and B are sounded together 4 beats/sec
a) 400 m/s b) 210 m/s
are heard. Freqencies of A and B are
c) 420 m/s d) 450 m/s respectively ( in Hz)
47. In case of super position of waves (at x = 0),
a) 40, 44 b) 88,80
y1  4 sin(1026t ) and y 2  2 sin(1014t )
c) 80,88 d) 44,40
i) the frequency of resulting wave is
510 Hz
DOPPLER’S EFFECT
ii) the amplitude of resulting wave varies
at frequency of 3 Hz 52. The speed at which a source of sound
should move so that a stationary observer
iii) the frequency of beats is 6 per second
11
iv) the ratio of maximum to minimum finds the apparent frequency equal to
12
intensity is 9
of the original frequency
The correct statements are
a) V/2 b) 2V
a) (i), (iv) only b) (ii), (iv) only
c) (i), (iii), (iv) only d) (i), (ii), (iii), (iv) c) V/4 d) V/11

VELOCITY INSTITUTE OF PHYSICS 25

 Waves
53. A whistling engine is approaching a stationary 57. A train is moving at 30 ms–1 in still air. The
observer with a velocity of 110 m/s. The frequency of the locomotive whistle is 500 Hz
velocity of sound is 330 m/s. The ratio of and the speed of sound is 345 ms–1 . The
frequencies as heard by the observer as the apparent wavelengths of sound infront of
engine approaches and receedes is
and behind the locomotive are respectively

a) 4 : 3 b) 4 : 1 a) 0.63 m, 0.80 m b) 0.63 m, 0.75 m

c) 3:6 d) 2 : 1 c) 0.60 m, 0.85 m d) 0.60 m, 0.75 m

54. Two aeroplanes ‘A’ and ‘B’ are moving away 58. A vehicle, with a horn of frequency n is
from one another with a speed of 720 kmph. moving with a velocity of 30 m/s in a direction
The frequency of the whistle emitted by ‘A’ perpendicular to the straight line joining the
is 1100 Hz. The apparent frequency of the observer and the vehicle. The observer
whistle as heard by the passenger of the perceives the sound to have a frequency
aeroplane ‘B’ is.(velocity of sound in air is
(n + n 1 ). If the velocity of sound in air is
350 ms–1 ).
300 m/s, then
a) 300 Hz b) 400 Hz
a) n1  10n b) n1  0
c) 500 Hz d) 600 Hz
c) n1  0.1n d) n1  0.1n
55. An engine is moving on a circular path of
radius 100 metre with a speed of 20 metre 59. A source of freqency 256 Hz rotates in
per second. The frequency observed by an circular path of radius 3 m at an angular
observer standing stationary at the centre speed of 10 rad/sec. The apparent freqency
of circular path when the engine blows a heard by an observer standing at the center
whistle of frequency 500 Hz is of circular track is

a) more than 500 Hz a) 250 Hz b) 252 Hz

b) less than 500 Hz c) 254 Hz d) 256 Hz

60. A source of sound is travelling towards

c) 500 Hz
stationary observer. The freqency of sound
d) no sound is heard heard by the observer is 25% more than that
of the actual freqency if speed of sound is V,
56. An observer moves towards a stationary that of the source is
source of sound, with a velocity one-fifth of
the velocity of sound. The percentage V V
a) b)
increase in the apparent frequency is 5 4

a) 5% b) 20%
V V
c) d)
3 2
c) zero d) 0.5%

VELOCITY INSTITUTE OF PHYSICS 26

 Waves
ECHO 64. A man fired a bullet in front of a mountain
61. A person is infront of a fort wall. The and he heard its echo after 2 seconds. After
person can hear echo of sound produced by travelling a distance of 85 m towards the
him when minimum distance between mountain, he fired another bullet and heard
person and wall is 16.75 m. The velocity of its echo after 1.5 seconds. The velocity of sound
sound in air is and distance between the mountain and the
a) 330 m/s b) 335 m/s man when the first bullet was fired are
c) 337.5 m/s d) 345 m/s
62. A person in an open car is approaching a a) 340 m/s, 340 m
vertical mountain with uniform velocity
b) 340 m/s, 140 m
90 kmph. The person blows horn, then hears
its echo after a time 4 s. The distance of the car c) 140 m/s, 340 m
from the mountain when the horn was blown
is. (Velocity of sound in air is 335 ms–1) d) 140 m/s, 140 m
a) 1440m b) 1240 m
65. A truck blowing horn of frequency 500 Hz
c) 720m d) 620m
travels towards a vertical mountain and
63. A person is in between two vertical
mountains. The person fires a gun then driver hears echo of frequency 600 Hz. If
hears the first echo from the nearby velocity of sound in air is 340 m/s then speed
mountain in a time 3 s and hears second of truck is
echo from far mountain in a time 4 s. The
a) 31 m/s b) 41m/s
person hears the third echo after
a) 5 s b) 6 s c) 51m/s d) 21m/s
c) 7 s d) 8 s

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 Waves
ASSIGNMENT
CLASSIFICATION OF WAVES & DISPLACEMENT 5. The speed of sound in hydrogen at STP is V.
RELATION IN A PROGRESSIVE WAVE
The speed of sound in a mixture containing
1. The equation of progressive wave is 3 parts of hydrogen and 2 parts of oxygen
y  0.01 sin(100t  x) where x, y are in meter at STP will be
and t in second, then V
V
i) Velocity of wave is 50 m/s a) b)
2 5
ii) Maximum velocity of particle is 1 m/s
V
iii) Wave length of wave is 2π meter.. c) 7V d)
7
a) only a,c are true b) only a,b are true
PRINCIPLE OF SUPERPOSITION OF WAVES
c) only b,c are ture d) a,b,c are true
2. A wave of angular frequency  propagates 6. A Sound wave with an amplitude of 3 cm
so that a certain phase of oscillation moves starts towards right from origin and gets
along x-axis, y-axis and z-axis with speeds reflected at a rigid wall after a second. If the
c1 , c2 and c3 respectively. The propagation vel oci ty of the w av e i s 340 ms-1 and it has a
constant k is wavelength of 2 m, the equations of incident
and reflected waves respectively are:

( iˆ  ˆj  kˆ ) a) y = 3×10-2 sin  (340 t - x),
a)
c12  c 22  c 32
y = –3×10-2 sin  (340t + x) towards left
ˆ ˆ  ˆ
b) c i  c j  c k b) y = 3×10-2 sin  (340 t + x),
1 2 3
y = –3×10-2 sin  (340t + x) towards left
1
c) (iˆ  ˆj  kˆ ) c) y = 3×10-2 sin  (340 t - x),
c
y = –3×10-2 sin  (340t - x) towards left
 ˆ ˆ ˆ
d) (c  c  c ) (i  j  k ) d) y = 3×10-2 sin  (340 t - x),
1 2 3
y = 3×10-2 sin  (340t + x) towards left
3. Two sound waves are represented by
7. Two loud speakers D
L 40 m
3 1 L1 and L1, driven 1
y1  sin t  cos t & y 2  sin t  cos t .
2 2 by a common 9m
The ratio of their amplitudes is L
oscillator and 2

a) 1 : 1 b) 3:2 amplifier, are arranged as shown. The

frequency of the oscillator is gradually
c) 2 : 3 d) 2 :1 increased from zero and the detector at D
records a series of maxima and minima. If
SPEED OF A TRAVELLING WAVE the speed of sound is 330 m/s then the
4. A pressure of 100 kPa causes a decrease in frequency at which the first maximum is
volume of water by 5 × 10 –3 percent. The observed
speed of sound in water is
a) 165 Hz b) 330 Hz
a) 1414 ms–1 b) 1000 ms –1
c) 495 Hz d) 660 Hz
c) 2000 ms–1 d) 3000 ms–1

VELOCITY INSTITUTE OF PHYSICS 28

 Waves
8. Sound signal is sent 11. A uniform rope of length 12 m and mass
through a composite 6 kg hangs vertically from a rigid support .
A block of mass 2 kg is attached at the free
tube as shown in
end of the rope. A transverse pulse of
figure. The radius of wavelength 0.06m is produced at the lower
the semicircle is r. Speed of sound in air is end of the rope. The wavelength of the pulse
V. The source of sound is capable to generate when it reaches the top of the rope is
frequencies in the range f1 to f2 (f2 > f1). If n is a) 0.06 m b) 0.12 m
an integer then frequency for maximum
intensity is given by c) 0.24 m d) 0.03 m

nV nV 12. A string of length l hangs freely from a rigid

a) b) r    2 
r support. The time required by a transverse
pulse to travel from bottom to half length of
nV nV
c) d)  r  2   the string is
r

REFLECTION OF WAVES & VIBRATIONS OF l

a) lg b) g
STRETCHED STRINGS

9. A wave pulse
on a string has 2l l
c) g d) 2 g
the dimension
shown in figure. 13. The length of a sonometer wire is 90 cm and
The wave speed the stationary wave setup in the wire is
is v = 1 cm/s. If point O is a free end. The represented by an equation
shape of wave at time t = 3 s is  x 
y  6 sin  cos( 250t ) where x, y are in cm
 30 
and t is in second. The number of loops is
a) b)
a) 1 b) 2

c) 4 d) 3

c) d) ORGAN PIPES

14. A tube of certain diameter and of length

10. The tension of a stretched string is increased 48 cm is open at both ends. Its fundamental
by 69%. In order to keep its frequency of frequency of resonance is found to be
vibration constant, its length must be 320Hz. If velocity of sound in air is
increased by 320 ms–1 the diameter of the tube is
a) 30% b) 20%
a) 1.33cm b) 2.33cm
c) 69% d) 69% c) 3.33cm d) 4.33cm

VELOCITY INSTITUTE OF PHYSICS 29

 Waves
15. A closed organ pipe has length l. The air in 19. A string oscillating at fundamental
it is vibrating in 3rd overtone with a frequency under a tension of 225 N
maximum amplitude of A. Find the produces 6 beats/sec with a sonometer. If
l the tension is 256 N, then again oscillating
amplitude at a distance of from closed at fundamental note it produces 6 beats per
14
second with the same sonometer. The
end of the pipe
frequency of the sonometer is
a) A b) zero a) 256 Hz b) 225 Hz

A c) 280 Hz d) 186 Hz
3
c) d) A
2 2 20. A closed organ pipe and an open pipe of the
same length produce 4 beats when they are set
16. The freqency of a stretched uniform wire of into vibrations simultaneously. If he length of
certain length is in resonance with the each of them were twice their initial lengths,
fundamental frequency of closed tube. If the number of beats produced will be
length of wire is decreased by 0.5 m, it is in
a) 2 b) 4
resonance with first overtone of closed pipe.
The initial length of wire is c) 1 d) 8

a) 0.5 m b) 0.75 m 21. The string of a sonometer is divided into two

parts using wedge. Total length of string is
c) 1 m d) 1.5 m 1 m and two parts differ by 2 mm. When
17. An open pipe resonates to a frequency f 1 sounded together they produce 2 beats/sec.
and a closed pipe resonates to a frequency The freqencies of two parts are
f1. If they are joined together to form a longer a) 501Hz, 503Hz b) 501Hz, 499Hz
tube, then it will resonate to a frequency of
c) 499Hz, 497Hz d) 497Hz, 495Hz
(neglect end corrections)
22. On vibrating an air column at 627° C and a
f1 f 2 f1 f 2 tuning fork simultaneously, 6 beats/sec are
a) 2 f  f b) f  2 f heard. The freqency of fork is less than that
2 1 2 1
of air column. No beats are heard at
– 48° C. The freqency of fork is
2 f1 f 2 f1  2 f 2
c) f  f d) f f a) 3Hz b) 6Hz
2 1 1 2

c) 10Hz d) 15Hz
BEATS
DOPPLER’S EFFECT
18. A tuning fork vibrating with a sonometer
wire of length 20 cm produces 5 beats per 23. One train is approaching an observer at rest
second. The beat frequency does not change and another is receding him with same
if the length of the wire is changed to velocity 4 m/s. Both the trains blow whistles
21 cm. The frequency of the tuning fork of same frequency of 243 Hz. The beat
must be frequency in Hz as heard by the observer
is : (Speed of sound in air = 320 m/s)
a) 200 Hz b) 210 Hz
a) 10 b) 6
c) 205 Hz d) 215 Hz c) 4 d) 1

VELOCITY INSTITUTE OF PHYSICS 30

 Waves
24. The frequency of the sound of a car horn as 28. The difference between apparent freqencies
recorded by an observer towards whom the of a source of sound as percieved by a
car is moving differs from the frequency of stationary observer during its approach and
the horn by 10%. Assuming the velocity of recession is 2% of the actual freqency. The
sound in air to be 330 ms–1, the velocity of speed of source is (V = 340 m/sec.)
the car is
a) 12 m/s b) 6.2 m/s
–1 –1
a) 36.7 ms b) 40 ms
c) 3.4 m/s d) 1.5 m/s
–1 –1
c) 30 ms d) 33 ms
ECHO
25. Two trains are approaching each other on
parallel tracks with same velocity. The 29. A person is infront of a vertical mountain
whistle sound produced by one train is fires a bullet and hears its echo after a time
heard by a passenger in another train. If 3 s. The person walks a distance 'd' towards
actual frequency of whistle is 620 Hz and mountain then fires another bullet and
apparent increase in its frequency is 100 Hz, hears its echo after a time 2 s. If velocity of
the velocity of one of the two trains is sound in air is 340 ms–1, the value of 'd' is
(Velocity of sound in air = 335 ms–1)
a) 85 cm b) 170 m
a) 90kmph b) 72 kmph
c) 255 m d) 340 m
c) 54kmph d) 36 kmph
30. A tuning fork of frequency 328 Hz is moved
26. A girl swings in a cradle with period π/4 towards a wall at a speed of 2 ms –1 . An
second and amplitude 2 m. A boy standing observer standing on the same side as the
infront of it blows a whistle of natural fork hears two sounds, one directly from the
frequency 1000 Hz. The minimum fork and the other reflected from the wall.
frequency as heard by the girl is (Velocity Number of beats per second is (Velocity of
of sound in air is 320 ms–a) sound in air 330 ms–1).

a) 850 Hz b) 1000 Hz a) 4 b) 5

c) 750 Hz d) 950 Hz c) 6 d) 7

27. A train is moving at 30 ms–1 in still air. The 31. A car running midway between parallel
frequency of the locomotive whistle is rows of buildings with velocity V c blow
horns and its echo from buildings is heard
500 Hz and the speed of sound is 345 ms–1.
after time t. The distance between the
The apparent wavelengths as heard by
parallel rows of buildings is ( speed of sound
stationary listeners in front of and behind
in air is V)
the locomotive if a wind of speed 10 ms –1
were blowing in the same direction as that a) d  t(V 2  Vc2 )
in which the locomotive is travelling is
b) d  t V 2  V c2
given by
t
c) d  2
a) 0.65 m, 0.73 m b) 0.60 m, 0.73 m V  V c2
t
d) d 
c) 0.65 m, 0.78 m d) 0.60 m, 0.71 m V  V c2
2

VELOCITY INSTITUTE OF PHYSICS 31

 Waves
MIXED CONCEPTS 37. s 1 and s 2 are two sound sources of
frequencies 338 Hz and 342 Hz respectively
32. A sonometer is set on the floor of a lift. When placed at a large distance apart. The velocity
the lift is at rest, the sonometer wire vibrates with which an observer should move from
with fundamental frequency 256 Hz. When s2 to s1 so that he may hear no beats will
9g be.....(velocity of sound in air = 340 m/s)
the lift goes up with acceleration a  , the
16 a) 1 m/s b) 2 m/s
frequency of vibration of the same wire
c) 3 m/s d) 4 m/s
changes to
38. On a quiet day, when a whistle crosses you
a) 512 Hz b) 320 Hz
then its pitch decreases in the ratio 4/5. If
c) 256 Hz d) 204 Hz the temperature on that day is 20°C, then
33. A sonometer wire, with a suspended mass the speed of whistle will be (velocity of
of M = 1 kg, is in resonance with a given sound at 0°C = 332 m/s)
tuning fork. The apparatus is taken to the
a) 28.2 m/s b) 38.2 m/s
moon where the acceleration due to gravity
is 1/6 that on earth. To obtain resonance on c) 18.2 m/s d) 48.2 m/s
the moon, the value of M should be 39. A stationary source emitting sound of
a) 1 kg b) 6 kg frequency 680Hz is at the origin. An
observer is moving with the velocity
c) 6 kg d) 36 kg
34. In a resonace air column experiment, first 2 (ˆi  ˆj ) m/s at a certain instant. If the speed
and second resonances are obtained at of sound in air is 340 m/s, then the apparent
lengths of air columns l1 and l2 , the third frequency received by him at that instant
resonance will be obtained at a length of is
a) 2l2  l1 b) l2  2l1 a) 680 Hz
c) l2  l1 d) 3l2  l1 b) 676 Hz
35. In a resonance column, first and second c) 684 Hz
resonance are obtained at depths 22.7 cm d) either 676 Hz or 684 Hz
and 70.2 cm, the third resonance will be
obtained at a depth of 40. The fundamental frequency of a sonometer
wire of length l is f o . A bridge is now
a) 117.7 cm b) 92.9 cm
introduced at a distance of l from the
c) 115.5 cm d) 113.5 cm
centre of the wire ( l < < l). The number of
36. Two different sound sources S1 and S2 have
beats heard per second if both sides of the
freqencies in the ratio 1:2. Source S 1 is
bridge are set to vibrate in their
approaching towards observer and S 2
fundamental mode is
receding from same observer. Speeds of both
S1 and S2 are V each and speed of sound air 8 f0 l f 0 l
is 330 m/s. If no beats are heard by the a) b)
l l
observer then the value of V is
a) 50 m/s b) 75 m/s 2 f0 l 4 f0 l
c) d)
c) 110 m/s d) 125m/s l l

VELOCITY INSTITUTE OF PHYSICS 32

 Waves
41. A sonometer wire of length L is plucked at 45. A stretched wire of length 114 cm is divided
a distance L/8 from one end then it vibrates into three segments whose frequencies are
with a minimum frequency n. If the same in the ratio 1 : 3 : 4, the lengths of the
wire plucked at a distance L/6 from another segments must be in the ratio
end the minimum frequency with which it
a) 18 : 24 : 72 b) 24 : 72 : 18
vibrates is
c) 24 : 18 : 72 d) 72 : 24 : 18
3 3
a) n b) n 46. The equatin of a wave on a stirng of
2 2
linear mass density 0.04 kg m–1 is given by
3n 4n
c) d)   t x 
4 3 y  0.02  2     . The tension
  0 . 04(s) 0 . 50 (m) 
42. Air column of 20 cm length in a resonance
in the string is
tube resonates with a certain tuning fork
when sounded at its upper open end. The a) 6.25 N b) 4.0 N
lower end of the tube is closed and adjustable
c) 12.5 d) 0.5 N
by changing the quantity of mercury filled
inside the tube. The temperature of the air is 47. Which of the following functions represent
27°C. The change in length of the air column a wave
required, if the temperature falls to 7°C and
a) ( x  vt ) 2 b) l n( x  vt )
the same tuning fork is again sounded at the
upper open end is 1
c) e ( x vt )2 d)
a) 1 mm b) 7 mm x  vt

c) 5 mm d) 13 mm 48. A simple harmonic wave of amplitude

8 units travels along positive x-axis. At any
43. Standing wave produced in a metal rod of
given instant of time, for a particle at a
length 1m is represented by the equation
distance of 10 cm from the orgin, the
x displacement is +6 units, and for a particle
y  106 sin sin 200t where x is in metre
2 at a distance of 25 cm fromt he origin, the
and t is in seconds. The maximum tensile displacement is +4 units. Calculate the
stress at the mid-point of the rod is (Young’s wavelength.
modulus of material of rod = 1012 N/m2).
a) 150 cm b) 250 cm
 6 2
a)  10 N/m b) 2  10 6 N/m 2 c) 350 cm d) 550 cm
2
49. Two wires are fixed on a sonometer. Their
 2
c)  10 6 N/m 2 d)  10 6 N/m 2 tensions are in the ratio 8:1, their lengths
2 2 3
are in the ratio 36 : 35 , the diameters are in
44. A sound wave travels with a velocity of
the ratio 4 : 1 and densities are in the ratio
300 ms–1 through a gas. 9 beats are produced
1 : 2. Find the frequencies of beats produced
in 3 sec when two waves pass through it
simultaneoulsy. If one of the waves has 2 m if the note of higher pitch has a frequnecy
wavelength, wavelength of the other wave is of 360 /s.

a) 1.98 m b) 2.04 m a) 20 Hz b) 10 Hz
c) 2.00 m d) 1.99 m c) 30 Hz d) 40 Hz

VELOCITY INSTITUTE OF PHYSICS 33

 Waves
50. A window whose area is 2 m 2 opens on a 54. In a Melde’s experiment when the tensiion
street where the street noise results at the is 100 gm and the tuning fork vibrates at
window an intensity level of 60 dB. How right angles to the direction of the string,
much acoustic power enters the window the later is thrown into four segments. If
through sound waves? now the tuning fork is set to vibrate along
the string find what additional weight
a) 1 µW b) 2 µW which will make the string vibrate in one
segment.
c) 3 µW d) 4 µW
a) 100 gm.wt b) 200 gm.wt
51. In a class of 100 students each shouting at
100 dB. Find noise level of class? c) 300 gm.wt d) 400 gm.wt
55. A pop- gun consists of a cylindrical barrel
a) 10 dB b) 100 dB
3 cm2 in cross- section closed at one end by
c) 12 dB d) 120 dB a cork and having a well fitting piston at
the other. If the piston is pushed slowly in ,
52. How long will it take sound waves to travel the cork is finally ejected, giving a pop, the
the distance l between the points A and B if frequency of which is found to be 512 Hz.
the air temperature between them varies Assuming that the initial distance between
linearly from T1 to T2 ? The velocity of sound the cork and the piston was 25 cm and that
propagation in air is equal to v   T , there is no leakage of air, calculate the force
where  is a constant. required to eject the cork. Atmospheric
pressure = 1 kg/cm2, v = 340 m/s
2l 4l a) 1.5 kg.wt b) 3 kg.wt
a) t   ( T  T ) b) t   ( T  T )
2 1 1 2 c) 6 kg.wt d) 8 kg.wt
56. A sources of sonic oscillations with
4l 2l frequenccy n = 1700 Hz and a receiver are
c) t   ( T T ) d) t   ( T  T )
1 2 1 2 located on the same normal to a wall. Both
the source and receiver are stationary, and
53. Two coherent narrow P the wall recedes from the source with
slits emitting of x velocity u = 6.0 cm/s. Find the beat frequency
2 registred by th receiver. The velocity of
O
wavelength λ in the S1 S2
D S
sound is equal to v = 340 m/s.
same phase are placed a) 0.2 Hz b) 0.3 Hz
parallel to each other at a small separation c) 0.4 Hz d) 0.6 Hz

of 2λ . The sound is detected by moving a 57. A locomotive approaching a crossing at a

speed of 80 miles/hr, sounds a whistle of
detector on the scree S at a distance D(>> λ )
frequency 400 Hz when 1 mile from the
from the slit S1 as shown in figure. Find the crossing. There is no wind, and the spee of
distance x such that the intensity at P is sound in air is 0.200 mile/s. What frequency
is heard by an observer 0.60 miles from the
equal to the intensity at O.
crossing on the straight road which crosses
a) 2 D b) 4 D the railroad at right angles?
a) 440 Hz b) 442 Hz
c) 3D d) 6D c) 444 Hz d) 446 Hz

VELOCITY INSTITUTE OF PHYSICS 34

 Waves
58. A mixture of two diatomic gases exists in a 60. A source of sound emits waves isotropically
closed cylinder. The volumes and velocities in three dimensions. If the intensity at a
in the two gases are v 1 , v 2 and c 1 , c 2 distance r0 from souce is I0, at what distance
respectively. Determine the velocities of the from the source is the intensity 0.100 I0 ?
gases in the mixture.
a) 1.16 r0 b) 2.16 r0
c) 3.16 r0 d) 4.16 r0
v1  v 2 v1  v 2
a) C1C 2 b) C1C2
v1c 22  v 2 c12 61. Two turning forks with natural frequencies
v1c12  v 2 c22
340 Hz each move relative to a stationary
observer. One fork moves away from the
v1  v 2 v1  v 2 listener, while the other moves towards him
c) C1C 2 2 2
d) C1C2 2 2
v1 c1  v 2 c2 v1 1c  v c
2 2 at the same speed. The listener hears beats
of frequency 3 Hz. Find the speed of the fork
59. A mixture of diatomic gases is obtained by (velocity of sound in air = 340 m/s).
mixing m1 & m2 masses of two gases, with
a) 0.5 m/s b) 1 m/s
velocities of sound in them c 1 & c 2
respectively. Determine the velocity of sound c) 1.5 m/s d) 2 m/s
in the mixture of gases. 62. A source of oscilaltions S is fixed to the
riverbed of a river with stream velocity  .
2 2
2
m c m c 2 m c m c
2 1 1 2 Two receivers R1 & R2 are fixed also to the
a) c  1 1 2 2
b) c  riverbed. If the source generates frequency
m1  m 2 m1  m 2
fs, what frequencies are received by receivers
R1 & R2?
m 2c 2  m1c2 c 22  c12
c) c  d) c  m 2 a) fs b) 1.2 fs
m1  m 2 m1  m 2
c) 1.4 fs d) 1.6 fs

VELOCITY INSTITUTE OF PHYSICS 35

 Waves
PREVIOUS YEAR NEET MCQS
CLASSIFICATION OF WAVES & DISPLACEMENT 6. The wave described by
RELATION IN A PROGRESSIVE WAVE y = 0.25 sin(10  x – 2  t),
1. A w av e t r av el l i n g i n t h e +v e x-direction where x and y are in meters & t in seconds,
having displacement along y-direction as is a wave travelling along the (2008)
1 m, wavelength 2π m and frequency of a) +ve x direction with frequency 1 Hz and
wavelength  = 0.2 m.
1 b) –ve x direction with amplitude 0.25 m and
Hz is represented by (2013)
π wavelength  = 0.2 m.
a) y = sin(10 x – 20 t) b) y = sin(2 x + 2 t) c) –ve x direction with frequency 1 Hz.
c) y = sin(x – 2t) d) y = sin(2  x – 2  t) d) +ve x direction with frequency  Hz and
2. The equation of a simple harmonic wave is wavelength  = 0.2 m.
7. A transverse wave propagating along x-axis

given by Y  3 sin ( 50t  x) , where x and y is represented by,
2 y(x, t) = 8.0sin(0.5  x–4  t–  /4)
are in metres and t is in seconds. The ratio where x is in metres and t is in seconds. The
of maximum particle velocity to the wave speed of the wave is (2006)
velocity is (Mains 2012) a) 8 m/s b) 4  m/s
c) 0.5  m/s d)  /4 m/s.
3
a) 2 b)  8. Which one of the following statements is
2
true? (2006)
2 a) both light & sound waves can travel in
c) 3 d) vacuum
3
b) both light & sound waves in air are
3. A transverse wave is represented by
transverse
y = Asin(  t – kx). For what value of the
c) the sound waves in air are longitudinal while
wavelength is the wave velocity equal to the the light waves are transverse
maximum particle velocity? (2010) d) both light and sound waves in air are
a) A / 2 b) A longitudinal.
c) 2A d) A 9. A wave travelling in positive X-direction
with a = 0.2 ms–2 , velocity = 360 ms–1 and
4. A wave in a string has an amplitude of
 = 60 m, then correct expression for the
2 cm. The wave travels in the +ve direction
wave is (2002)
of x axis with a speed of 128 m/s. and it is
  x 
noted that 5 complete waves fit in 4 m length a) y  0.2 sin  2  6t  
of the string. The equation describing the   60 
wave is (2009)   x 
a) y = (0.02) m sin (15.7 x – 2010t) b) y  0.2 sin   6t   
  60  
b) y = (0.02) m sin (15.7 x + 2010t)
  x 
c) y = (0.02) m sin (7.85 x – 1005t) c) y  0.2 sin  2  6t   
  60  
d) y = (0.02) m sin (7.85 x + 1005t)
5. A point performs simple harmonic   x 
d) y  0.2 sin   6t   
oscillation of period T and the equation of   60  
motion is given by x = a sin(  t +  /6). After 10. The equation of a wave is represented by
the elapse of what fraction of the time period  x
the velocity of the point will be equal to half y  10  4 sin 100t   , then the velocity of
 10 
of its maximum velocity? (2008) wave will be (2001)
a) T/3 b) T/12 a) 100 m/s b) 4 m/s
c) T/8 d) T/6 c) 1000 m/s d) 10 m/s

VELOCITY INSTITUTE OF PHYSICS 36

 Waves
11. A transverse wave is represented by the 18. Equation of progressive wave is given by
2  t x 
equation y  y 0 sin

( vt  x). For what value y  4 sin       where y, x are in cm
 5 9 6
of  , is the maximum particle velocity equal and t is in seconds. Then which of the
to two times the wave velocity? (1998) following is correct? (1988)
y0 y0 a) v = 5 cm b)  = 18 cm
a)   b)  
2 3 c) a = 0.04 cm d) f = 50 Hz
c)   2y0 d)   y 0 SPEED OF A TRAVELLING WAVE
12. In a sinusoidal wave, the time required for a 19. 4.0 g of a gas occupies 22.4 litres at NTP. The
particular point to move from maximum specific heat capacity of the gas at constant
displacement to zero displacement is volume is 5.0 J K–1 mol–1. If the speed of sound
0.170 s. The frequency of wave is (1998) in this gas at NTP is 952 m s–1, then the heat
a) 0.73 Hz b) 0.36 Hz capacity at constant pressure is
c) 1.47 Hz d) 2.94 Hz (Take gas constant R = 8.3 J K–1 mol–1)(2015)
13. The equation of a sound wave is a) 7.0 J K–1 mol–1 b) 8.5 J K–1 mol–1
–1 –1
y = 0.0015 sin (62.4x + 316t). The wavelength c) 8.0 J K mol d) 7.5 J K–1 mol–1
of this wave is (1996) 20. Sound waves travel at 350 m/s through a
a) 0.3 unit warm air and at 3500 m/s through brass.
b) 0.2 unit The wavelength of a 700 Hz acoustic wave
c) 0.1 unit as it enters brass from warm air (2011)
d) cannot be calculated a) decrease by a factor 10
14. A hospital uses an ultrasonic scanner to b) increase by a factor 20
locate tumours in a tissue. The operating c) increase by a factor 10
frequency of the scanner is 4.2 MHz. d) decrease by a factor 20
The speed of sound in a tissue is 1.7 km/s. 21. The temperature at which the speed of sound
The wavelength of sound in the tissue is becomes double as was at 27°C is (1993)
close to (1995) a) 273°C b) 0°C
a) 4 × 10–3 m b) 8 × 10–3 m c) 927°C d) 1027°C
c) 4 × 10–4 m d) 8 × 10–4 m. 22. The velocity of sound in any gas depends
15. Which one of the following represents a upon
wave? (1994) a) wavelength of sound only (1988)
a) y  A sin(t  kx) b) density and elasticity of gas
c) intensity of sound waves only
b) y  A cos( at  bx  c ) d) amplitude and frequency of sound
c) y  A sin kx PRINCIPLE OF SUPERPOSITION OF WAVES
d) y  A sin t 23. Two waves are represented by the equations
16. The frequency of sinusodial wave y1 = asin(  t + kx + 0.57) m
y = 0.40cos [2000t + 0.80] would be (1992) and y2 = acos(  t + kx) m,
a) 1000  Hz b) 2000 Hz where x is in meter and t in sec. The phase
difference between them is (2011)
1000 a) 1.0 radian b) 1.25 radian
c) 20 Hz d) Hz
 c) 1.57 radian d) 0.57 radian
17. With the propogation of a longitudinal wave 24. Two periodic waves of intensities I1 and I2
through a material medium, the quantities pass through a region at the same time in
transmitted in the propogation direction are the same direction. The sum of the maximum
a) energy, momentum and mass (1992) and minimum intensities is (2008)
b) energy 2
a) ( I 1  I 2 ) b) 2(I1 + I2)
c) energy and mass
d) energy and linear momentum c) I1 + I2 d) ( I 1  I 2 ) 2

VELOCITY INSTITUTE OF PHYSICS 37

 Waves
25. A point source emits sound equally in all 32. A wave of frequency 100 Hz travels along a
directions in a non-absorbing medium. string towards its fixed end. When this
Two points P and Q are at distances of 2 m wave travels back, after reflection, a node
and 3 m respectively from the source. The is formed at a distance of 10 cm from the
ratio of the intensities of the waves at P fixed end. The speed of the wave (incident
and Q is (2005)
and reflected) is (1994)
a) 3 : 2 b) 2 : 3
a) 20 m/s b) 40 m/s
c) 9 : 4 d) 4 : 9
26. The phase difference between two waves, c) 5 m/s d) 10 m/s
represented by 33. A stationary wave is represented by,
y1 = 10–6 sin[100t + (x/50) + 0.5] m y = A sin(100t) cos (0.01x), where y and A are
y2 = 10–6 cos[100t + (x/50)] m, in millimetres, t is in seconds and x is in
where x is expressed in metres and t is metres. The velocity of the wave is (1994)
expressed in seconds, is approximately (2004) 4
a) 10 m/s b) not derivable
a) 1.07 radians b) 2.07 radians c) 1 m/s d) 102 m/s
c) 0.5 radians d) 1.5 radians 34. Wave has simple harmonic motion whose
27. Two waves having equation period is 4 seconds while another wave
x1  a sin(t  kx  1 ) , x 2  a sin(t  kx  2 ) which also possesses simple harmonic
If in the resultant wave the frequency and motion has its period 3 second. If both are
amplitude remain equal to amplitude of combined, then the resultant wave will have
superimposing waves, the phase difference the period equal to (1993)
between them is (2001) a) 4 s b) 5 s
 2 c) 12 s d) 3 s
a) b)
6 3 35. Velocity of sound waves in air is 330 m/s.
 
c) d) For a particular sound wave in air, a path
4 3 difference of 40 cm is equivalent to phase
28. The equations of two waves acting in
difference of 1.6  . The frequency of this
perpendicular directions are given as
wave is (1990)
x  a cos(t   ) and y  a cos(t   ) , where
 a) 165 Hz b) 150 Hz
    , the resultant wave represents c) 660 Hz d) 330 Hz
2
a) a parabola b) a circle (2000) 36. If the amplitude of sound is doubled
c) an ellipse d) a straight line and the frequency reduced to one fourth,
29. A standing wave having 3 nodes and 2 the intensity of sound at the same point
antinodes is formed between two atoms will be (1989)
having a distance 1.21 Å bewteen them. The a) increasing by a factor of 2
wavelength of the standing wave is (1998) b) decreasing by a factor of 2
a) 6.05 Å b) 2.42 Å c) decreasing by a factor of 4
c) 1.21 Å d) 3.63 Å
d) unchanged
30. Standing waves are produced in 10 m long
stretched string. If the string vibrates in REFLECTION OF WAVES & VIBRATIONS OF
5 segments and wave velocity is 20 m/s, STRETCHED STRINGS
the frequency is (1997) 37. A tuning fork with frequency 800 Hz
a) 5 Hz b) 10 Hz produces resonance in a resonance column
c) 2 Hz d) 4 Hz tube with upper end open and lower end
31. Two sound waves having a phase difference closed by water surface. Successive
of 60° have path difference of (1996)
resonance are observed at length 9.75 cm,
 
a) b) 31.25 cm and 52.75 cm. The speed in air is
6 3
 a) 500 m/s b) 156 m/s (2019)
c) 2 d) c) 344 m/s d) 172 m/s
2
VELOCITY INSTITUTE OF PHYSICS 38
 Waves
38. A tuning fork is used to produce resonance 43. The length of the wire between two ends
in a glass tube. The length of the air column of a sonometer is 100 cm. What should be
in this tube can be adjusted by a variable the positions of two bridges below the wire
piston. At room temperature of 27°C two so that the three segments of the wire have
successive resonances are produced at 20 cm their fundamental frequencies in the ratio
and 73 cm of column length. If the frequency 1 : 3 : 5. (Karnataka 2013)
of the tuning fork is 320 Hz, the velocity of
1500 500 1500 300
sound in air at 27°C is (2018) a) cm, cm b) cm, cm
a) 330 m s–1 b) 339 m s–1 23 23 23 23
c) 350 m s–1 d) 300 m s–1 300 1500 1500 2000
c) cm, cm d) cm, cm
39. The fundamental frequency in an open 23 23 23 23
organ pipe is equal to the third harmonic of 44. When a string is divided into three segments
a closed organ pipe. If the length of the of length l 1 , l 2 and l 3 the fundamental
closed organ pipe is 20 cm, the length of the frequencies of these three segments are 1 , 2
open organ pipe is (2018)
and 3 respectively. The original fundamental
a) 13.2 cm b) 8 cm
c) 12.5 cm d) 16 cm frequency ( ) of the string is (2012)
40. A uniform rope of length L and mass m 1 a)   1   2   3
hangs vertically from a rigid support. A
block of mass m2 is attached to the free end b)   1   2   3
of the rope. A transverse pulse of 1 1 1 1
wavelength 1 is produced at the lower end c)       
1 2 3
of the rope. The wavelength of the pulse
when it reaches the top of the rope is 2 . 1 1 1 1
d)       
The ratio 2 / 1 is (2016) 1 2 3

45. Two identical piano wires, kept under the

m2 m1  m2
a) b) same tension T have a fundamental frequency
m1 m1
of 600Hz. The fractional increase in the
m1 m1  m2 tension of one of the wires which will lead to
c) d) occurrence of 6 beats/s when both the wires
m2 m2
41. A string is stretched between fixed points oscillate together would be (Mains 2011)
separated by 75.0 cm. It is observed to have a) 0.01 b) 0.02
resonant frequencies of 420 Hz and 315 Hz. c) 0.03 d) 0.04
There are no other resonant frequencies 46. Each of the two strings of length 51.6 cm
between these two. The lowest resonant and 49.1 cm are tensioned separately by
frequency for this string is (2015) 20 N force. Mass per unit length of both the
a) 10.5 Hz b) 105 Hz strings is same and equal to 1 g/m. When
c) 155 Hz d) 205 Hz both the strings vibrate simultaneously the
42. If n 1 , n 2 and n 3 are the fundamental number of beats is (2009)
frequencies of three segments into which a a) 7 b) 8
string is divided, then the original c) 3 d) 5
fundamental frequency n of the string is 47. If the tension and diameter of a sonometer
given by (2014)
wire of fundamental frequency n is doubled
1 1 1 1 and density is halved then its fundamental
a) n  n  n  n
1 2 3 frequency will become (2001)
1 1 1 1 n
b)    a) b)
n n1 n2 n3 4 2n

c) n  n1  n2  n3 n
c) n d)
d) n = n1 + n2 + n3 2

VELOCITY INSTITUTE OF PHYSICS 39

 Waves
48. A string is cut into three parts, having 55. The fundamental frequency of a closed
fundamental frequencies n1, n2, n3 respectively. organ pipe of length 20 cm is equal to the
Then original fundamental frequency n second overtone of an organ pipe open at
related by the expression as (2000) both the ends. The length of organ pipe open
1 1 1 1 at both the ends is (2015)
a)    b) n = n1 × n2 × n3 a) 120 cm b) 140 cm
n n1 n2 n3
c) 80 cm d) 100 cm
n1  n2  n3
c) n = n1 + n2 + n3 d) n  56. The number of possible natural oscillations
3
of air column in a pipe closed at one end of
49. The length of a sonometer wire AB is 110 cm.
length 85 cm whose frequencies lie below
Where should the two bridges be placed from
1250 Hz are (Velocity of sound = 340 m s–1)
A to divide the wire in 3 segments whose
a) 4 b) 5 (2014)
fundamental frequencies are in the ratio
c) 7 d) 6
of 1 : 2 : 3? (1995)
57. If we study the vibration of a pipe open at
a) 60 cm and 90 cm b) 30 cm and 60 cm
both ends, then the following statement is
c) 30 cm and 90 cm d) 40 cm and 80 cm.
not true. (2013)
50. A stretched string resonates with tuning fork
a) All harmonics of the fundamental
frequency 512 Hz when length of the string
frequency will be generated.
is 0.5 m. The length of the string required to
b) Pressure change will be maximum at both
vibrate resonantly with a tuning fork of
ends.
frequency 256 Hz would be (1993)
a) 0.25 m b) 0.5 m c) Open end will be antinode.
c) 1 m d) 2 m d) Odd harmonics of the fundamental
51. A 5.5 metre length of string has a mass of frequency will be generated.
0.035 kg. If the tension in the string in 77 N, 58. A cylindrical tube, open at both ends has
the speed of a wave on the string is (1989) fundamental frequency f in air. The tube is
a) 110 m s –1
b) 165 m s –1 dipped vertically in water, so that half of it
c) 77 m s–1 d) 102 m s–1 is in water. The fundamental frequency of
air column is now (1997)
ORGAN PIPES a) f/2 b) 3f/4
52. The two nearest harmonics of a tube closed c) 2f d) f
at one end and open at other end are 220 Hz 59. A closed organ pipe (closed at one end) is
and 260 Hz. What is the fundamental excited to support the third overtone. It is
frequency of the system? (2017) found that air in the pipe has (1991)
a) 20 Hz b) 30 Hz a) three nodes and three antinodes
c) 40 Hz d) 10 Hz b) three nodes and four antinodes
53. The second overtone of an open organ pipe c) four nodes and three antinodes
has the same frequency as the first overtone d) four nodes and four antinodes
of a closed pipe L metre long. The length of
the open pipe will be (2016)
BEATS
a) L b) 2L 60. In a guitar, two strings A and B made of
c) L / 2 d) 4L same material are slightly out of tune and
54. An air column, closed at one end and open at produce beats of frequency 6 Hz. When
the other, resonates with a tuning fork when tension in B is slightly decreased, the beat
the smallest length of the column is 50 cm. frequency increases to 7 Hz. If the frequency
The next larger length of the column of A is 530 Hz, the original frequency of B
resonating with the same tuning fork is (2016) will be (2020)
a) 150 cm b) 200 cm a) 523 Hz b) 524 Hz
c) 66.7 cm d) 100 cm c) 536 Hz d) 537 Hz

VELOCITY INSTITUTE OF PHYSICS 40

 Waves
61. Three sound waves of equal amplitudes 67. Two sound waves with wavelengths 5.0 m
have frequencies (n – 1), n, (n + 1). They and 5.5. m respectively, each propagate in a
superimpose to give beats. The number of gas with velocity 330 m/s. We expect the
beats produced per second will be (2016) following number of beats per second.(2006)
a) 1 b) 4 a) 6 b) 12
c) 3 d) 2 c) 0 d) 1
62. A source of unknown frequency gives 68. Two stationary sources each emitting
4 beats/s when sounded with a source of waves of wavelength  , an observer moves
known frequency 250 Hz. The second from one source to another with velocity u.
harmonic of the source of unknown Then number of beats heard by him (2000)
frequency gives five beats per second, when 2u u u
sounded with a source of frequency 513 Hz. a) b) c) u d)
  2
The unknown frequency is
69. Two waves of lengths 50 cm and 51 cm
a) 240 Hz b) 260 Hz (2013) produced 12 beats per sec. The velocity of
c) 254 Hz d) 246 Hz sound is (1999)
63. Two sources P and Q produce notes of a) 340 m/s b) 331 m/s
frequency 660 Hz each. A listener moves from c) 306 m/s d) 360 m/s
P to Q with a speed of 1 ms–1. If the speed of 70. A source of sound gives 5 beats per second,
sound is 330 m/s, then the number of beats when sounded with another source of
heard by the listener per second will be frequency 100 second–1. The second harmonic
a) 4 b) 8 of the source, together with a source of
c) 2 d) zero frequency 205 sec–1 gives 5 beats per second.
64. Two sources of sound placed close to each What is the frequency of the source? (1995)
other, are emitting progressive waves given a) 105 second –1
b) 205 second –1

by y1 = 4sin600  t and y 2 = 5sin608  t. An c) 95 second–1 d) 100 second–1

observer located near these two sources of 71. A source of frequency v gives 5 beats/second
sound will hear (2012) when sounded with a source of frequency
a) 4 beats per second with intensity ratio 200 Hz. The second harmonic of frequency
25 : 16 between waxing and waning. 2v of source gives 10 beats/second when
b) 8 beats per second with intensity ratio sounded with a source of frequency 420 Hz.
25 : 16 between waxing and waning. The value of v is (1994)
c) 8 beats per second with intensity ratio 81 : 1 a) 205 Hz b) 195 Hz
between waxing and waning. c) 200 Hz d) 210 Hz
d) 4 beats per second with intensity ratio 81 : 1 72. For production of beats the two sources
between waxing and waning. must have (1992)
65. A tuning fork of frequency 512 Hz makes a) different frequencies and same amplitude
4 beats per second with the vibrating string b) different frequencies
of a piano. The beat frequency decreases to c) different frequencies, same amplitude and
2 beats per sec when the tension in the piano same phase
string is slightly increased. The frequency d) different frequencies and same phase
of the piano string before increasing the DOPPLER’S EFFECT
tension was (2010) 73. Two cars moving in opposite directions
a) 510 Hz b) 514 Hz approach each other with speed of 22 m s–1
c) 516 Hz d) 508 Hz and 16.5 m s–1 respectively. The driver of the
66. Two vibrating tuning fork’s produce waves first car blows a horn having a frequency
given by y1 = 4 sin500  t and y2 = 2 sin506  t. 400 Hz. The frequency heard by the driver of
Number of beats produced per minute is(2006) second car is (velocity of sound is 340 m s–1)
a) 360 b) 180 a) 361 Hz b) 411 Hz (2017)
c) 60 d) 3 c) 448 Hz d) 350 Hz

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 Waves
74. A siren emitting a sound of frequency 79. A car is moving towards a high cliff. The
800 Hz moves away from an observer driver sounds a horn of frequency f. The
towards a cliff at a speed of 15 m s–1. Then, reflected sound heard by the driver has
frequency 2f. If v is the velocity of sound,
the frequency of sound that the observer
then the velocity of the car, in the same
hears in the echo reflected from the cliff is
velocity units, will be (2004)
(Take velocity of sound in air = 330 m s–1)
a) v / 2 b) v/3
a) 838 Hz b) 885 Hz (2016)
c) 765 Hz d) 800 Hz c) v/4 d) v/2
75. A source of sound S 80. An observer moves towards a stationary
source of sound with a speed 1/5 th of the
emitting waves of
speed of sound. The wavelength and
frequency 100 Hz
frequency of the source emitted are  and f
and an observer O respectively. The apparent frequency and
are located at some wavelength recorded by the observer are
distance from each other. The source is moving respectively (2003)
with a speed of 19.4 m s–1 at an angle of 60° a) 1.2 f, 1.2  b) 1.2 f, 
with the source observer line as shown in c) f, 1.2  d) 0.8 f, 0.8 
the figure. The observer is at rest. The 81. A whistle revolves in a circle with angular
apparent frequency observed by the observer speed ω = 20 rad/s using a string of length
(velocity of sound in air 330 m s–1), is (2015) 50 cm. If the frequency of sound from the
a) 106 Hz b) 97 Hz whistle is 385 Hz, then what is the minimum
c) 100 Hz d) 103 Hz frequency heard by an observer which is far
76. A speeding motorcyclist sees traffic jam ahead away from the centre (velocity of sound =
340 m/s) (2002)
him. He slows down to 36 km hour –1 . He
a) 385 Hz b) 374 Hz
finds that traffic has eased and a car moving c) 394 Hz d) 333 Hz
ahead of him at 18 km hour–1 is honking at 82. A vehicle, with a horn of frequency n is
a frequency of 1392 Hz. If the speed of sound moving with a velocity of 30 m/s in a
is 343 m s–1, the frequency of the honk as direction perpendicular to the straight line
heard by him will be (2014) joining the observer and the vehicle. The
a) 1332 Hz b) 1372 Hz observer perceives the sound to have a
c) 1412 Hz d) 1454 Hz frequency n + n1. Then (if the sound velocity
in air is 300 m/s) (1998)
77. A train moving at a speed of 220 m s –1
a) n1 = 0.1n b) n1 = 0
towards a stationary object, emits a sound
c) n1 = 10n d) n1 = –0.1n
of frequency 1000 Hz. Some of the sound 83. A star, which is emitting radiation at a
reaching the object gets reflected back to the wavelength of 5000 Å, is approaching the
train as echo. The frequency of the echo as earth with a velocity of 1.5 × 10 4 m/s. The
detected by the driver of the train is (Speed change in wavelength of the radiation as
of sound in air is 330 ms–1) (Mains 2012) received on the earth is (1995)
a) 3500 Hz b) 4000 Hz a) 25 Å b) 100 Å
c) 5000 Hz d) 3000 Hz c) zero d) 2.5 Å
84. Two trains move towards each other with
78. The driver of a car travelling with speed
the same speed. The speed of sound is
30 m/s towards a hill sounds a horn of
340 m/s. If the height of the tone of the
frequency 600 Hz. If the velocity of sound whistle of one of them heard on the other
in air is 330 m/s, the frequency of reflected changes to 9/8 times, then the speed of each
sound as heard by driver is (2009) train should be (1991)
a) 555.5 Hz b) 720 Hz a) 20 m/s b) 2 m/s
c) 500 Hz d) 550 Hz c) 200 m/s d) 2000 m/s

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 Waves
PREVIOUS YEAR AIIMS MCQS
CLASSIFICATION OF WAVES & DISPLACEMENT 8. For a wave propagating in a medium,
1 RELATION IN A PROGRESSIVE WAVE identify the property that is independent of
1. The equation of a wave is given by the others.
a) velocity (2006)
 2t 
y  10 sin    . If the displacement is b) wavelength
 30  c) frequency
5 cm at t = 0, then the total phase at t = 7.5 s d) all options depend on each other
will be 9. The equaton of a progressive wave is given
2  by y  5 sin(100t  0.4x) where y and x are
a) rad b) rad (1996)
3 3 in m and t is in s. (2015)
 2 1) The amplitude of the wave is 5 m.
c) rad d) rad
2 5 2) The wavelength of the wave is 5 m.
2. The waves in which the particles of the 3) The frequency of the wave is 50 Hz.
medium vibrate in a direction perpendicular 4) The velocity of the wave is 250 m s–1.
to the direction of wave motion are known as Which of the following statements are
a) propagated waves (1998) correct?
b) longitudinal waves a) (1), (2) and (3) b) (2) and (3)
c) transverse waves c) (1) and (4) d) all are correct
d) none of these 10. The displacement of a particle executing
3. The number of waves, contained in unit SHM is given by y  0.25 sin 200 t cm . The
length of the medium, is called (1998) maximum speed of the particle is (2016)
a) wave pulse a) 200 cm s–1 b) 100 cm s–1
b) wave number c) 50 cm s–1 d) 5.25 cm s–1
c) elastic wave 11. A wave is represented by the equation
d) electromagnetic wave y  0.5 sin( 10 t  x ) metre. It is a travelling
4. A transverse stationary wave passes wave propagating along +x direction with
through a string with the equation velocity
y  10 sin  ( 0.02 x  2.00 t ) , where x is in meters a) 10 m s–1 b) 20 m s–1 (2016)
and t in seconds. The maximum velocity of c) 5 m s –1
d) none of these
particle in wave motion is (2000)
a) 63 b) 78 SPEED OF A TRAVELLING WAVE
c) 100 d) 121 12. If at same temperature and pressure, the
5. If equation of sound wave is densities for two diatomic gases are d1 and
y  0.0015 sin(62.4 x  316t ) , d2 respectively, then the ratio of velocities
then its wavelength will be (2002) of sound in these gases will be (1996)
a) 0.2 unit b) 0.3 unit d2 d2
c) 0.1 unit d) 2 unit a) 2d b) d
1 1
6. The waves produced by a motorboat sailing
in water are (2004)
2d1 d1
a) transverse c) d d) d
b) longitudinal 2 2

c) longitudinal and transverse 13. Newton’s formula for the velocity of sound
d) stationary in gases, is (1998)
7. A stone thrown into still water, creats a
circular wave pattern moving radially 2p p
a) v  b) v 
outwards. If r is the distance measured from  
the centre of the pattern, the amplitude of the
wave varies as (2006)  3 p
c) v  p d) v 
a) r 1 / 2
b) r 1
c) r 2
d) r 3 / 2 2 

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 Waves
14. The velocities of sound at the same 20. Two waves represented by y  a sin(t  kx)
temperature in two monoatomic gases of and y  a cos(t  kx) are superposed. The
densities 1 and  2 are v1 and v2 respectively.. resultant wave will have an amplitude
If 1 /  2  4 , then the value of v1 / v2 is (2002) a) a b) 2 a (2014)
a) 1 / 4 b) 2
c) 2 a d) zero
c) 1 / 2 d) 4
REFLECTION OF WAVES & VIBRATIONS OF
PRINCIPLE OF SUPERPOSITION OF WAVES STRETCHED STRINGS
15. Whenever stationary waves are set up, in
21. Standing waves are produced in 10 m long
any medium, then (1997)
stretched string. If the string vibrates in 5
a) condensations occur at nodes segments and wave velocity is 20 m/s, its
b) refractions occur at antinodes frequency is (1998)
c) maximum strain is experienced at nodes a) 5 Hz b) 4 Hz
d) no strain is experienced at antinodes
c) 2 Hz d) 10 Hz
16. Ratio of intensities of two waves is 9 : 1. If
these two are superimposed, what is the 22. If vibrations of a string are to be increased
ratio of maximum and minimum intensities? by a factor two, then tension in the string
must be made (1999)
a) 9 : 1 b) 3 : 1 (2000)
a) four times b) half
c) 4 : 1 d) 5 : 3
c) twice d) eight times
17. The graph between wave number ( ) and
23. A string in a musical instrument is 50 cm
angular frequency ( ) is (2002)
long and its fundamental frequency is
800 Hz. If a frequency of 1000 Hz is to be
produced, then required length of string is
a) b) a) 62.5 cm b) 40 cm (2002)
c) 50 cm d) 37.5 cm
24. When a guitar string is sounded with 440 Hz
tuning fork, a beat frequency of 5 Hz is heard.
If the experiment is repeated with a tuning
c) d)
fork of 437 Hz, the beat frequency is 8 Hz.
The string frequency (Hz) is (2006)
18. Five sinusoidal waves have the same a) 445 b) 435
frequency 500 Hz but their amplitudes are in
c) 429 d) 448
1 1
the ratio 2 : : : 1 : 1 and their phase angles 25. A uniform string is vibrating with a
2 2
fundamental frequency f. The new frequency,
π π π
0, , , and π respectively. The phase if radius and length both are doubled
6 3 2
would be
angle of resultant wave obtained by the
superposition of these five waves is (2010) a) 2 f b) 3 f (2010)
a) 30° b) 45° c) f / 4 d) f / 3
c) 60° d) 90°
26. A 5.5 metre length of string has a mass of
19. Two sinusoidal waves of intensity I having 0.035 kg. If the tension in the string is 77 N,
same frequency and same amplitude the velocity of the wave on the string is
interferes constructively at a point. The (2014)
resultant intensity at a point will be (2012)
a) 210 m s–1 b) 40 m s–1
a) I b) 2I
c) 110 m s–1 d) 55 m s–1
c) 4I d) 8I

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 Waves
ORGAN PIPES 35. The second overtone of an open pipe has the
27. The frequency of a tuning fork is 256 Hz. It same frequency as the first overtone of a
will not resonate with a fork of frequency closed pipe 2 m long. The length of the open
a) 738 Hz b) 256H z (1994) pipe is (2010)
a) 8 m b) 4 m
c) 768 Hz d) 512 Hz
c) 2 m d) 1 m
28. A tube closed at one end containing air
produces fundamental note of frequency BEATS
512 Hz. If the tube is open at both the ends, 36. When a stretched wire and a tuning fork
the fundamental frequency will be (1995) are sounded together, 5 beats per second are
a) 1024 Hz b) 256 Hz produced, when length of wire is 95 cm or
c) 1280 Hz d) 768 Hz 100 cm, frequency of the fork is (2017)
29. The tension in piano wire is 10 N. What a) 90 Hz b) 100 Hz
c) 105 Hz d) 195 Hz
should be the tension in the wire to produce
a note of double the frequency? (1995) DOPPLER’S EFFECT
a) 40 N b) 5 N 37. An observer standing by the side of a road
c) 80 N d) 20 N hears the siren of an ambulance, which is
30. A closed organ pipe and an open organ pipe moving away from him. If the actual
of the same length produce four beats, when frequency of the siren is 2000 Hz, then the
frequency heard by the observer will be
sounded together. If the length of the closed
a) 2000 Hz b) 1990 Hz (1996)
organ pipe is increased, then the number of
c) 4000 Hz d) 2100 Hz
beats will (1996)
38. If a star is moving towards the earth, then
a) remains same b) increase
spectrum lines are shifted towards (1997)
c) decrease d) first (b) then (a) a) red b) infrared
31. A resonance air column of length 20 cm c) blue d) green
resonates with a tuning fork of frequency 39. A source of frequency 240 Hz is moving
450 Hz. Ignoring the correction, the velocity towards an observer with a velocity of
of sound in air will be (1999) 20 m/s. The observer is now moving towards
a) 920 m/s b) 720 m/s the source with a velocity of 20 m/s.
c) 820 m/s d) 360 m/s Apparent frequency heard by observer, if
32. If fundamental frequency is 50 Hz and next velocity of sound is 340 m/s, is (1996, 2001)
successive frequencies are 150 Hz and a) 268 Hz b) 270 Hz
250 Hz then it is (2001) c) 360 Hz d) 240 Hz
a) a pipe closed at both end 40. A siren emitting sound of frequency 800 Hz
b) a pipe closed at one end is going away from a static listener with a
c) an open pipe speed of 30 m/s. Frequency of the sound to
d) a stretched pipe be heard by the listener is (take, velocity of
33. An organ pipe closed at one end has sound as 330 m/s) (2002, 2007)
fundamental frequency of 1500 Hz. The a) 733.3 Hz b) 481.2 Hz
maximum number of overtones generated c) 644.8 Hz d) 286.5 Hz
by this pipe which a normal person can 41. What is your observation when two
sources are emitting sound with frequency
hear is
499 Hz and 501 Hz? (2011)
a) 14 b) 13 (2004)
a) Frequency of 500 Hz is heard with change
c) 6 d) 9
in intensity take place twice.
34. Two closed organ pipes of length 100 cm and
b) Frequency of 500 Hz is heard with change
101 cm produce 16 beats in 20 sec. When in intensity take place once.
each pipe is sounded in its fundamental c) Frequency of 2 Hz is heard with change in
mode, calculate the velocity of sound. (2008) intensity take place once.
a) 303 m s–1 b) 332 m s–1 d) Frequency of 2 Hz is heard with change in
–1
c) 323.2 m s d) 300 m s–1 intensity take place twice.
VELOCITY INSTITUTE OF PHYSICS 45
 Waves
ECHO 47. Assertion : In the relation f 
1 T
, where
42. If man were standing unsymmetrically 2l 
between parallel cliffs, claps his hands and symbols have standard meaning,  represents
starts hearing a series of echoes at intervals linear mass density. (2008)
of 1 s. If speed of sound in air is 340 m s–1, Reason : The frequency has the dimensions of
the distance between two cliffs would be inverse of time.
48. Assertion : Transverse sound wave does not
a) 340 m b) 510 m (2011)
occur in gases. (2011)
c) 170 m d) 680 m Reason : Gases can not sustain shearing strain.
49. Assertion : To hear distinct beats, difference
Assertion And Reason in frequencies of two sources should be less
43. Assertion : When two vibrating tuning forks than 10. (2014)
having frequencies 256 Hz and 512 Hz are held Reason : More the number of beats per sec
near each other, beats cannot be heard. (1994) more difficult to hear them.
50. Assertion : The fundamental frequency of an
Reason : The principle of superposition is valid open pipe increases as the temperature is
only if the frequencies of the oscillators are increased. (2015)
nearly equal. Reason : This is because as the temperature
44. Assertion : The flash of lightening is seen increases, the velocity of sound increases more
before the sound of thunder is heard. (2002) rapidly than length of the pipe.
51. Assertion : Water waves in a river are not
Reason : Speed of sound is greater than speed polarized. (2018)
of light. Reason : Water waves are longitudinal as well
45. Assertion : When a beetle moves along the sand as transverse in nature.
within a few tens of centimeters of a sand 52. Assertion : In a string wave, during reflection
scorpion, the scorpion immediately turns from fix boundary, the reflected waves is
towards the beetle and dashes towards it. (2003) inverted. (2018)
Reason : The force on string by clmap is in
Reason : When a beetle disturbs the sand, it
downward direction while string is pulling the
sends pulses along the sand’s surface. One set
clamp in upward direction.
of pulses is longitudinal while the other set is
53. Assertion : The speed of sound in a gas is not
transverse.
affected by change in pressure provided the
46. Assertion : Sound waves cannot propagate temperature of the gas remains constant. (2018)
through vacuum but light waves can. Reason : The speed of sound is inversely
Reason : Sound waves cannot be polarised but proportional to the square root of the density
light waves can be. (1997, 2007) of the gas.

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 Waves
PREVIOUS YEAR JIPMER MCQS
CLASSIFICATION OF WAVES & DISPLACEMENT PRINCIPLE OF SUPERPOSITION OF WAVES
1 RELATION IN A PROGRESSIVE WAVE
7. Two identical sinusoidal waves each of
1. The equation of a wave travelling in a string amplitude 5 mm with a phase difference of
can be written as y  3 cos  100t  x  . Its
π/2 are travelling in the same direction in
wavelength is (y in cm) (2012)
a string. The amplitude of the resultant
a) 3 cm b) 100 cm
wave (in mm) is (2013)
c) 2 cm d) 5 cm
a) zero b) 5 2
2. Which of the following statements is
true? (2011) 5
c) d) 2.5
a) Both light and sound waves are transverse. 2
b) The sound waves are longitudinal while the
REFLECTION OF WAVES & VIBRATIONS OF
light waves are transverse. STRETCHED STRINGS
c) Both light & sound waves are longitudinal.
d) Both light and sound waves can travel in 8. The length of a sonometer wire AB is
vacuum. 110 cm. Where should the two bridges be
placed from A to divide the wire in three
SPEED OF A TRAVELLING WAVE segments whose fundamental frequencies
3. An earthquake generates both transverse (S) are in the ratio of 1:2:3 (2016)
and longitudinal (P) sound waves in earth.
a) 30 cm and 90 cm b) 40 cm and 80 cm
The speed of ‘S’ waves about 4.5 km s–1 and
that of ‘P’ waves is about 8.0 km s –1 . A c) 60 cm and 90 cm d) 30 cm and 60 cm
seismograph records P and S waves from 9. Two uniform strings A and B made of steel
an earthquake. The first P wave arrives are made to vibrate under the same tension.
4.0 min before the first S wave. The epicenter If the first overtone of A is equal to the
of the earthquake is located at a distance of second overtone of B and if the radius of A
about (2016) is twice that of B, the ratio of the lengths of
a) 50000 km b) 2500 km the strings is (2012)
c) 25 km d) 250 km a) 1 : 2 b) 1 : 3
4. The speed of sound through oxygen gas at
c) 1 : 4 d) 1 : 6
T K is v ms–1. If the temperature becomes 2T
and oxygen gas dissociated into atomic 10. A string in a musical instrument is 50 cm
oxygen, the speed of sound. (2014) long and its fundamental frequency is
a) remains the same b) becomes 2v 800 Hz. If a frequency of 1000 Hz is to be
c) becomes d) none of these produced, then required length of string is
2v
5. The time of reverberation of room A is one a) 62.5 cm b) 40 cm (2010)
second. What will be the time (in seconds) of c) 50 cm d) 37.5 cm
reverberation of a room, having all the
dimension s double of those of room A? (2011) ORGAN PIPES
a) 2 b) 4 11. An organ pipe open at one end is vibrating
c) 1 / 2 d) 1+ in first overtone and is in resonance with
6. The speed of sound in hydrogen at NTP is another pipe open at both ends and
1270 m s–1. Then, the speed in a mixture of vibrating in third harmonic. The ratio of
hydrogen and oxygen in the ratio 4 : 1 by lengths of two pipes is
volume will be (2010)
a) 3 : 8 b) 8 : 3 (2015)
a) 950 m s–1 b) 830 m s–1
c) 1 : 2 d) 4 : 1
c) 635 m s–1 d) 317 m s–1

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 Waves
12. Air is blown at the mouth of an open tube DOPPLER’S EFFECT
of length 25 cm and diameter 2 cm. If the
18. A source of sound emitting a tone of
velocity of sound in air is 330 m s –1 , then
frequency 200 Hz moves towards an
the emitted frequencies are (in Hz) (2012)
observer with a velocity v equal to the
a) 660, 1320, 2640 b) 660, 1000, 3300 velocity of sound. If the observer also moves
c) 302, 664, 1320 d) 330, 990, 1690 away from the source with the same velocity
13. A closed organ pipe of length 20 cm is v, the apparent frequency heard by the
sounded with tuning fork in resonance. observer is (2015)
What is the frequency of tuning fork? a) 50 Hz b) 100 Hz
(Take v = 332 m s–1)
c) 150 Hz d) 200 Hz
a) 300 Hz b) 350 Hz (2011)
c) 375 Hz d) 415 Hz 19. An engine running at speed v/10 sounds a
whistle of frequency 600 Hz. A passenger
14. The third overtone of an open organ pipe of
in a train coming from the opposite side at
length l0 has the same frequency as the third speed v/15 experiences this whistle to be of
overtone of a closed pipe of length lc. The frequency v. If v is speed of sound in air and
l0 there is no wind, v is nearest to (2012)
ratio l is equal to (2010)
c a) 710 Hz b) 630 Hz
a) 2 b) 3 / 2 c) 580 Hz d) 510 Hz
c) 5 / 3 d) 8 / 7
20. If a source of sound of frequency v and a
BEATS listener approach each other with a velocity
15. A closed organ pipe and an open organ pipe equal to (1/20) of velocity of sound, the
of same length produce 2 beats sec–1 s when apparent frequency heard by the listener is
they are set into vibrations together in
 21   20 
fundamental mode. The length of open pipe a)  v b)  v (2011)
 19   21 
is now halved and that of closed pipe is
doubled. The number of beats produced will
 21   19 
be (2014) c)  v d)  v
 20   20 
a) 7 b) 4
c) 8 d) 2 ECHO
16. Two sound sources emitting sound each of
21. A boy standing between two cliffs claps and
wavelength  are fixed at points A and B. hear two echoes, the first after 2 seconds and
A listener moves with velocity u from A to the second after 3 seconds. The velocity of
B. The number of beats heard by him per sound is 360 m/s. The distance in meter
second is (2014) between the two cliffs is (2016)
2u u a) 900 b) 360
a) b)
 
c) 1400 d) 1200
u 2
c) d) 22. A car is moving with a speed of 72 km h–1
3 u towards a hill. Car blows horn at a distance
17. Two waves of wavelengths 50 cm and of 1800 m from the hill. If echo is heard after
51 cm produced 12 beats per second. The 10 second, the speed of sound (in ms–1) is
velocity of sound is (2013) a) 300 b) 320 (2012)
–1 –1
a) 340 m s b) 331 m s
c) 340 d) 360
c) 306 m s–1 d) 360 m s–1

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 Waves
NCERT UNSOLVED
1. A string of mass 2.50 kg is under a tension of 8. A transverse harmonic wave on a string is
200 N. The length of the stretched string is
described by yx , t   3.0 sin36t  0.018 x   / 4  ,
20.0m. If the transverse jerk is struck at one
end of the string, how lo ng does the where x and y are in cm and t in s. The positive
disturbance take to reach the other end? direction of x is from left to right.
2. A stone dropped from the top of a tower of a) is this a travelling wave or a stationary
height 300 m splashes into the water of a pond wave? If it is travelling, what are the speed
near the base of the tower. When is the splash and direction of its propagation?
hard at the top? Given that the speed of sound
b) What are its amplitude and frequency ?
in air is 340 m s–1 ? (g = 9.8 m–2)
c) what is the initial phase at the origin?
3. A steel wire has a length of 12.0 m and a mass
of 2.10 kg. What should be the tension in the d) What is the least distance between two
wire so that speed of a transverse wave on the successive crests in the wave?
wire equals to the speed of sound in dry air at 9. For the wave described in question no. 8, plot
20 °C = 343 m s–1. the displacement (y) versus (t) graphs for x =
0, 2 and 4 cm. What are the shapes of these
P
4. Use the formula v  to explain why the graphs? In which aspects does the oscillatory

motion in travelling wave differ from one
speed of sound in air
point to another : amplitude, frequency or
a) is independent of pressure, phase?
b) increases with temperature, 10. For the travelling harmonic wave,
c) increases with humidity. yx , t   2.0 cos 2 10t  0.0080x  0.35

5. You have learnt that a traveling wave in one where x and y are in cm and t in s. Calculate the
dimension is represented by a function y = f(x , phase difference between oscillatory motion of
t) where x and t must appear in the two points separated by a distance of
combination (x – vt) to (x + vt). i.e. y = f(x  vt). a) 4 m b) 0.5 m c)  / 2 d) 3 / 4
Is the converse true ? Examine if the following
11. The transverse displacement of a string
functions for y can possible represent a
(clamped at its two ends) is given by
travelling wave
a) (x - vt)2 b) log [(x + vt)/x0]  2 
yx , t   0.06 sin x  cos120t 
 3 
c) 1/(x + vt)
where x and y are in m and t in s. The length
6. A bat emits ultrasonic sound of frequency of the string is 1.5 m and its mass is 3.0 × 10–2
1000 kHz in air. If this sound meets a water surface, kg. Answer the following :
what is the wavelength of (a) the reflected sound,
a) Does the function represent a travelling
(b) the transmitted sound? speed of sound in air
wave or a stationary wave?
is 340 m s–1 and in water 1486 m s–1.
b) Interpret the wave as a superposition of two
7. A hospital uses an ultrasonic scanner to locate
waves travelling in opposite direction. What
tumours in a tissue. What is the wavelength of
are the wavelength, frequency, and speed
sound in the tissue in which the speed of sound
of each wave?
is 1.7 km s–1 ? The operating frequency of the
scanner is 4.2 MHz. c) Determine the tension in the string.

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 Waves
12. i) For the wave on a string described in 18. Two sitar strings A and B playing the note ‘Ga’
question no. 11, do all the points on the string are slight out of tune and produce beats A is
oscillate with the same (a) frequency, (b) slightly reduced and the beat frequency is found
phase, (c) amplitude? Explain your to reduce to 3 Hz. If the original frequency of A
answers. is 324 Hz, what is the frequency of B?
ii) What is the amplitude of a point 0.375 m 19. Explain why (or how) :
away from on end?
a) In a sound wave, a displacement node is a
13. Given below are some functions of x and t to pressure antinode and vice versa,
represent the displacement (transverse or
b) bats can ascertain distances, directions, nature
longitudinal) of an elastic wave. State which
and sizes of the obstacles without any “eyes”.
of these represent (i) a travelling wave,
(ii) a stationary wave or (iii) none at all : c) a violin note and sitar note may have the
same frequency, yet we can distinguish
a) y  2 cos3x  sin10t  between the two notes,

b) y  2 x  vt d) solids can support both longitudinal and

transverse waves, but only longitudinal
c) y  3 sin5x  0.5t   4 cos5x  0.5t  waves can propagate in gases, and

d) y  cos x sin t  cos 2 x sin 2t e) the shape of a pulse gets distorted during
propagation in a dispersive medium.
14. A wire stretched between two rigid supports
vibrates in its fundamental mode with a 20. A train, standing at the outer signal of a railway
frequency of 45 Hz. The mass of the wire is station blows a whistle of frequency 400 Hz in
3.5 × 10–2 kg and its linear mass density is still air.
4.0 × 10–2 kg m–1. What is (a) the speed of a i) What is the frequency of the whistle for a
transverse wave on the string, and (b) the platform observer when the train
tension in the string?
a) approaches platform with a speed of 10 m s–
15. A metre-long tube open at one end, with a 1
.
movable piston at the other end, sows
b) recedes from platform with speed of 10 m s–
resonance with a fixed frequency source 1
?
(a tuning fork of frequency 340 Hz) when the
tube length is 25.5 cm or 79.3 cm. Estimate ii) What is the speed of sound in each case?
the speed of sound in air at the temperature of Speed of sound in still air can be taken as 340
the experiment. The edge effects may be m s–1.
neglected. 21. A train, standing in a station-yard, blows a
16. A steel rod 100 cm long is clamped at its whistle of frequency 400 Hz in still air. The wind
middle. The fundamental f requency of starts blowing in the direction from the yard
longitudinal vibrations of the rod are given to to the station with a speed of 10 m s–1. What are
be 2.53 kHz. What is speed of sound in steel ? the frequency, wavelength, and speed of sound
for an observer standing on the station
17. A pipe 20 cm long is closed at one end. Which
platform? Is the situation exactly identical to the
harmonic mode of the pipe is resonantly excited
case when the air is still and the observer runs
by a 430 Hz source? Will the same source be in
towards the yard at a speed of 10 m s–1? The
resonance with the pipe if both ends are open?
speed of sound in still air can be taken as
(speed of sound in air is 340 m s–1)
340 m s–1.

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 Waves
22. A travelling harmonic wave on a string is 25. A SONAR system fixed in a submarine
described by yx , t   7.5 sin0.0050x  12t   / 4  operator at a frequency 40.0 kHz. An enemy
a) What are the displacement and velocity of submarine moves towards the SONAR wit a
oscillation of a point at x = 1 cm, and t = 1 s
speed of 360 km h–1. What is the frequency of
? Is this velocity equal to the velocity of wave
propagation? sound reflected by the submarine ? Take the
b) Locate the points of the string which have the speed of sound in water to be 1450 m s–1.
same transverse displacement & velocity as
the x = 1 cm point at t = 2 s, 5 s and 11 s. 26. Earthquakes generate sound waves inside the
23. A narrow sound pulse (for example, a short
earth. Unlike a gas, the earth can experience
pip by a whistle) is sent across a medium.
a) Does the pulse have a definite (i) frequency, both transverse (S) and longitudinal (P) sound
(ii) wavelength, (iii) speed of propagation? waves. Typically the speed of S wave is about
b) If the pulse rate is 1 after every 20 s, that is the 4.0 km s–1, and that of P wave is 8.0 km s–1. A
whistle is blown for a split of second after
seismograph records P and S waves from an
every 20 s, is the frequency of the note
produced by whistle equal to 1/20 or 0.05 Hz? earthquake. The first P wave arrives 4 min
24. One end of a long string of linear mass density before the first S wave. Assuming the waves
8.0 × 10–3 kg m–1 is connected to an electrically travel in straight line, at what distance does the
driven tuning fork of frequency 256 Hz. The
earthquake occur?
other end passes over a pulley and is tied to a
pan containing a mass of 90 kg. The pulley end
27. A bat is flitting about in a cave, navigating via
absorbs all the incoming energy so that reflected
waves at this end have negligible amplitude. At ultrasonic beeps. Assume that the sound
t = 0, the left end (fork end) of the string x = 0 emission frequency of the bat is 40 kHz.
has zero transverse displacement (y = 0) and is During one fast swoop directly towards a flat
moving along positive y-direction. The
wall surface, the bat is moving at 0.03 times
amplitude of the wave is 5.0 cm. Write down
the transverse displacement y as function of x the speed of the sound in air. What frequency
and t that describes the wave on the string. does the bat hear reflected off the wall ?

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 Waves
NCERT EXEMPLAR
1. Water waves produced by a motor boat sailing 6. Which of the following statements is true for
in water are wave motion?

a) neither longitudinal nor transverse a) Mechanical transverse waves can propagate

through all mediums.
b) both longitudinal and transverse
b) Longitudinal waves can propagate through
c) only longitudinal
solids only
d) only transverse
c) Mechanical transverse waves can propagate
2. Sound waves of wavelength  travelling in a through solids only
medium with a speed of v m s –1 enter into
d) Longitudinal waves can propagate through
another medium where its speed is 2v m s–1.
vacuum.
Wavelength of sound waves in the second
medium is 7. A sound wave is passing through air column
in the form of compression and rarefaction.
 In consecutive compressions and rarefactions,
a)  b)
2
a) density remains constant
c) 2 d) 4
b) Boyle’s law is obeyed
3. Speed of sound wave in air
c) bulk modulus of air oscillates
a) is independent of temperature
d) there is no transfer of heat
b) increases with pressure
8. Equation of a plane progressive wave is given
c) increases with increase in humidity
 x
by y  0.6 sin 2  t   . On reflection from a
d) decreases with increase in humidity  2

4. Change in temperature of the medium changes 2

denser medium its amplitude becomes  
3
a) frequency of sound waves
of the amplitude of the incident wave. The
b) amplitude of sound waves equation of the reflected wave is

c) wavelength of sound waves

 x
a) y  0.6 sin 2  t  
d) loudness of sound waves  2

5. With propagation of longitudinal waves

 x
through a medium, the quantity transmitted is b) y  0.4 sin 2  t  
 2
a) matter
 x
b) energy c) y  0.4 sin 2  t  
 2
c) energy and matter
 x
d) energy, matter and momentum d) y  0.4 sin 2  t  
 2

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 Waves
9. A string of mass 2.5 kg is under a tension of 13. Speed of sound waves in a fluid depends upon
200 N. The length of the stretched string is a) directly on density of the medium
20 m. If the transverse jerk is struck at one
b) square of Bulk modulus of the medium
end of the string, the disturbance will reach
the other end in c) inversely on the square root of density

a) one second b) 0.5 second d) directly on the square root of bulk modulus
of the medium.
c) 2 second d) data given is insufficient
14. During propagation of a plane progressive
10. A train whistling at constant frequency is mechanical wave
moving towards a station at a constant speed
a) all particles are vibrating in the same phase.
v. The train goes past a stationary observer on
the station. The frequency n of the sound as b) amplitude of all the particles are equal.
heard by the observer is plotted as a function c) particles of the medium executes S.H.M.
of time t. Identify the expected curve.
d) wave velocity depends upon the nature of
the medium.
a) b) 15. The transverse displacement of a string
(clamped at its both ends) is given by
yx , t   0.06 sin 2x / 3  cos120t 
The length of the string is 1.5 m. All the points
c) d) on the string between two consecutive nodes
vibrate with
11. A transverse harmonic wave on a string is a) same frequency
described by
b) same phase
a) The wave is travelling from right to left.
c) same energy
b) The speed of the wave is 20 m s–1.
d) different amplitude.
c) Frequency of the wave is 5.7 Hz.
16. A train, standing in a station yard, blows a
d) The least distance between two successive
whistle of frequency 400 Hz in still air. The wind
crests in the wave is 2.5 cm.
starts blowing in the direction from the yard to
12. The displacement of a string is given by the station with a speed of 10 m s–1. Given that
yx , t   0.06 sin 2x / 3  cos120t  the speed of sound in still air is 340 m s–1.
where x & y are in m and t in s. The length of a) the frequency of sound as heard by an
a) It represents a progressive wave of observer standing on the platform is 400 Hz.
frequency 60 Hz b) the speed of sound for the observer standing
b) It represents a stationary wave of frequency on the platform is 350 m s–1.
60 Hz. c) the frequency of sound as heard by the
c) It is the result of superposition of two waves observer standing on the platform will
of wavelength 3 m, frequency 60 Hz each increase
travelling with a speed of 180 m s –1 in d) the frequency of sound heard by the
opposite direction observer standing on the platform will
d) Amplitude of this wave is constant. decrease.

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 Waves
17. Which of the following statements are true for 25. A steel wire has a length of 12 m and a mass of
a stationary wave? 2.10 kg. What will be the speed of a transverse
a) Every particle has a fixed amplitude which wave on this wire when a tension of 2.06 × 104
is different from the amplitude of its nearest N is applied?
particle.
26. A pipe 20 cm long is closed at one end. Which
b) All the particles cross their mean position harmonic mode of the pipe is resonantly
at the same time. excited by a source of 1237.5 Hz ? (sound
c) All the particles are oscillating with same velocity in air = 330 m s–1)
amplitude
27. A train standing at the outer signal of a railway
d) There is no net transfer of energy across any
station blows whistle of frequency 400 Hz in still
plane.
air. The train begins to move with a speed of
e) There are some particles which are always 10 m s –1 towards the platform. What is the
at rest. frequency of the sound for an observer standing
18. A sonometer wire is vibrating in resonance on platform? (sound velocity in air = 330 m s–1)
with a tuning fork. Keeping the tension applied
same, the length of the wire id doubled. Under 28. The wave pattern on a stretched string is shown
what conditions would the tuning fork still be in figure. Interpret what kind of wave this is
in resonance with the wire? and find its wavelength.
19. An organ pipe of length L open at both ends is
found to vibrate in its first harmonic when
sounded with a tuning fork of 480 Hz. What
should be the length of a pipe closed at one
end, so that it also vibrates in its first harmonic
with the same tuning fork?
20. A tuning fork A, marked 512 Hz, produces 5
beats per second, when sounded with another
unmarked tuning fork B. If B is loaded with
wax the number of beats is again 5 per second.
What is the frequency of the tuning fork B
when not loaded? 29. The patterns of standing waves formed on a
21. The displacement of an elastic wave is given stretched string at two instants of time are
by the function y  3 sin t  4 cos t. shown in figure. The velocity of two waves
where y is in cm and t is in second. Calculate superimposing to form stationary waves is
the resultant amplitude. 360 m s–1 and their frequencies are 256 Hz.
22. A sitar wire is replaced by another wire of same
length and material but of three times the
earlier radius. If the tension in the wire remains
the same, by what factor will the frequency 0
change?
23. At what temperature (in °C) will the speed of
a) Calculate the time at which the second curve
sound in air be 3 times its value at 0 °C?
is plotted.
24. When two waves of almost equal frequencies
n1 and n2 reach at a point simultaneously, what b) Mark nodes and antinodes on the curve
is the time interval between successive c) Calculate the distance between A’ and C’
maxima?
VELOCITY INSTITUTE OF PHYSICS 54
 Waves
30. A tuning fork vibrating 34. Given below are some functions of x and t to
represent the displacement of an elastic wave.
with a frequency of
a) y = 5 cos(4x) sin (20t)
512 Hz is kept close to
b) y = 4 sin (5x - t/2) + 3 cos(5x - t/2)
the open end of a tube
c) y = 10 cos[(252 - 250)  t]cos[(252 + 250)  t]
filled with water, as
d) y = 100 cos (100  t + 0.5x)
shown in figure. The water
State which of these represent
level in the tube is gradually
a) a travelling wave along-x direction
lowered. When the water level is 17 cm below
b) a stationary wave
the open end, maximum intensity of sound is
heard. If the room temperature is 27°C, calculate c) beats
a) speed of sound in air at room temperature d) a travelling wave along +x direction. Given
reasons for your answers.
b) speed of sound in air at 0°C
35. In the given progressive wave
c) if the water in the tube is replaced with
mercury, will there be any difference in your y  5 sin100t  0.4x 
observations?
where y and x are in m, t is in s. What is the
31. Show that when a string fixed at its two ends
vibrates in 1 loop, 2 loops, 3 loops and 4 loops, a) amplitude b) wavelength
the frequencies are in the ratio 1:2:3:4. c) frequency d) wave velocity
32. The earth has a radius of 6400 km. The inner e) particle velocity amplitude.
core of 1000 km radius is solid. Outside it,
there is a region from 1000 km to a radius of 36. For the harmonic travelling wave
3500 km which is in molten state. Then again
y  2 cos 2 10t  0.0080 x  3.5
from 3500 km to 6400 km, the earth is solid.
Only longitudinal (P) waves can travel inside where x and y are in cm and t in second. What
a liquid. Assume that the P wave has a speed is the phase difference between the oscillatory
–1 –1
of 8 km s in solid parts and 5 km s in liquid motion at two points separated by a distance of
parts of the earth. An earthquake occurs at
a) 4 m
some place close to the surface of the earth.
Calculate the time after which it will be b) 0.5 m
recorded in a seismometer at a diametrically
opposite point on the earth if wave travels c)  / 2
along diameter?
d) 3 / 4 (at a given instant of time)
33. If c is r.m.s. speed of molecules in a gas and v
e) What is the phase difference between the
is the speed of sound waves in the gas, show
oscillation of a particle located at x = 100 cm
that c/v is constant and independent of
at t = T s and t = 5 s?
temperature for all diatomic gases.

VELOCITY INSTITUTE OF PHYSICS 55