PW – 1
WAV E S
C1
Wave
A wave is a disturbance that travels or propagates and transports energy and momentum without the
transport of matter. The ripples on a pond, the sound we hear, visible light, radio and TV signals are the
examples of waves.
Mechanical waves such as water waves or sound waves require material medium for their propagation.
These waves travels within or on the surface of material with elastic properties. These waves are governed
by Newton’s Laws.
Electromagnetic waves, such as light and TV signals, are non-mechanical and can propagate through
vacuum.
Transverse and Longitudinal Waves
If a particle of the medium in which the wave is travelling moves perpendicular to the direction of wave
propagation, the wave is called transverse. Example of transverse waves are EM waves, waves on the
string.
If a particle of the medium move parallel to the direction of wave propagation, the wave is said to be
longitudinal. Example of longitudinal wave are sound wave in any medium. Note that sound wave in solid
may be transverse or may be longitudinal, depending upon the mode of excitation.
Non-mechanical waves are always transverse but the converse is not true.
Mechanism of propagation of mechanical waves :
Mechanical waves are related to the elastic properties of the medium. In transverse waves, the constituents
of the medium oscillate perpendicular to wave motion causing change in shape. That is, each element of the
medium in subject to shearing stress. Solids and strings have shear modulus, that is they sustain shearing
stress. Fluids have no shape of their own - they yield to shearing stress. This is why transverse waves are
possible in solids and strings (under tension) but not in fluids. However, solids as well as fluids have bulk
modulus, that is, they can sustain compressive strain. Since longitudinal waves involve compressive stress
(pressure), they can be propagated through solids and fluids. Thus a steel bar possessing both bulk and
sheer elastic moduli can propagate longitudinal as well as transverse waves. But air can propagate only
longitudinal pressure waves (sound). When a medium such as steel bar propagaes both longitudinal and
transverse waves, their speeds can be different since they arise from different elastic moduli.
C2
Characteristics of a Wave
Three physical characteristics are important in describing a wave : wave length, frequency and wave speed.
Wave length (denoted by ) : One wavelength is the minimum distance between any two identical points
on a wave - for example, adjacent crests or adjacent troughs, as in figure, which is a graph of displacement
versus position for a sinusoidal wave at a specific time.
Frequency : The frequency of sinusoidal waves is the same as the frequency of simple harmonic motion of
a particle of the medium. The number of complete vibrations of a point that occur in one second or the
number of wavelengths that pass a given point in one second is called frequency. The period T of the wave
is the minimum time it takes a particle of the medium to undergo one complete oscillation, and is equal to
1
. Figure shows position versus time for a particle of the medium as a
f
sinusoidal wave is passing through its position.
the inverse of the frequency : T 
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The period is the time between instants when the particle has identical displacement and velocities.
Wave speed (denoted by v) : Waves travel through the medium with a specific wave speed, which
depends on the properties of the medium being disturbed.
Another important parameter for the wave (in figure) is the amplitude of the wave denoted by A. This is the
maximum displacement of a particle of the medium from the equilibrium position.
Relation between wavelength, frequency and wave speed :
Since in one period T the wave advances by one wavelength , therefore, the wave velocity is v 

 f
T
Practice Problems :
1.
What is the difference between the two figures ?
2.
A narrow sound pulse (for example, a short pip by a whistle) is sent across a medium. (a) Does the
pulse have a definite (i) frequency, (ii) wavelength, (iii) speed of propagation ? (b) If the pulse rate is
1 after every 20 s. (that is the whistle is blown for a split of second after every 20 s), is the frequency
of the note produced by the whistle equal to 1/20 or 0.05 Hz ?
3.
Earthquakes generate sound wave inside the earth. Unlike a gas, the earth can experience both
transverse (S) and longitudinal (P) sound waves. Typically the speed of S wave is about 4.0 km s–1,
and that of P wave is 8.0 km s–1. A seismograph records P and S waves from an earthquake. The first
P wave arrives 4 min before the first S wave. Assuming the waves travel in straight, line, at what
distance does the earthquake occur ?
[Answers : (3) 1920 km]
C3
Equation of a Travelling Wave
The equation of a wave traveling along the positive x-axis is given by y(x, t) = f(x – vt). The function y is
called the wave function depends on the two variables position x and time t which is read “y as a function
of x and t”.
Meaning of y :
Einstein Classes,
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Figure represents the shape and position of the pulse at time t = 0. At this time, the shape of the pulse,
whatever it may be, can be represented by some mathematical function that we will write as y(x, 0) = f(x).
This function describes the vertical position y of the element of the string located at each value of x at time
t = 0. Because the speed of the pulse is v, the pulse has traveled to the right at a distance vt at the time t. We
adopt a simplification model in which the shape of the pulse does not change with time (In reality, the pulse
changes its shape and gradually spreads out during the motion. This effect is called dispersion and is
common to many mechanical waves; however we adopt a simplification model that ignores this effect).
Thus, at time t, the shape of the pulse is the same as it was at time t = 0, as in figure. Consequently, an
element of the string at x at this time has the same y position as an element located at x – vt has at time
t=0:
y(x, t) = y(x – vt, 0)
In general, then, we can represent the displacement y for all positions and times, measured in a stationary
frame with the origin at O, as
y(x, t) = f(x – vt)
Similarly, if the wave pulse travels to the left (negative x-axis), the displacement of the string is
y(x, t) = f(x + vt)
Consider a point P on the string, identified by a particular value of its x coordinates. As the pulse passes
through P, the y coordinates of this point increases, reaches a maximum, and then decreases to zero. The
wave function y(x, t) represents the y coordinates of any point P located at position x at any time t.
Furthermore, if t is fixed (e.g., in the case of taking a snapshot of the pulse), then the wave function y as a
function of x, sometimes called the waveform, defines a curve representing the actual geometric shape of
the pulse at that time.
In general, the wave motion in one dimension is given by y = f(x ± vt)
Practice Problems :
1.
2.
You have learnt that a travelling wave in one dimension is represented by a function y = f(x, t) where
x and t must appear in the combination x – vt or x + vt, i.e., y = f(x ± vt). Is the converse true ?
Examine if the following functions for y can possibly represent a travelling wave :
(a)
(x – vt)2
(b)
log [(x + vt)/x0]
(c)
1/(x + vt)
(d)
sin (x – vt)
(e)
cos (x – vt)
(f)
tan (x – vt)
(g)
x  vt
A wave pulse moving along the x-axis is represented by the wave function
y ( x, t ) 
2.0
( x  3.0t ) 2  1
where x and y are measured in centimeters and t is in seconds. Plot the waveform at t = 0, t = 1.0 s,
and t = 2.0 s. Also find the direction of propagation of wave pulse, amplitude and velocity ?
[Answers (1) The converse is not true. An obvious requirement for an acceptable function for a
travelling wave is that it should be finite everywhere and at all times. Only function (a), (d), (e)
satisfies this condition, the remaining functions cannot possibly represent a travelling wave]
C4
Sine Harmonic Travelling Wave
The wave function that represents a sine harmonic travelling wave is given by
y(x, t) = Asin[(kx ± t) + ]
The negative sign is used when the wave travels along the positive x-axis, and vice-versa.
Meaning of Standard Symbols :
y(x, t)
:
displacement as a function of position x and time t
A
:
amplitude of a wave

:
angular frequency of the wave
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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k
:
angular wave number or propagation constant
kx ± t + 
:
phase of the wave

:
initial phase or phase at x = 0 and t = 0
Relation between wave number (k) and wave length () : k 
2

Relation between angular frequency, frequency and time period :  
Wave velocity : v 
2
 2 f
T


 f  .
T
k
Wave Velocity and Particle Velocity
Wave velocity is the velocity of the disturbance which propagates through a medium. It only depends on the
properties of the medium and is independent of time and position.
Particle velocity is the rate at which particle’s displacement vary as a function of time, i.e.,
y
  A cos (kx  t   ) .
t
Acceleration of the Particle :
The acceleration of the particle is obtained by differentiating above equation w.r.t. time
 2y
t 2
   2 A sin (kx  t   ) .
Practice Problems :
1.
A wave equation which gives the displacement along y-direction is given by
y = 10–4 sin (60t + 2x)
where x and y are in m and t in s. This represents a wave
2.
3.
(a)
travelling with velocity of 30 m/s in the negative x-direction
(b)
of wavelength () m
(c)
of frequency (30/) Hz
(d)
all the above
A transverse sinusoidal wave of amplitude a, wavelength  and frequency f is traveling on a stretched
string. The maximum speed of any point on the string is v/10, where v is the speed of propagation of
the wave. If a = 10–3 m and v = 10 ms–1, then  and f are given by
(a)
 = 2 × 10–2 m
(b)
 = 10–3 m
(c)
f = 103/(2) Hz
(d)
both (a) and (c) are correct
A transverse wave is described by the equation y = A sin 2(ft – x/). The maximum particle velocity
is equal to n times the wave velocity then the relation between A and  is
(a)
n = A
(b)
n = 2A
(c)
n = 3A
(d)
n = 4A
[Answers : (1) d (2) d (3) b]
C5
Wave speed on stretched string : The speed of a wave on a stretched string is set by properties of the
string. The speed on a string with tension T and linear density µ is v 
T
.
µ
Practice Problems :
1.
Using the dimensional method or otherwise find the wave speed on a stretched string of mass per
unit length µ and under the tension T ? Express this speed in terms of cross-sectional area A of the
string and density of the string .
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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2.
Consider two strings of the same material but different cross-sectional radius in the ratio 1 : 2. Find
the ratio of the tension in the string such that the wave speed in both string is same ?
3.
A string of mass per unit length µ is suspended vertically from a roof and length of the string is L. A
pulse is produced at the lowermost point.
(a)
Find the speed of the pulse at the distance x from the lower end.
(b)
Does it depend on the mass per unit length ? Comment ?
(c)
What is the time taken by the pulse to reach uppermost point ?
4.
Consider a wire of Young’s Modulus of elasticity Y, density  and coefficient of linear expansion .
The wire is rigidly clamped between two points and the temperature is raised by T. Find the wave
speed of the transverse wave produced on the wire.
C6
Energy Transmitted by a Wave
The average power transmitted by the wave is
Pav 
dE av 1
dx
 µ 2 A 2 v where v 
is the wave velocity..
dt
2
dt
The mass per unit length of a wire is given by µ =  a where  is the density and a is the cross-sectional area.
The intensity of the wave is given by I 
Pav 1 2 2
  A v
a
2
Note that the power and intensity are proportional to the square of the frequency and square of the amplitude.
Practice Problems :
1.
Derive the expression for the average power and average intensity carried by the wave on a string of
mass per unit length µ. The angular frequency, amplitude and speed of the wave are , A and v
respectively. Also prove that propagation of average kinetic energy per unit time and average
potential energy per unit time carried by the wave are equal.
2.
A string with linear mass density µ = 5.00 × 10–2kg/m is under a tension of 80.0 N. How much power
must be supplied to the string to generate sinusoidal waves at a frequency of 60.0 Hz and an
amplitude of 6.00 cm ?
3.
Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string that has a linear mass
density to 4.00 × 10–2 kg/m. If the source can deliver a maximum power of 300 W and the string is
under a tension of 100 N, what is the highest vibrational frequency at which the source can
operate ?
4.
A is singing a note and at the same time B is singing a note with exactly one-eighth the frequency of
the note of A. The energies of the two sounds are equal. The amplitude of the note of B is
(a)
Same as that of A
(b)
Twice that of A
(c)
Four times that of A
(d)
Eight times that of A
[Answers : (2) 512 W (3) 55.1 Hz (4) d]
C7
Velocity of Longitudinal Waves
The velocity of longitudinal wave is given by
E
where E is the modulus of elasticity of the medium and

 is the density of the medium in which wave will propagate. If the longitudinal wave will propagate in
solid then E = Y (Young’s modulus of elasticity) and if the longitudinal wave will propagate in fluid then
E = B (Bulk modulus of elasticity).
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C8
Velocity of Sound Wave
The speed of longitudinal waves in a fluid is given by v 
where B is the bulk modulus defined as B  
B

V
p
, where
is the fractional change in volume
V
V / V
produced by the change in pressure p.
The propagation of sound in gas is an adiabatic process. For gas the bulk modulus in adiabatic condition is
given by B = p, where  is an adibatic exponent.
Thus, velocity of sound in air is given by v 
Using ideal gas equation
p RT
, v

p M
p

RT
M
Substituting the values of  and M for air we obtain the approximate value of speed of sound in air at
absolute temperature T as v  20 T
Practice Problems :
1.
The ratio (Vs/Vrms) of the velocity of sound in a gas (Vs) and the root mean square velocity (Vrms) of its
molecular at the same temperature is : ( = ratio of the two specific heats of the gas)
(a)
2.
3/ 
(c)
3
(d)
1
3
(2 / 7)
(b)
(1 / 7)
(c)
( 3) / 5
(d)
( 6) / 5
The temperature of air is increased from 300 K to 301 K. The fractional change in velocity of sound
is
(a)
4.
(b)
The ratio of the speed of sound in nitrogen gas to that in helium gas, at 300 K is
(a)
3.
/3
1/300
(b)
1/600
(c)
1/900
(d)
1/1200
The speed of sound in a gas at NTP is 300 m/sec. If the pressure is increased four times, without
change in temperature, the velocity of sound will be
(a)
150 m/sec
(b)
300 m/sec
(c)
600 m/sec
(d)
1200 m/sec
5.
Estimate the speed of sound in air at standard temperature and pressure. The mass of 1 mole of air
is 29.0 × 10–3 kg.
6.
What is the percentage change in velocity of sound if (i) percentage change in temperature is 1%
(ii) percentage change in temperature is 21% (iii) percentage change in pressure is 100% keeping the
temperature constant.
7.
Find the velocity of sound in a mixture of gases consists of one mole of He gas and two moles of O2
gas at the temprature of 00C.
[Answers : (1) a (2) c (3) b (4) b (5) 280 m s–1 (6) (i) 0.5% (ii) 10% (iii) 0%]
C9
Relationship Between Pressure Waves and Displacement Waves
The displacement accompanying a sound wave in air are longitudinal displacements of small elements of
the fluid from their equilibrium positions. Such displacement results if the source of the waves oscillates in
air, for example diaphragm of a loud-speaker. As the vibrating source moves forward, it compresses the
medium past it, increasing the density locally. This part of the medium compresses the layer next to it by
collision. The compression travels in the medium at a speed which depends on the elastic and inertia
properties of the medium. As the source moves back, it drags the medium and produces a rarefaction in the
layer. The layer next to it is then dragged back and thus the rarefaction pulse passes forward. In this way,
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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compression and rarefaction pulses are produced which travel in the medium.
A longitudinal wave in a fluid (liquid or gas) can be described either in terms of the longitudinal
displacement suffered by the particles of the medium or in terms of the excess pressure generated due to the
compression or rarefaction.
For a harmonic wave, the longitudinal displacement (y) is given by y = A sin (kx – wt)
p  B
We know
.
V
V
Since change in volume is produced by the displacement of the particles, therefore,
V y

V
x
Thus p   B
y
= –BAk cos (kx – wt)
x
or
p = –p0 cos (kx – wt)
where p0 = BAk is the pressure amplitude.
Note that displacement wave and pressure wave amplitudes are /2 out of phase.
Practice Problems :
1.
A sinusoidal sound wave is described by the displacement
s(x, t) = (2.00 µm) cos[(15.7 m–1)x – (858 s–1)t]
(a) Find the amplitude, wavelength, and speed of this wave. (b) Determine the instantaneous
displacement of an element of air at the position x = 0.050 0 m at t = 3.00 ms. (c) Determine the
maximum speed of an element’s oscillatory motion.
2.
Calculate the pressure amplitude of a 2.00-kHz sound wave in air if the displacement amplitude is
equal to 2.00 × 10–8 m.
3.
An experimenter wishes to generate in air a sound wave that has a displacement amplitude of
5.50 × 10–6 m. The pressure amplitude is to be limited to 0.840 N/m2. What is the minimum
wavelength the sound wave can have ?
4.
A sound wave traveling in air has a pressure amplitude of 4.00 N/m2 and a frequency of 5.00 kHz.
Take P = 0 at the point x = 0 when t = 0. (a) What is P at x = 0 when t = 2.00 × 10–4 s ? (b) What is
P at x = 0.020 0 m when t = 0 ?
5.
Write an expression that describes the pressure variation as a function of position and time for a
sinusoidal sound wave in air, if  = 0.100 m and Pmax = 0.200 N/m2.
[Answers : (1) (a) 2.00 µm, 40.0 cm, 54.6 m/s (b) –0.433 µm (c) 1.72 mm/s (3) 5.81 m
(5) P = (0.200 N/m2) sin(62.8x/m – 2.16 × 104t/s)]
C10
Intensity of sound wave and Sound level
The intensity I of a sound wave at a surface is the average rate per unit area at which energy is transferred
P
, where P is the time rate of energy transfer (power) of the
A
sound wave and A is the area of the surface intercepting the sound. The intensity I carried by the sound
by the wave through or onto the surface : I 
wave is given by
p 02
where p0 is the pressure amplitude and  is the density of medium in which the wave
2v
will propagate with speed v.
Sound Level in Descibels : The sound level  in decibels (dB) is defined as   (10dB ) log
I
, where I0
I0
( = 10–12 W/m2) is a reference intensity to which all intensities are compared. For every factor of 10
increases in intensity, 10 dB is added to the sound level.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 8
Practice Problems :
1.
Two sound waves move in the same direction in the same medium. The pressure amplitudes of the
waves are equal but the wavelength of the first wave is double the second. Let the average power
transmitted across a cross-section by the first wave be P1 and that by the second wave be P2. Find the
value of
P1
.
P2
2.
A sound wave of frequency 300 Hz has an intensity of 1.00 µ W/m2. What is the amplitude of the air
oscillations caused by this wave ?
3.
Two sounds differ in sound level by 1.00 dB. What is the ratio of the greater intensity to the smaller
intensity ?
4.
A certain sound is increased in sound level by 30 dB. By what multiple is (a) its intensity increased
and (b) its pressure amplitude increased ?
5.
(a) If two sound waves, one in air and one in (fresh) water, are equal in intensity, what is the ratio of
the pressure amplitude of the wave in water to that of the wave in air ? Assume the water and the air
are at 200C. (b) If the pressure amplitudes are equal instead, what is the ratio of the intensities of the
waves ? The density of air is 1.21 kg/m3 and for water 0.998 × 103 kg/m3 at 200C.
6.
Find the ratios (greater to smaller) of (a) the intensities, (b) the pressure amplitudes, and (c) the
particle displacement amplitudes for two sounds whose sound levels differ by 37 dB.
7.
Derive the formula of intensity carried by the sound wave.
[Answers : (1) 1 (2) 36.8 nm (4) (a) 1000 (b) 32 (5) (a) 59.7 (b) 2.81 × 10–4]
C11
Variation of intensity with distance
(a)
For point source, the wavefronts are spherical and the intensity at the distance r from the source
of power P is
(b)
4 r 2
.
For line source, the wavefronts are cylinderical and the intensity at the distance r from the source
of power P is
(c)
P
P
where L is the length of the source.
2rL
For plane wave front, the intensity does not change with distance.
Practice Problems :
1.
(a) Show that the intensity I of a wave is the product of the wave’s energy per unit volume u and its
speed v. (b) Radio waves travel at a speed of 3.00 × 108 m/s. Find u for a radio wave 480 km from a
50,000 W source, assuming the wavefronts are spherical.
2.
A source emits sound wave isotropically. The intensity of the waves 2.50 m from the source is
1.91 × 10–4 W/m2. Assuming that the energy of the waves is conserved, find the power of the source.
3.
A sound wave travels out uniformly in all directions from a point source. (a) Justify the following
expression for the displacement s of the transmitting medium at any distance r from the source :
b
sin k (r  vt ) , where b is a constant. Consider the speed, direction of propagation, periodicity,,
r
and intensity of the wave. (b) What is the dimension of the constant b ?
s
4.
A point source emits 30.0 W of sound isotropically. A small microphone intercepts the sound in an
area of 0.750 cm2, 200 m from the source. Calculate (a) the sound intensity there and (b) the power
intercepted by the microphone.
5.
An electric spark jumps along a straight line of length L = 10m, emitting a pulse of sound that travels
radially outward from the spark. The power of the emission is Ps = 1.6 × 104 W. (a) What is the
intensity I of the sound when it reaches a distance r = 12 m from the spark ? (b) At what time rate Pd
is sound energy intercepted by an acoustic detrector of area Ad = 2.0 cm2, aimed at the spark and
located a distance r = 12 m from the spark ?
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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6.
The sound level 46 m in front of the speaker systems was  2 = 120 dB. What is the ratio of the
intensity I2 of the band at that spot to the intensity I1 of a jackhammer operating at sound level
 1 = 92 dB ?
[Answers : (1) (b) 5.76 × 10–17 J/m3 (2) 15.0 mW (3) (b) L2 (5) (a) 21 W/m2 (b) 4.2 mW (6) 630]
C12
Superposition of Waves
When two or more waves traverse in the same medium, the displacement of any particle of the medium is
the sum of the displacement that the individual waves would give it.
The wave do not alter one another and each propagates through the medium as if the other were not there.
C13
Interference : Adding waves that differ in Phase only :
If the two waves have different amplitudes a1 and a2, respectively, the resultant amplitude is given by
A  a12  a 22  2a1a 2 cos 
where  is the constant phase difference.
If
 = 2n
where n = 0, 1, 2, .....
Amax = a1 + a2
If
 = (2n – 1) 
where n = 1, 2, 3, .....
Amin = a1 – a2
Since intensity is proportional to the square of the amplitude. Therefore,
I  I 1  I 2  2 I 1I 2 cos 
If
 = 2n
I max 
If
where n = 0, 1, 2,.....
I
1
 I2
 = (2n – 1) 
I min 
I
1

2
where n = 1, 2, 3,....
 I2

2
Practice Problems :
1.
Can independent sources produce interference pattern ?
2.
What is the result intensity if waves from incoherent sources are superimposed ?
3.
Waves generated from two incoherent sources of intensity level 10dB and 90 dB are superimposed.
What is the resultant intensity in dB ?
4.
Two speakers placed 3.00 m apart are driven in phase by the same oscillator. A listeners originally at
point O, which is located 8.00 m from the center of the line connecting the two speakers. The listener
then moves to point P, which is a perpendicular distance 0.350 m from O before reaching the first
cancellation of waves, resulting in a minimum in sound intensity. What is the frequency of the
oscillator ?
5.
Two waves travelling in the same direction along a stretched string. The waves are 90.00 out of phase.
Each wave has an amplitude of 4.00 cm. Find the amplitude of the resultant wave.
6.
Two sinusoidal waves are described by the wave functions y1 = (5.00 m) sin [(4.00x – 1200t)] and
y2 = (5.00 m) sin [(4.00x – 1200t – 0.250)] where x, y1, and y2 are in meters t is in seconds. (a) What
is the amplitude of the resultant wave ? (b) What is the frequency of the resultant wave ?
7.
Two identical sinusoidal waves with wavelengths of 3.00 m travel in the same direction at a speed of
2.00 m/s. The second wave originates from the same point as the first, but at a later time. Determine
the minium possible time interval between the starting moments of the two waves if the amplitude of
the resultant wave is the same as that of each of the two initial waves.
[Answers : (4) 1.3 kHz (5) (a) 9.24 m (b) 600 Hz]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 10
C14
Beats : Adding Wave That Differ in Frequency Only :
If two or more waves of slightly different frequencies (frequency different should be very small as
compared to the frequency of waves) are superimposed, the intensity of the resulting wave have
alternate maxima and minima such that the intensity of the resultant wave varies periodically with time. The
number of minima or maxima in one second is called the beat frequency.
Practice Problems :
1.
When two tuning forks A and B are sounded together x beat/s are heard. Frequency of A is n. Now
when one prong of fork B is loaded with a little wax, the number of beat/s decreases. The frequency
of fork B is
(a)
2.
(b)
n–x
(c)
n – x2
(d)
n – 2x
The speed of sound in a gas in which two waves of wavelengths 50 cm and 50.4 cm produce 6 beats
per second is
(a)
3.
n+x
338 m/s
(b)
350 m/s
(c)
378 m/s
(d)
400 m/s
Consider two waves of slightly different frequencies f1 and f2 with equal amplitudes A. Derive for
(a) frequency of the resultant wave (b) beat frequency (c) frequency of amplitude variation
(d) frequency of intensity variation (e) intensity of the resultant wave as a function of time. Also
draw the variation of intensity and amplitude of resultant wave with time. Also draw the variation of
resultant displacement with time at a given position.
[Answers : (1) a (2) c]
C15
Standing Waves : Adding Waves That Differ In Direction Only :
Consider two waves y1 and y2 that have the same amplitude, wavelength and frequency but travel in opposite directions.
y1 = a sin (kx – t)
y2 = a sin(kx + t)
Using superposition principle
y = y1 + y2 = a[sin(kx – t) + sin(kx + t)]
or
y = 2a sin kx cos t
or
y = A cos t where A = 2a sin kx
The above equation shows that the string executes simple harmonic motion such that every point on the
string vibrates in same phase with same frequency but different amplitudes which depends on the position
x of the point along the string.
This type of wave motion represented by equation is called a standing wave because it appears to travel
neither to the left nor to the right.
There are positions along the string for which the amplitude of oscillation is always zero (called nodes),
and other positions where the amplitudes of oscillation is always 2a (called antinodes)


. The distance between two successive antinodes is
.
2
2
Also, nodes and antinodes occur alternatively and equally spaced from each other.
The distance between two successive node is
Practice Problems :
1.
For the stationary wave
 x 
y  4 sin 
 cos( 96 t ) , the distance between a node and the next antinode is
 15 
(a)
2.
7.5 units
(b)
15 units
(c)
22.5 units
(d)
30 units
A standing wave having 3 nodes and 2 antinodes is formed between two atoms having a distance
1.21 Å between them. The wavelength of the standing wave is
(a)
1.21 Å
(b)
6.08 Å
(c)
3.63 Å
(d)
2.42 Å
[Answers : (1) a (2) a]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 11
C16
Standing Wave Pattern on the String
Standing waves on a string can be set up by reflection of traveling waves from the ends of the string. If an
end is fixed, it must be the position of a node. This limits the frequencies at which standing waves will
occur on a given string. Each possible frequency is a resonant frequency, and the corresponding standing
wave pattern is a resonant frequency, and the corresponding standing wave pattern is an oscillation mode.
For a stretched string of length L with fixed ends, the resonant frequences are f 
n = 1, 2, 3,..... (where v =
v
v
n
, for

2L
T
).
µ
The oscillation mode corresponding to n = 1 is called the fundamental mode or the first harmonic; the mode
corresponding to n = 2 is the second harmonic; and so on.
Practice Problems :
1.
Two vibrating strings of the same material but length L and 2L have radii 2r and r respectively. They
are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one
of length L with frequency v1 and the other with frequency v2. The ratio v1/v2 is given by
(a)
2.
(a)
3.
2
(b)
4
(c)
8
(d)
1
A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes
between the two bridges when a mass of 9 kg is suspended from the wire. When this mass is replaced
by a mass M, the wire resonates with the same tuning form forming three antinodes for the same
positions of the bridges. The value of M is
25 kg
(b)
5 kg
(c)
12.5 kg
(d)
1/25 kg
In order to double the frequency of the fundamental note emitted by a stretched string, the length is
reduced to 3/4 th of the original length and the tension is changed. The factor by which the tension is
to be changed is
(a)
3/8
(b)
2/3
(c)
8/9
(d)
9/4
[Answers : (1) d (2) a (3) d]
C17
Standing wave pattern in pipes
Standing sound wave pattern can be set up in pipes. A pipe open at both ends will resonate at frequencies
f
v nv
, n = 1, 2, 3,....

 2L
where v is the speed of sound in the gas in the pipe. For a pipe closed at one end and open at the another, the
resonant frequencies are f 
v nv

, n = 1, 3, 5, .....
 4L
Practice Problems :
1.
Velocity of sound in air is 320 m/s. A pipe closed at one end has a length of 1 m. Neglecting end
correction, the air column in the pipe cannot resonate for sound of frequency
(a)
2.
80 Hz
(b)
240 Hz
(c)
320 Hz
(d)
400 Hz
If the fundamental frequency of a pipe closed at one end is 512 Hz, the fundamental frequency of a
pipe of the same dimensions but open at both ends will be
(a)
1024 Hz
(b)
512 Hz
(c)
256 Hz
(d)
128 Hz
[Answers : (1) c (2) a]
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 12
C18
Reflection and Transmission of Waves :
When a traveling pulse reaches a boundary, part of all of the pulses is reflected. Any part not reflected is
said to be transmitted through the boundary.
When the pulse reaches the fixed boundary, it is reflected. The reflected pulse has exactly the same
amplitude as the incoming pulse but is inverted. The inversion can be explained as follows. The pulse is
created initially with an upward and then downward force on the free end of the string.As the pulse arrives
at the fixed end of the string, the string first produces an upward force on the support. By Newton’s third
law, the support exerts an equal and opposite reaction force on the string. Thus, the positive shape of the
pulse results in a downward and then upward force on the string as the entirety of the pulse encounters the
rigid end. Thus, reflection at a rigid end causes the pulse to invert on reflection.
Reflection of the pulse at the fixed boundary or rigid boundary
Now consider a second idealized situation in which reflection is total and transmission is zero. In this
simplification model, the pulse arrives at the end of a string that is perfectly free to move vertically. The
tension at the free end is maintained by tying the string to a ring of negligible mass that is free to slide
vertically on a frictionless post. Again, the pulse is reflected, but this time it is not inverted. As the pulse
reaches the post, it exerts a force on the free end, causing the ring to accelerate upward. In the process, the
ring reaches the top of its motion and is then returned to its original position by the downward component
of the tension force. Thus, the ring experiences the same motion as if it were raised and lowered by hand.
This produces a reflected pulse that is not inverted and whose amplitude is the same as that of the incoming
pulse.
Reflection of the pulse at the free end
Finally, we may have a situation in which the boundary is intermediate between these two extreme cases;
that is, it is neither completely rigid nor completely free. In this case, part of the wave is transmitted and part
is reflected. For instance, suppose a string is attached to a more dense string as in figure. When a pulse
traveling on the first string reaches the boundary between the two strings, part of the pulse is reflected and
inverted and part is transmitted to the more dense string. Both the reflected and transmitted pulses have
smaller amplitude than the incident pulse. The inversion in the reflected wave is similar to the behaviour of
a pulse meeting a rigid boundary. As the pulse travels from the initial string to the more dense string, the
junction acts more like a rigid end than a free end. Thus, the reflected pulse is inverted.
Figure : (a) A pulse travelling to the right on a light string attached to a heavier string. (b) Part of the incident
pulse is reflected (and inverted), and part is transmitted to the heavier string.
When a pulse traveling on a dense string strikes the boundary of a less dense string, as in figure, again part
is reflected and part transmitted. This time, however, the reflection pulse is not inverted. As the pulse
travels from the dense string to the less dense one, the junction acts more like a free end than a rigid end.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 13
Figure : (a) A pulse traveling to the right on a heavy string attached to a lighter string. (b) The incident pulse is
partially reflected and partially transmitted. In this case, the reflected pulse is not inverted.
The above observations may be stated as follows :
1.
If the wave will incident on a fixed boundary or rigid boundary then the reflected wave will be inverted that
means there is a phase change of . If the wave will incident on a loose boundary or free boundary then the
reflected wave will not be inverted that means there is no phase change.
2.
If a wave enters a region from higher wave velocity (rarer medium) to smaller wave velocity (denser
medium), the reflected wave is inverted that means there is a phase change of . If a wave enters a region
from lower wave velocity (denser medium) to larger wave velocity (rarer medium), the reflected wave is
not inverted that means there is no phase change.
Reflection of sound wave expressed in terms of pressure wave : In case of reflection of longitudinal
pressure waves, it suffers a phase change of  from a free or open end and no change in phase from rigid
boundaries. When a sound wave gets reflected from a rigid boundary, a compression pulse reflects as a
compression pulse and a rarefaction pulse reflect as a rarefaction pulse.
If ai is the amplitude of the incident wave in the medium 1, then the amplitudes of the reflected (ar) and
transmitted (at) waves in the medium 1 and 2 respectively, are given by
 v  v1 


 and a t  a i  2 v 2  .
a r  a i  2


 v 2  v1 
 v 2  v1 
Practice Problems :
1.
Consider a wave which is represented by y = a sin (t – k1x) is incident from rarer medium to denser
medium. If amplitude of reflected wave is ar, amplitude of transmitted wave is at and wave number
of the transmitted wave is k2 then find (a) equation of reflected wave (b) equation of transmitted
wave (c) ar (d) at.
2.
Consider a wave which is represented by y = a sin (t – k1x) is incident from denser medium to rarer
medium. If amplitude of reflected wave is ar, amplitude of transmitted wave is at and wave number
of the transmitted wave is k2 then find (a) equation of reflected wave (b) equation of transmitted
wave (c) ar (d) at.
3.
Consider a wave which is represented by y = a sin (t – kx) is incident on a rigid boundary. If there
is no transmission of waves then what is the equation of the reflected wave ?
4.
Consider a wave which is represented by y = a sin (t – kx) is incident on a free boundary. If there is
no transmission of waves then what is the equation of the reflected wave ?
5.
Consider a pressure wave which is represented by P = P0 sin (t – kx) is incident on a rigid
boundary. If there is no transmission of waves then what is the equation of the reflected wave ?
6.
Consider a wave which is represented by P = P0 sin (t – kx) is incident on a free boundary. If
there is no transmission of waves then what is the equation of the reflected wave ?
7.
A man standing in front of a mountain at a certain distance beats a drum at regular intervals. The
drumming rate is gradually increased and he finds that the echo is not heard distinctly when the rate
becomes 40 per minute. He then moves nearer to the mountain by 90 m and finds the echo is again
not heard when the drumming rate becomes 60 per minute. Calculate (a) the distance between the
mountain and the initial position of the man (b) the velocity of sound.
[Answers : (1) (a) y r = –a rsin(t + k1x) (b) y t = a tsin(t – k2x) (c)
(2) (a) yr = arsin(t + k1x) (b) yt = atsin(t – k2x) (c)
2k 1
k 2  k1
a
a (d)
k1  k 2
k1  k 2
2k 1
k1  k 2
a (3) –asin(t + kx)
a (d)
k1  k 2
k1  k 2
(4) asin(t + kx) (5) P0 sin(t + kx) (6) – P0 sin(t + kx) (7) (a) 270 m (b) 360 m/s]
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 14
C19
Doppler Effect :
The Doppler effect is a change in the observed frequency of a wave when the source or the detector moves
relative to the transmitting medium (such as air). For sound the observed frequency f  is given in terms of
the source frequency f by f   f
v  v0
, where v0 is the speed of the detector relative to the medium, vS is
v  vS
that of the source, and v is the speed of sound in the medium. The signs are chosen such that f  tends to be
greater for motion (of detector of source) “toward” and less for motion “away”.
Wind Effect :
The above formulae can be modified by taking the wind effects into account. The velocity of sound should
be taken as : v + vw or v – vw if the wind is blowing in the same or opposite direction as source to observer.
Practice Problems :
1.
A vehicle with a horn of frequency n is moving with a velocity of 30 m/s in a direction perpendicular
to the straight line joining the observer and the vehicle. The observer perceives the sound to have a
frequency n + n1. Then (if the sound velocity in air is 300 m/s)
(a)
2.
(b)
n1 = 0
(c)
n1 = –0.1 n
(d)
n1 = 0.1 n
A train moves towards a stationary observer with speed 34 m/s. The train sounds a whistle and its
frequency registered by the observer is f1. If the train’s speed is reduced to 17 m/s, the frequency
registered is f2. If the speed of sound is 340 m/s then the ratio f1/f2 is
(a)
3.
n1 = 10 n
18/19
(b)
1/2
(c)
2
(d)
19/18
A siren placed at a railway platform is emitting sound of frequency 5 kHz. A passenger sitting in a
moving train A records a frequency of 5.5 kHz while the train approaches the siren. During his
return journey in a different train B he records a frequency of 6.0 kHz while approaching the same
siren. The ratio of the velocity of train B to that of train A is
(a)
242/252
(b)
2
(c)
5/6
(d)
11/6
[Answers : (1) b (2) d (3) b]
Einstein Classes,
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 15
SINGLE CORRECT CHOICE TYPE
1.
2.
3.
A glass tube of 1.0 m length is filled with water; the
water can be drained out slowly at the bottom of
the tube. If a vibrating tuning fork of frequency
500 Hz is brought at the upper end of the tube and
the velocity of sound is 330 m/s, then the total
number of resonances obtained will be
(a)
4
(b)
3
(c)
2
(d)
1
Two stretched wires are in unison. If the tension is
one of the wires is increased by 1%, 3 beats are
produced in 2 s. The initial frequency of each wire
is
(a)
150 Hz
(b)
200 Hz
(c)
300 Hz
(d)
450 Hz
6.
NL
Ll
(b)
N 
l

1 

Ll 
546 L 
(c)
N
l

1 

L
546 L  
(d)
N 
l

1 

Ll 
546L 
Two wave pulse (the shape of one being inverted
with respect to the other) travel in opposite
direction on a string to approach each other. What
will happen to them ?
A man is standing between two parallel cliffs and
fires a gun. If he hears first and second echoes
after 1.5 s and 3.5 s respectively, the distance
between the cliffs is (Velocity of sound in air =
340 ms–1)
(a)
(a)
1190 m
(b)
850 m
(c)
595 m
(d)
510 m
They will collide and vanish after
collision
(b)
The pulses will pass through each other
without any change in their shape
(c)
The pulses will reflect, that is, the pulse
going towards right will move to left
after collision and vice-versa
(d)
The pulses will pass through each other
but their shapes will be modified.
7.
8.
4.
Two pulses in a stretched string whose centres are
initially 8 cm apart are moving towards each other
as shown in the figure. The speed of each pulse is 2
cm/s. After 2 seconds, the total energy of the pulses
will be
(a)
5.
(a)
A light pointer fixed to one prong of a tuning fork
touches a vertical plate. The fork is set vibrating
and the plate is allowed to fall freely. 8 complete
oscillations are counted when the plate falls through
10 cm. The frequency of the tuning fork is
(a)
28 Hz
(b)
56 Hz
(c)
70 Hz
(d)
80 Hz
A copper wire of Young’s Modulus Y, coefficient of
linear expansion  and densiry  is held at the two
ends by rigid supports. At initial temperature, the
wire is just taut, with negligible tension. The speed
of transverse waves in this wire if the change in
temperature is t is given by
(a)
Yt

(b)
2Yt

(c)
Yt
2
(d)
3Yt
2
zero
(b)
purely kinetic
(c)
purely potential
(d)
partly kinetic and partly potential
A hollow metallic tube of length L, closed at one
end, produces resonance with a tuning fork of
frequency N. The entire tube is carefully heated so
that in the equilibrium condition, its length changes
by l. If  is the linear coefficient of expansion of the
metal, the tube will resonate to a frequency of,
assuming the temperature change to be small and
initial temperature is 00C.
Einstein Classes,
9.
A piezo-electric quartz plate of thickness d is
vibrating in resonant conditions. The Young’s
Modulus of elasticity of quartz is Y and density is
 Its fundamental frequency
(a)
1
2d
Y

(b)
1
2d
2Y

(c)
1
2d
Y
2
(d)
1
2d
3Y
2
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10.
The figure is a composite of three shapshots, each
of a wave traveling along a particular string. The
phases for the waves are given by (a) 2x – 4t,
(b) 4x – 8t, and (c) 8x – 16t. Which phase
corresponds to which wave in the figure ?
16.
17.
11.
(a)
a-2, b-3, c-1
(b)
a-1, b-2, c-3
(c)
a-3, b-2, c-1
(d)
none of these
A sinusoidal wave travels along a string. The time
for a particular point to move from maximum
displacement to zero is 0.170 s. The wave length is
1.40 m. The wave speed is
(a)
(c)
12.
13.
14.
15.
2.06 m/s
3.00 m/s
(b)
(d)
3.50 m/s
(a)
0.90 y1
(b)
0.83 y1
(c)
0.69 y1
(d)
0.77 y1
Two sinusoidal waves with identical wavelengths
and amplitudes travel in opposite directions along
a string with a speed of 10 cm/s. If the time
interval between instants when the string is flat is
0.50 s, the wavelength of the waves is
1 cm
(b)
(c)
3 cm
(d)
18.
19.
2 cm
none of these
Two identical piano wires have a fundamental
frequency of 600 Hz when kept under the same
tension. The fractional increase in the tension of
one wire will lead to the occurrence of 6 beats/s when
both wires oscillate
(a)
0.02
(b)
0.03
(c)
0.04
(d)
0.05
A girl is sitting near the open window of a train
that is moving at a velocity of 10.00 m/s to the east.
The girl’s uncle stands near the tracks and watches
the train move away. The locomotive whistle emits
sound at frequency 500.0 Hz. The air is still. Let
the frequency heared by the uncle is f1. Now a wind
begins to blow from the east at 10.00 m/s. In this
situation the frequency heared by the uncle is f2.
Then the value of f2 – f1 is
(a)
0.4 Hz
(b)
0.3 Hz
(c)
0.2 Hz
(d)
0 Hz
Einstein Classes,
(a)
400 m/s
(b)
450 m/s
(c)
550 m/s
(d)
none of these

Consider TV = constant is the process in gas
(Molecular Weight = M) for the propagation of
sound wave. Velocity of sound wave at
temperature T is given by
(a)
(  1)RT
M
(b)
2(  1)RT
M
(c)
3(  1)RT
2M
(d)
(  1)RT
2M
1.06 m/s
Three sinusoidal waves of the same frequency travel
along a string in the positive direction of an x-axis.
Their amplitudes are y1, y1/2 and y1/3, and their
phase constants are 0, /2, and , respectively. The
amplitude of the resultant wave is
(a)
The velocity of sound in the gaseous mixture
consists of 1 mol of He gas and 2 moles of nitrogen
gas at the temperature of 270C is
20.
The pressure for 6 × 10–3 m3 of certain liquid is
decreased from 10 m of water to 9 m of water. Due
to the this the fractional increase in volume of the
liquid is 10–6. The density of the liquid is 540 kg/m3.
The velocity of sound in this liquid is
(a)
500 m/s
(b)
1000 m/s
(c)
1500 m/s
(d)
none
A uniform rope of length 12 m and mass 6 kg hangs
vertically from a rigid support. A block of mass
2 kg is attached to the free end of the rope. A
transverse pulse of wavelength 0.06 m is produced
at the lower end of the rope. The wavelength of the
pulse when it reaches the top of the rope is
(a)
0.12 m
(b)
0.06 m
(c)
0.18 m
(d)
0.03 m
A sonometer wire has a length of 114 cm between
two fixed ends. Two bridges be placed so as to
divide the wire into three segments whose
fundamental frequencies are in the ratio 1 : 3 : 4
then the minimum length of the the segment is
(a)
24 cm
(b)
21 cm
(c)
18 cm
(d)
15 cm
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21.
22.
23.
24.
25.
A sonometer wire fixed at one end has a solid mass
M hanging form its other end to produce tension in
it. It is found that 70 cm length of the wire
produces a certain fundamental frequency when
plucked. When the same mass M is immersed in
water, it is found that the length of the wire has to
be changed by 5 cm in order that it will produce
the same fundamental frequency. The density of the
material of mass M is
(a)
4.26 × 103 kg/m3
(b)
5.26 × 103 kg/m3
(c)
6.26 × 103 kg/m3
(d)
7.26 × 103 kg/m3
A pipe of length 1.5 m closed at one end is filled
with a diatomic gas and it resonates in its
fundamental mode with a tuning fork. Another pipe
of the same length but open at both ends is filled
with air and it also resonates in its fundamental
mode with the same tuning fork. Given that the
velocity of sound in air is 360 m/s at 300C where the
experiment is performed. The gas is
26.
27.
When a train is approaching the observer, the
frequency of the whistle is 100 cps while when it
has passed the observer, it is 50 cps. The frequency
when the observer moves with the train is
(a)
44.67 Hz
(b)
55.67 Hz
(c)
66.67 Hz
(d)
77.67 Hz
Two tuning forks with natural frequencies 340 Hz
each move relative to a stationary observer . One
fork moves away from the observer, while the other
moves towards him at the same speed. The observer
hears beats of frequency 3 Hz. If the velocity of
sound in air is 340 m/s then the speed of tuning fork
is
(a)
1.5 m/s
(b)
2 m/s
(c)
2.5 m/s
(d)
3 m/s
(a)
O2
(b)
N2
The maximum pressure amplitude pm that the
human ear can tolerate in loud sound is about
28 Pa (which is very much less than the normal air
pressure of about 10 5 Pa). The displacement
amplitude sm for such a sound in air of density
 = 1.21 kg/m3, at a frequency of 1000 Hz and a
speed of 343 m/s is
(c)
H2
(d)
none
(a)
1 cm
(b)
2 cm
(c)
3 cm
(d)
none
A ‘pop’ gun consists of a tube 25 cm long closed at
one end by a cork and at the other end by a tightly
fitted piston. The piston is pushed slowly in.When
the pressure rises to one and half times the
atmospheric pressure, the cork is violently blown
out. If the velocity of sound in air is 340 m/s then
the frequency of the ‘pop’ caused by its ejection is
(a)
410 Hz
(c)
610 Hz
(b)
510 Hz
(d)
710 Hz
28.
29.
The speed of sound in a certain metal is V. One end
of a long pipe of that metal of length L is struck a
hard blow. A listener at the other end hears two
sounds, one from the wave that travels along the
pipe and the other from the wave that travels
through the air. If v is the speed of sound in air,
time interval t elasped between the arrivals of the
two sounds is
0
A column of air at 51 C and a tuning fork produce
4 beats per sec when sounded together. As the
temperature of the air column is decreased, the
number of beats per sec tends to decrease and when
the temperature is 160C the two produce 1 beat per
sec. The frequency of the tuning-fork is
(a)
30 Hz
(b)
40 Hz
(c)
50 Hz
(d)
60 Hz
A string 25 cm long and having a mass of 2.5 gm is
under tension.A pipe closed at one end is 40 cm long.
When the string is set vibrating in its first overtone
and the air in the pipe in its fundamental frequency.
8 beats per sec are heard. It is observed that
decreasing the tension in the string decreases the
beat frequency. If the speed of sound in air is 320
m/s. The tension in the string is
(a)
25 N
(b)
27 N
(c)
29 N
(d)
31 N
Einstein Classes,
30.
(a)
Vv
L

 vV 
(b)
Vv
2L 

 vV 
(c)
Vv
L

 2 vV 
(d)
Vv
L

 2 vV 
A continuous sinusoidal longitudinal wave is sent
along a very long coiled spring from an oscillating
source attached on it. The frequency of the source
is 25 Hz, and at any time the distance between
sucessive points of maximum expansion in the
spring is 24 cm. The maximum longitudinal
displacement of a particle in the spring is 0.30 cm
and the wave moves in the negative direction of an
x-axis. Place x = 0 at the source and take the
displacement there is to be zero when t = 0. The
equation for the wave is
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
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31.
(a)
 

0.3 cos x  50t 
12


(b)
 

0.3 sin  x  50t 
 12

(c)
 

0.3 cos x  50t 
 24

(d)
 

0.3 sin x  50t 
 24

A sonometer wire under tension of 64 N vibrating
in its fundamental mode is in resonance with a
vibrating tuning fork. The vibrating portion of the
sonometer wire has a length of 10 cm and mass
1 gm. The vibrating tuning fork is now moved away
from the vibrating wire at a constant speed and an
observer standing near the sonometer hears one
beat per sec. The speed with which the tuning fork
is moved, if the speed of sound in air is 300 m/s, is
(a)
0.75 m/s
(b)
1 m/s
(c)
1.25 m/s
(d)
1.50 m/s
ANSWERS (SINGLE CORRECT CHOICE TYPE)
Einstein Classes,
1.
b
16.
d
2.
c
17.
a
3.
b
18.
d
4.
b
19.
a
5.
b
20.
c
6.
b
21.
d
7.
b
22.
d
8.
a
23.
b
9.
a
24.
c
10.
a
25.
b
11.
a
26.
c
12.
b
27.
a
13.
d
28.
d
14.
a
29.
a
15.
a
30.
b
31.
a
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
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EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
7.
The length L of the air column is
Comprehension-1
Two radio stations broadcast their programmes at
the same amplitude A. and at slightly different
frequencies 1 and 2 respectively, where
2 – 1 = 103 Hz. A detector receives the signals from
the two stations simultaneously. It can only detect
signals of intensity > 2A2
1.
2.
3.
4.
5.
6.
8.
The maximum intensity observed is
(a)
3A2
(b)
4A2
(c)
9A2
(d)
16A2
(a)
15
m
16
(b)
14
m
15
(c)
7
m
8
(d)
17
m
14
The amplitude of pressure variation at the middle
of the column is
(a)
P0
(b)
(c)
P0
2
(d)
P0
2
The frequency of resultant wave is
(a)
1000 Hz
(b)
500 Hz
(c)
2000 Hz
(d)
can’t be calculated as the data is
insufficient
9.
The frequency of intensity variation is
10.
(a)
1000 Hz
(b)
500 Hz
(c)
2000 Hz
(d)
can’t be calculated as the data is
insufficient
1000 Hz
(b)
500 Hz
(c)
2000 Hz
(d)
can’t be calculated as the data is
insufficient
(a)
P0
(b)
P0 – P0
(c)
P0 + P0
(d)
none
The minimum pressure at the closed-end of the pipe
is
(a)
P0
(b)
P0 – P0
(c)
P0 + P0
(d)
none
The period of a pulsating variable star may be
estimated by considering the star to be executing
radial longitudinal pulsations in the fundamental
standing wave mode; that is, the star’s radius
varies periodically with time, with a displacement
antinode at the star’s surface.
11.
The time-interval between successive maxima of the
intensity of the signal received by the detector is
–3
The maximum pressure at the open-end of the pipe
is
Comprehension-3
The frequency of amplitude variation is
(a)
3
P0
2
We can expect the center of the star to be
(a)
a displacement node
(b)
a displacement antinode
–3
(a)
10 s
(b)
0.5 × 10 s
(c)
may be a displacement or antinode
(c)
10 × 10–3s
(d)
20 × 10–3s
(d)
none
The time for which the detector remains idle in each
cycle of the intensities of the signal is
(a)
2 × 10–4 s
(b)
3 × 10–4 s
(c)
4 × 10–4 s
(d)
5 × 10–4 s
12.
By analogy with a pipe with one open end, the
period of pulsation T is given by
(a)
2R
v
(b)
4R
v
(c)
5R
v
(d)
6R
v
Comprehension-2
The air column in a pipe closed at one end is made
to vibrate in its second overtone by a tuning fork of
frequency 440 Hz. The speed of sound in air is
330 m s–1. End corrections may be neglected. Let P0
denote the mean pressure at any point in the pipe,
and P 0 the maximum amplitude of pressure
variation.
Einstein Classes,
where R is the equilibrium radius of the star and v
is the average sound speed in the material of the
star.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
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13.
Temperature of the air and water = 200C; Density
of river water = 103 kg/m3, Bulk modulus of the
water = 2.088 × 10 9 Pa; Gas constant
R = 8.31 J/mol-K; Mean molar mass of
air = 28.8 × 10–3kg/mol; adiabatic exponent of air is
1.4.
Typical white dwarf stars are composed of
material with a bulk modulus of 1.33 × 1022 Pa and
a density of 1010 kg/m3. They have radii equal to
9.0 × 10–3 solar radius. The approximate pulsation
period of a white dwarf is
(a)
11s
(b)
22s
(c)
33s
(d)
44s
18.
Comprehension-4
A long wire PQR is made by joining two wires PQ
and QR of equal radii. PQ has length 4.8 m and
mass 0.06 kg. QR has length 2.56 m and mass
0.2 kg. The wire PQR is under a tension of 80 N. A
sinusodial wave of amplitude 3.5 cm is sent along
the wire PQ from the end P. No power is dissipated
during the propagation of the wave.
The incident wave and reflected wave in the wire
PQ will superimposed and gives a standing wave
pattern of waves whose envelop is shown in figure
19.
20.
21.
15.
16.
17.
(a)
100000 Hz
(b)
100696 Hz
(c)
103039 Hz
(d)
none
The frequency detected by the stationary observer
in the river, whom the boat is approaching, is
(a)
100000 Hz
(b)
100696 Hz
(c)
103039 Hz
(d)
none
Now the transmitter is pulled up into the air. The
wavelength of sound in air is
(a)
14.45 mm
(b)
7.22 mm
(c)
3.44 mm
(d)
1.72 mm
The air is blowing with a speed of 5 m/s in the
direction opposite to the river stream. The
frequency detected by an stationary observer
outside the water whom the boat is approaching
Standing wave ratio,  = Amax/Amin
(a)
100000 Hz
(b)
100696 Hz
% reflection =
(c)
103039 Hz
(d)
none
MATRIX-MATCH TYPE
Average power in the reflected wave
 100
Average power in the incident wave
14.
The frequency detected by the observer placed in
the boat is
Matching-1
The equation for the displacement of a stretched
string is given by
The time taken by the wave to reach the other end
R of the wire is
(a)
0.07 s
(b)
0.14 s
(c)
0.21 s
(d)
0.28 s
x 
 t
y  4 sin 2


 0.02 100 
where y and x are in cm and t in sec.
The amplitude of the reflected wave after the
incident wave pulse cross the joint Q is
(a)
1.5 cm
(b)
2 cm
(c)
2.5 cm
(d)
3 cm
Column - A
Column - B
(A)
frequency in Hz
(p)
50
(B)
maximum particle speed (q)
in m/sec
4
(C)
maximum particle
acceleration in m/s2
(r)
100
(D)
wave speed in m/sec
(s)
4002
The value of  is
(a)
1.5
(b)
2
(c)
2.5
(d)
3
The percentage reflection at the point Q is
approximately
(a)
43%
(b)
50%
(c)
55%
(d)
67%
Comprehension-5
Matching-2
The vibration of a string of length 60 cm fixed at
both ends are represented by the equation
 x 
y  4 sin  cos(96t )
A boat is travelling in a river with a speed of
 15 
10 m/s along the stream flowing with a speed of
where x and y are in cm and t in sec.
2 m/s. From this boat, a sound transmitter is
lowered into the river through a rigid support. The
Column - A
Column - B
wavelength of the sound emitted from the
(A)
The maximum
(p)
zero
transmitter inside the water is 14.45 mm. Assume
displacement at x = 5 cm
that attenuation of sound in water and air is
in cm
negligible.
Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 21
(B)
String is vibrating in the
overtone
(q)
23 cm
(C)
Velocity of the particle
at x = 7.5 cm and
t = 0.25 s is in cm/s
(r)
3
(D)
The time in second when (s)
the string will become
straight for the first time
1/192
MULTIPLE CORRECT CHOICE TYPE
1.
2.
3.
4.
5.
6.
A wire of mass 9.8 × 10–3 kg per metre passes over a
frictionless pulley fixed on the top of an inclined
frictionless plane which makes an angle of 300 with
the horizontal. Masses M1 and M2 are tied at the
two ends of the wire. The mass M1 rests on the plane
and the mass M2 hangs vertically downwards. The
whole system is in equilibrium. Now a transverse
wave propagates along the wire with a velocity of
100 m/s. Then
(a)
M2 = 10 kg
(b)
M1 = 15 kg
(c)
M1 = 20 kg
(d)
M2 = 30 kg
7.
8.
If two sound waves, y1 = 0.3 sin 596[t – x/330] and
y2 = 0.5 sin 604[t – x/330] are superimposed.
A string that is stretched between fixed supports
separated by 75.0 cm has resonant frequencies of
420 and 315 Hz, with no intermediate resonant
frequencies.
(a)
the lowest resonant frequency is 105 Hz
(b)
the wave speed is 158 m/s
(c)
the lowest resonant frequency is 158 Hz
(d)
the wave speed is 105 m/s
A transverse sinusoidal wave of amplitude a, wavelength  and frequency f is traveling on a stretched
string. The maximum speed of any point on the
string is v/10, where v is the speed of propagation
of the wave. If a = 10–3 m and v = 10 ms–1, then 
and f are given by
(a)
 = 2 × 10–2 m
(b)
 = 10–3 m
(c)
f = 103/(2) Hz
(d)
f = 103/() Hz
Standing waves can be produced
(a)
on a string clamped at both the ends
(b)
on a string clamped at one end and free
at the other
(a)
the frequency of resultant wave is 300 Hz
(b)
the frequency at which the amplitude of
resultant waves varies is 2Hz
(c)
when incident wave gets reflected from a
wall
(c)
the frequency at which beats are
produced 4Hz
(d)
in the organ pipe
(d)
the ratio of maximum and minimum
intensities of beats is 16
A turning fork having frequency of 340 Hz is
vibrated just above a cylindrical tube. The height
of the tube is 120 cm. Water is slowly poured in. If
velocity of sound in air is 340 m/s then the
different height of water required for resonance is
(a)
45 cm
(b)
95 cm
(c)
75 cm
(d)
50 cm
9.
10.
Which one of the following does not represent a
travelling wave ?
(a)
tan (kx – t)
(b)
(c)
log (x + vt)
(d)
kx  t
e ( x  vt )
2
A transverse wave is travelling along a string from
left to right. Figure represents the shape of the string
(snap shot) at a given instant. Choose the correct
statement from the following
The first overtone of an open organ pipe of length
L1 beats with the first overtone of a closed organ
pipe of length L2 with a beat frequency of 2.2 Hz.
The fundamental frequency of the closed organ pipe
is 110 Hz. Then
(a)
L1 = 1.0067 m
(b)
L2 = 0.9937 m
(c)
L1 = 0.9937 m
(d)
L2 = 0.75 m
A source of sound of frequency 256 Hz is moving
rapidly towards a wall with a velocity of 5 m/s. The
number of beats per sec will be heard if sound
travels at a speed of 330 m/s
(a)
0 Hz
(b)
7.7 Hz
(c)
12.2 Hz
(d)
2.5 Hz
Einstein Classes,
(a)
Points at D, E & F have an upward
velocity.
(b)
Points at C & G have zero velocity.
(c)
Points at A & E have maximum
magnitude of velocity.
(d)
All the above
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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11.
The figure indicates the direction of motion of a
sound source and a detector of six situations in
stationary air :
Source
(a)

Detector
 0 speed
(b)

 0 speed
(c)


(d)


(e)


(f)


16.
Choose the correct statement from the
following :
12.
13.
14.
15.
(a)
The minimum frequency detected by the
detector is in case f.
(b)
The frequency detected by the detector
in case c may be equal to actual
frequency.
(c)
The maximum frequency detected by the
detector is in case e.
(d)
All are correct
17.
2
(b)
the wavelength of the resulting sound in
air is 1 m
(c)
the velocity of propagation of transverse
waves along the wire is 330 m/s
(d)
the wavelength of the resulting sound in
air is 2 m
A wire of density 9 × 10 3 kg/m3 and young’s
modulus of elasticity 9 × 10 10 N/m2 is stretched
between two clamps 1 m apart and is subjected to
an extension of 4.9 × 10–4m. The possible frequency
of transverse vibrations in the wire is
(a)
35 Hz
(b)
70 Hz
(c)
45 Hz
(d)
90 Hz
An under-water swimmer sends a sound signal to
the surface. The velocity of sound in water is
1500 m/s and in air is 360 m/s. If it produces 5 beats
per second when compared with the fundamental
tone of a pipe of 20 cm length closed at one end, the
wavelength of sound in water is
(a)
3.30 m
(b)
3.33 m
(c)
3.35 m
(d)
3.37 m
The equation y = A sin (t – kx) represents a wave
Assertion-Reason Type
(a)
with amplitude A and frequency (/2)
(b)
with amplitude (A/2) and frequency
(/)
(c)
which is harmonic-progressive
Each question contains STATEMENT-1 (Assertion)
and STATEMENT-2 (Reason). Each question has
4 choices (A), (B), (C) and (D) out of which ONLY
ONE is correct.
(d)
complex wave
The (x, y) coordinates of the corners of a square
plate are (0, 0), (L, 0), (L, L) and (0, L). The edges
of the plate are clamped and transverse standing
waves are set up in it. If u(x, y) denotes the
displacement of the plate at the point (x, y) at some
instant of time, the possible expression(s) for u is
(are) (a = positive constant)
(a)
a cos (x/2L) cos (y/2L)
(b)
a sin (x/L) sin (y/L)
(c)
a sin (x/L) sin (2y/L)
(d)
none of these
y(x, t) = 0.8/[(4x + 5t)2 + 5] represents a moving
pulse, where x and y are in metre and t in second.
Then
(a)
in 2 s it will travel a distance of 2.5 m
(b)
its maximum displacement is 0.16 m
(c)
it is a symmetric pulse
(d)
all are correct
A sonometer wire 100 cm in length has a
fundamental frequency of 330 Hz. The velocity of
sound in air is 330 m/s.
(a)
the velocity of propagation of transverse
waves along the wire is 660 m/s
Einstein Classes,
1.
(A)
Statement-1 is True, Statement-2 is True;
Statement-2 is a correct explanation
for Statement-1
(B)
Statement-1 is True, Statement-2 is True;
Statement-2 is NOT a correct
explanation for Statement-1
(C)
Statement-1 is True, Statement-2 is False
(D)
Statement-1 is False, Statement-2 is True
STATEMENT-1 : Doppler’s effect of sound will be
observed at moon.
STATEMENT-2 : Doppler’s effect of light will be
observed at moon.
2.
STATEMENT-1 : Sound wave is always a
longitudinal wave.
STATEMENT-2 : Longitudinal waves cannot be
polarised.
3.
STATEMENT-1 : A plane wave of sound travelling
in air is incident upon a phane water surface. The
angle of incidence is 600. Assuming Snell’s law to
be valid for sound waves, it follows that the sound
wave will be refracted into water away from the
normal.
STATEMENT-2 : Velocity of sound in water is
greater than the velocity of sound in air
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New Delhi – 110 018, Ph. : 9312629035, 8527112111
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4.
STATEMENT-1 : The graph for the variation of
velocity of sound with pressure at constant
temperature is parabolic.
STATEMENT-2 : Velocity of sound equals to
9.
STATEMENT-1 : The quality of sound from an
open pipe is different than from a closed pipe of
the same fundamental frequency.
STATEMENT-2 : Quality of sound depends on
number of overtones present.
P

10.
STATEMENT-1 : Resonance column method is
used to determine the speed of sound in air.
STATEMENT-1 : An organ pipe is in resonance
with a tuning fork of frequency ‘f ’ . If the
temperature of the gas will change then we must
observe the phenomena of beats.
STATEMENT-2 : In this method, a vibrated
tuning fork is held near the open end of the tube in
such a way that the prongs vibrate parallel to the
length of the tube.
STATEMENT-2 : Frequency of organ pipe will be
independent of temperature variations if the
coefficient of linear expansion of its material is
(1/546)/0C.
where the symbols have their usual meanings.
5.
6.
11.
STATEMENT-1 : Reflection of a displacement wave
from a denser medium or rigid support or fixed end
or closed end, there is inversion of reflected
displacement wave i.e., there is a phase change of
.
STATEMENT-2 : Reflection of longitudinal
pressure waves, it suffers a phase change of  from
a free or open end and no change in phase from
rigid boundaries.
7.
STATEMENT-1 : A sonometer wire is in resonance
with a tuning fork of frequency ‘f ’ . If the
temperature of the wire will change by t then we
can observe the phenomena of beats with beat
frequency ½ft where  is the coefficient of
linear expansion of the wire.
STATEMENT-1 : A string of length L fixed at both
ends vibrates in its fundamental mode at a
frequency f and a maximum amplitude A. Take the
origin at one end of the string and the X-axis along
the string. Take the Y-axis along the direction of
the displacement. Take t = 0 at the instant when the
middle point of the string passes through its mean
position and is going towards the positive
y-direction. The equation describing the standing
wave is Asin(x/L)sin(2ft)
STATEMENT-2 : In a standing wave, all the
particles between two successive notes reach their
extreme positions together, thus moving in phase.
In a travelling wave, the phases of nearby particles
are always different.
STATEMENT-2 : Phenomena of beats of sound
wave is independent of temperature.
8.
STATEMENT-1 : The frequency of organ pipe
depends on the radius of the pipe.
STATEMENT-2 : It is due to end correction.
(Answers) EXCERCISE BASED ON NEW PATTERN
COMPREHENSION TYPE
1.
b
2.
d
3.
a
4.
b
5.
a
6.
d
7.
a
8.
b
9.
a
10.
b
11.
a
12.
b
13.
b
14.
b
15.
a
16.
c
17.
a
18.
a
19.
b
20.
c
21.
c
2.
[A-q; B-r; C-p; D-s]
a, c, d
5.
a, b
9.
a, b, c
13.
b, c
14.
a, b, c, d
6.
MATRIX-MATCH TYPE
1.
[A-p; B-q; C-s; D-p]
MULTIPLE CORRECT CHOICE TYPE
1.
a, c
2.
a, b, c, d
3.
a, b
6.
a, b
7.
a, c
8.
a, b, c, d
10.
a, b, c, d 11.
a, c
12.
b, c
15.
a, b
a, b
17.
a, d
16.
4.
ASSERTION-REASON TYPE
1.
D
2.
D
3.
A
4.
D
5.
B
7.
C
8.
A
9.
A
10.
D
11.
B
Einstein Classes,
B
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 24
INITIAL STEP EXERCISE
(SUBJECTIVE)
1.
A wire of uniform cross-section is stretched between
two points 1 m apart. The wire is fixed at one end
and a weight of 9 kg is hung over a pulley at the
other hand produces fundamental frequency of
750 Hz. (a) What is the velocity of transverse waves
propagating in the wire ? (b) If now the suspended
weight is submerged in a liquid of density (5/9) that
of the weight, what will be the velocity and
frequency of the wave propagating along the wire.
6.
An aluminium wire of cross-sectional area
1 × 10–6 m2 is joined to a copper wire of the same
cross-section. This compound wire is stretched on
a sonometer, pulled by a weight of 10 kg. The total
length of the compound wire between the two
bridges is 1.5 m of which the aluminium wire is
0.6 m and the rest is the copper wire. Transverse
vibrations are set up in the wire by using an
external force of variable frequency. Find the
lowest frequency of excitation for which standing
waves are formed such that the joint in the wire is
a node. What is the total number of nodes observed
at this frequency excluding the two at the ends of
the wire ? The density of aluminium is
2.6 × 103 kg/m3 and that of copper 1.0401 × 104 kg/
m3.
2.
A rifle shot is fired in a valley formed between two
parallel mountains. The echo from one mountain
is heard after 2 s and the echo from the other
mountain is heard 2 s later. If velocity of sound is
360 m/s (a) What is the width of the valley ? (b) Is it
possible to hear the subsequent echoes from the two
mountains simultaneously at the same point ? If
so, after what time ?
3.
A whistle of frequency 540 Hz rotates in a circle of
radius 2 m at an angular speed of 15 rad/sec. What
is the lowest and highest frequency heard by a
listener, a long distance away at rest with respect to
the centre of the circle (v = 330 m/s). Can the
apparent frequency be ever equal to actual ?
4.
A transverse sinusoidal wave of wavelength 20 cm
is moving along a string in the positive x-direction.
The transverse displacement of the string particle
at x = 0 as a function of time is shown in fig. Make
a rough sketch of one wavelength of the wave
(the portion between x = 0 and x = 20 cm)at time
t = 0. (b) What is the speed of the wave ? (c) Write
the equation for the wave with all the constants
evaluated. (d) What is the transverse velocity of the
particle at x = 0 at t = 5.0 s ?
7.
A metal wire of diameter 1 mm is held on two knife
edges separates by a distance of 50 cm. The tension
in the wire is 100 N. The wire vibrating with its
fundamental frequency and a vibrating tuning fork
together produce 5 beats per sec. The tension in the
wire is then reduced to 81 N. When the two are
excited, beats are heard again at the same rate.
Calculate (a) the frequency of the fork and (b) the
density of the material of the wire.
8.
AB is a cylinder of length 1 m fitted with a thin
flexible diaphragm C at middle and two other thin
flexible diaphragms A and B at the ends. The
portions AC and BC contain hydrogen and oxygen
gases respectively. The diaphragms A and B are set
into vibrations of the same frequency. What is the
minimum frequency of these vibrations for which
diaphragm C is a node ? Under the condition of the
experiment the velocity of sound in hydrogen is 1100
m/s and oxygen 300 m/s.
In fig., two pulses travel along a string in
opposite directions. The wave speed v is 2.0m/s and
the pulses are 6.0 cm apart at t = 0. (a) Sketch the
wave patterns when t is equal to 5.0, 10, 15, 20, and
25 ms. (b) In what form (or type) is the energy of
the pulses at t = 15 ms ?
9.
A source of sound is moving along a circular orbit
of radius 3 m with an angular velocity of 10 rad/s.
A sound detector located far away from the source
is executing linear simple harmonic motion along
the line BD with amplitude BC = CD = 6 m. The
frequency of oscillation of the detector is (5/) per
sec. The source is at the point A when the detector
is at the point B. If the source emits a continuous
sound wave of frequency 340 Hz, find the
maximum and the minimum frequencies recorded
by the detector [velocity of sound = 330 m/s.].
5.
Einstein Classes,
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 25
10.
A band playing music at a frequency f is moving
towards a wall at a speed v B . A motorist is
following the band with a speed vm. If v is the speed
of sound, obtain an expression for the beat
frequency heard by the motorist.
11.
The type of rubber band used inside some
baseballs and golf balls obeys Hooke’s law over a
wide range of elongation of the band. A segment of
this material has an unstretched length l and a mass
m. When a force F is supplied, the band stretches
an additional length l. (a) What is the speed
(in terms of m, l, and the spring constant k) of
transverse waves on this stretched rubber band ?
(b) Using your answer to (a), show that the time
required for a transverse pulse to travel the length
of the rubber band is proportional to
12.
A source of sound emitting a 1200 Hz note travels
along a straight line at a speed of 170 m/s. A
detector is placed at a distance of 200 m from the
line of motion of the source. (a) Find the frequency
of sound received by the detector at the instant when
the source gets closest to it. (b) Find the distance
between the source and the detector at the instant
it detects the frequency 1200 Hz. Velocity of sound
in air = 340 m/s.
1 / l if
l << l and is constant if l >> l.
FINAL STEP EXERCISE
(SUBJECTIVE)
1.
There are three sources of sound of equal
intensities with frequenct 400, 401 and 402 Hz.
What is the beat frequency heard if all are sounded
simultaneously ?
2.
A metallic rod of length 1 m is rigidly clamped at
its mid point. Longitudinal stationary waves are set
up in the rod in such a way that there are two nodes
on either side of the mid-point. The amplitude of
an antinode is 2 × 10–6 m. Write the equation of
motion at a point 2 cm from the mid-point and those
of constituent waves in the rod. (Y = 2 × 1011 N/m2
and  = 8 × 103 kg/m3)
3.
Show that the maximum kinetic energy in each loop
of a standing wave produced by two traveling waves
of identical amplitudes is 22µy2mfv.
4.
A source of sonic oscillations with frequency
f0 = 1700 Hz and a receiver are located at the same
point. At the moment t = 0 the source starts
receding from the receiver with constant
acceleration a = 10.0 m/s2. Assuming the velocity of
sound to be equal to v = 340 m/s, find the
oscillation frequency registered by the stationary
receiver t = 10.0 second after the start of motion.
5.
The displacement of the medium in a sound wave
is given by the equation
y = A cos (ax + bt)
where A, a and b are positive constants. The wave
is reflected by an obstacle situated at x = 0. The
intensity of the reflected wave is 0.64 times that of
the incident wave (a) What is the wavelength and
frequency of the incident wave ? (b) Write the
equation for the reflected wave. (c) In the resultant
wave formed after reflection, find the maximum
Einstein Classes,
and minimum values of the particle speed in the
medium. (d) Express the resultant waves as a
superposition of a standing wave and a travelling
wave. What are the positions of the antinodes of
the standing wave ? What is the direction of
propagation of the travelling wave ?
6.
A uniform horizontal rod of length 40 cm and mass
M is supported by two idential wires as shown in
figure. A point mass of 4.8 kg be placed on the rod
at the distance 5 cm from the left end such that the
same tuning fork may excites the wire on left into
its fundamental vibrations and that on right into
its first overtone. Find the value of M ?
7.
The following equations represent transverse
waves :
z1 = a cos (kx – t)
z2 = a cos (kx + t)
z3 = a cos (ky – t)
Identify the combinations of the waves which will
produce (i) standing wave (ii) a wave going in the
direction making an angle of 450 with the positive x
and positive y axis. In each case, find the positions
at which the resultant intensity is always zero.
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111
PW – 26
8.
9.
A small sphere of radius R is arranged to pulsate
so that its radius varies is simple harmonic motion
between a minimum of R – R and a maximum of
R + R with frequency f. This produces sound
waves in the surrounding air of density  and bulk
modulus B.
(a)
Find the intensity of sound waves at the
surface of the sphere (the amplitude of
oscillation is same as that of air at the
surface of the sphere)
(b)
Find the total acoustic power radiated by
the sphere.
(c)
At a distance d >> R from the centre of
the sphere find the amplitude, pressure
amplitude and intensity of sound wave.
(0, –100 m) starts moving with velocity 5m/s î and
10m/s ĵ respectively. Find the frequency received
by the observer when the source crosses the origin.
10.
A 2 m string is fixed at one end ans is vibrating in
its third harmonic with amplitude 3 cm and
frequency 100 Hz. (a) At what time is its kinetic
energy maximum ? What is the shape of the string
at this time ? (b) Find the maximum kinetic energy
of the string ?
11.
A string 120 cm in length sustains a standing wave
with the points of the string at which the
displacement amplitude is equal to 3.5 mm being
separated by 15 cm. Find the maximum
displacement amplitude. To which overtone do these
oscillations correspond ?
At t = 0, a source of sound (frequency = 100 Hz)
and observer are situated at (–100 m, 0) and
ANSWERS SUBJECTIVE (INITIAL STEP EXERCISE)
1.
1500 m/s, 500 Hz
2.
(a)
3.
495 Hz, 594 Hz
6.
161.8 Hz, 3
7.
(a)
8.
1650 Hz
9.
442 Hz, 255 Hz
10.
 vv 
f  2 m2  2v B
 v  vB 
12.
(a)
1080 m (b)
6s
95 Hz
1600 Hz
(b)
12.74 × 103 kg/m3
(b)
224 m
ANSWERS SUBJECTIVE (FINAL STEP EXERCISE)
1.
2Hz
2.
10–6 sin [25000t + 5x], 10–6 sin [2500t – 5x]
4.
1350 Hz
5.
(a)
(i)
(ii)
 3 5 
,
,
.... , negative x-axis
2a 2a 2a
z1 and z2
z1 and z3
6.
1.2 kg
8.
(a)
I  2  2 B f 2 (  R ) 2
(b)
P  8  3 B f 2 R 2 (  R ) 2
(c)
A  ( R / d )R , Pmax  2 B ( Rf / d )R , I  2 2 B (fR / d ) 2 ( R ) 2
9.
11.
7.
(b)
(c)
(d)
b 2
,
2 a
– 0.8A cos (bt – ax)
0, 1.8bA
–1.6 A sin ax sin bt + 0.2A cos (bt + ax)
97 Hz
5 mm, 3
Einstein Classes,
10.
(a)
2.5 × 10–3 s, straight
(b)
89 m J
Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road
New Delhi – 110 018, Ph. : 9312629035, 8527112111