STRING WAVE

PART - I : SUBJECTIVE QUESTIONS

SECTION (A) : EQUATION OF TRAVELLING WAVE (INCLUDING SINE WAVE)
A 1. Consider the wave y = (5 mm) sin (1 cm –1) x – (60 s–1) t]. Find (a) the amplitude (b) the wave number, (c) the
wavelength, (d) the frequency, (e) the time period and (f) the wave velocity.

A 2. The string shown in figure is driven at a frequency of 5.00 Hz.

The amplitude of the motion is 12.0 cm, and the wave speed is 20.0 m/s. y
Furthermore, the wave is such that y = 0 at x = 0 and t = 0. Determine
(a) the angular frequency and (b) wave number for this wave. (c) Write an
x
expression for the wave function. Calculate (d) the maximum transverse
speed and (e) the maximum transverse acceleration of a
point on the string. x=0

A 3. The sketch in the figure shows displacement time curve of a sinusoidal wave at x = 8 m.Taking velocity of
wave v = 6m/s along postive x-axis, write the equation of the wave.

2 8 14
t (in sec)
0
½m

A 4. A transverse wave is travelling along a string from left to right. The fig.
represents the shape of the string (snap-shot) at a given instant. At this
instant (a) which points have an upward velocity (b) which points will have
downward velocity (c) which points have zero velocity (d) which points have
maximum magnitude of velocity.

SECTION (B) : SPEED OF A WAVE ON A STRING

B 1. A piano string having a mass per unit length equal to 5.00 × 10–3 kg/m is under a tension of 1350 N. Find the
speed with which a wave travels on this string.

B 2. In the arrangement shown in figure, the string has mass of 4.5 g. How
much time will it take for a transverse disturbance produced at the floor to
reach the pulley ? Take g = 10 m/s2.

B 3. 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. What is the wavelength of the pulse when it reaches the top of the rope?

B 4. A particle on a stretched string supporting a travelling wave, takes 5.0 ms to move from its mean position to
the extreme position. The distance between two consecutive particles, which are at their mean positions, is
2.0 cm. Find the frequency, the wavelength and the wave speed.

B 5. Two wires of different densities but same area of cross-section are soldered together at one end and are
stretched to a tension T. The velocity of a transverse wave in the first wire is double of that in the second wire.
Find the ratio of the density of the first wire to that of the second wire.
PSC SIR
 STRING WAVE
B 6. A 4.0 kg block is suspended from the ceiling of an elevator through a string having a linear mass density of
19.2 × 10–3 kg/m. Find the speed (with respect to the string) with which a wave pulse can proceed on the
string if the elevator accelerates up at the rate of 2.0 m/s2. Take g = 10 m/s2.

SECTION (C) : POWER TRANSMITTED ALONG THE STRING

C 1. A 6.00 m segment of a long string has a mass of 180 g. A high-speed photograph shows that the segment
contains four complete cycles of a wave. The string is vibrating sinusoidally with a frequency of 50.0 Hz and
a peak-to-valley displacement of 15.0 cm. (The “peak-to-valley” displacement is the vertical distance from the
farthest positive displacement to the farthest negative displacement.) (a) Write the function that describes
this wave traveling in the positive x direction. (b) Determine the power being supplied to the string.

C 2. A transverse wave of amplitude 0.50 mm and frequency 100 Hz is produced on a wire stretched to a tension
of 100 N. If the wave speed is 100 m/s, what average power is the source transmitting to the wire?

C 3. A tuning fork of frequency 440 Hz is attached to a long string of linear mass density 0.01 kg/m kept under a
tension of 49 N. The fork produces transverse waves of amplitude 0.50 mm on the string. (a) Find the wave
speed and the wavelength of the waves. (b) At what average rate is the tuning fork transmitting energy to the
string?

SECTION (D) : INTERFERENCE, REFLECTION, TRANSMISSION

2
D 1. The equation of a plane wave travelling along positive direction of x-axis is y = asin
(t – x) When this

wave is reflected at a rigid surface and its amplitude becomes 80%, then find the equation of the
reflected wave

D 2. A series of pulses, each of amplitude 0.150 m, are sent on a string that is attached to a post at one end. The
pulses are reflected at the post and travel back along the string without loss of amplitude. When two waves
are present on the same string. The net displacement of a give point is the sum of the displacements of the
individual waves at the point. What is the net displacement at point on the string where two pulses are
crossing, (a) if the string is rigidly attached to the post ? (b) If the end at which reflection occurs is free to
slide up and down?

D 3. Two waves, each having a frequency of 100 Hz and a wavelength of 2.0 cm, are travelling in the same
direction on a string. What is the phase difference between the waves (a) if the second wave was produced
0.015 s later than the first one at the same place, (b) if the two waves were produced a distance 4.0 cm
behind the second one? (c) If each of the waves has an amplitude of 2.0 mm, what would be the amplitudes
of the resultant waves in part (a) and (b) ?

SECTION (E) : STANDING WAVES AND RESONANCE

E 1. What are (a) the lowest frequency, (b) the second lowest frequency, and (c) the third lowest frequency for
standing waves on a wire that is 10.0 m long has a mass of 100 g. and is stretched under a tension of 250 N
which is fixed at both ends ?

E 2. A nylon guitar string has a linear density of 7.20 g/m and is under a tension
of 150 N. The fixed supports are distance D = 90.0 cm apart. The string is D
oscillating in the standing wave pattern shown in figure. Calculate the
(a) speed. (b) wavelength, and (c) frequency of the traveling waves whose
superposition gives this standing wave.

E 3. The length of the wire shown in figure between the pulleys is 1.5 m and its
mass is 12.0 g. Find the frequency of vibration with which the wire vibrates in
two loops leaving the middle point of the wire between the pulleys at rest.

PSC SIR
 STRING WAVE
E 4. A string oscillates according to the equation

  1  
y’ = (0.50 cm) sin  cm  x  cos [(40  s–1)t].
 3  
What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose
superposition gives this oscillation? (c) what is the distance between nodes? (d) What is the transverse
9
speed of a particle of the string at the position x = 1.5 cm when t = 8
s?

E 5. A string vibrates in 4 loops with a frequency of 400 Hz .

(a) What is its fundamental frequency ?
(b) What frequency will cause it to vibrate into 7 loops .

E 6. The vibration of a string of length 60 cm is represented by the equation,

y = 3 cos (x/20) cos (72t) where x & y are in cm and t in sec.
(i) Write down the component waves whose superposition gives the above wave.
(ii) Where are the nodes and antinodes located along the string.
(iii) What is the velocity of the particle of the string at the position x = 5 cm & t = 0.25 sec.

PART - II : OBJECTIVE QUESTIONS

SECTION (A) : EQUATION OF TRAVELLING WAVE (INCLUDING SINE WAVE)
A 1. For the wave shown in figure, the equation for the wave, travelling along +x axis with velocity 350 ms–
1
when its position at t = 0 is as shown

314
(A) 0.05 sin ( x – 27500 t)
4

379
(B) 0.05 sin ( x – 27000 t)
5

314
(C) 1 sin ( x – 27500 t)
4

289
(D) 0.05 sin ( x + 25700 t)
5

A 2. The displacement of a wave disturbance propagating in the positive x-direction is given by y = 1/(1 + x 2)
at time t = 0 and y = 1/[1 + (x – 1) 2] at t = 2 seconds where x and y are in metres. The shape of the wave
disturbance does not change during the propagation. The velocity of the wave is [JEE - 90]
(A) 2.5 m/s (B) 0.25 m/s (C) 0.5 m/s (D) 5 m/s

A 3. A transverse wave is described by the equation Y = Y0 sin 2 (ft – x/). The maximum particle velocity
is equal to four times the wave velocity if [JEE - 84]
(A)  =  Y0/4 (B)  =  Y0/2 (C)  =  Y0 (D)  = 2 Y0


A 4. A travelling wave on a string is given by y = A sin [x +  t + ]. The displacement and velocity of
6
oscillation of a point  = 0.56 /cm,  = 12/sec, A = 7.5 cm, x = 1 cm and t = 1s is
(A) 4.6 cm, 46.5 cm s –1 (B) 3.75 cm, 77.94 cm s–1
(C) 1.76 cm, 7.5 cm s –1 (D) 7.5 cm, 75 cm s –1

A 5. A transverse wave of amplitude 0.50m, wavelength 1m and frequency 2 hertz is propagating in a string in the
negative x-direction. The expression form of the wave is [REE - 89]
(A) y(x, t) = 0.5 sin (2x – 4t) (B) y(x, t) = 0.5 cos (2x + 4t)
(C) y(x, t) = 0.5 sin (x – 2t) (D) y(x, t) = 0.5 cos (2x – 2t)

PSC SIR
 STRING WAVE
A 6. Both the strings, show in figure, are made of same material and have same
cross-section. The pulleys are light. The wave speed of a transverse wave in the
string AB is v1 and in CD it is v2. The v1/v2 is

(A) 1 (B) 2 (C) 2 (D) 1/ 2

A 7. Two blocks each having a mass of 3.2 kg are connected by a wire CD and the
system is suspended from the ceiling by another wire AB (figure). The linear mass
density of the wire AB is 10 g/m and that of CD is 8 g/m. The speed of a transverse
wave pulse produced in AB and in CD are :
(A) 79 m/s and 63 m/s (B) 63 m/s and 79 m/s
(C) 63 m/s in both (D) 79 m/s in both

A 8. Two stretched wires A and B of the same lengths vibrate independently. If the radius, density and tension of
wire A are respectively twice those of wire B, then the fundamental frequency of vibration of A relative to that
of B is [REE - 90]
(A) 1 : 1 (B) 1 : 2 (C) 1 : 4 (D) 1 : 8

A 9. Three consecutive flash photographs of a travelling wave on a string are reproduced in the figure here.
The following observations are made. Mark the one which is correct. (Mass per unit length of the string
= 3 g/cm.)

(A) displacement amplitude of the wave is 0.25 m, wavelength is 1 m, wave speed is 2.5 m/s and
the frequency of the driving force is 0.2/s.
(B) displacement amplitude of the wave is 2.0 m, wavelength is 2 m, wave speed is 0.4 m/s and the
frequency of the driving force is 0.7/s.
(C) displacement amplitude of the wave is 0.25 m, wavelength is 2 m, wave speed is 5 m/s and the
frequency of the driving force is 2.5 /s.
(D) displacement amplitude of the wave is 0.5 m, wavelength is 2 m, wave speed is 2.5 m/s and the
frequency of the driving force is 0.2/s.

A 10. A heavy ball is suspended from the ceiling of a motor car through a light string. A transverse pulse travels at
a speed of 60 cm/s on the string when the car is at rest and 62 cm/s when the car accelerates on a horizontal
road. Tthen acceleration of the car is : (Take g = 10 m/s2.)
(A) 2.7 m/s2 (B) 3.7 m/s2 (C) 2.4 m/s2 (D) 1.4 m/s2

A 11. A steel wire of mass 4.0 g and length 80 cm is fixed at the two ends. The tension in the wire is 50 N. The
wavelength of the fourth harmonic of the fundamental will be :
(A) 80 cm (B) 60 cm (C) 40 cm (D) 20 cm

A 12. A copper wire is held at the two ends by rigid supports. At 30ºC the wire is just taut, with negligible
tension. The speed of transverse waves in this wire at 10ºC is :
( = 1.7 × 10–5 / ºC, Y = 1.3 × 1011 N/m 2, d = 9 × 103 kg/m 3 ). [JEE - 79]
(A) 80 m/sec (B) 90 m/sec (C) 100 m/sec (D) 70 m/sec

A 13. Two small boats are 10m apart on a lake. Each pops up and down with a period of 4.0 seconds due to
wave motion on the surface of water. When one boat is at its highest point, the other boat is at its
lowest point. Both boats are always within a single cycle of the waves. The speed of the waves is :
(A) 2.5 m/s (B) 5.0 m/s (C) 14 m/s (D) 40 m/s

PSC SIR
 STRING WAVE
A 14*. The plane wave represented by an equation of the form y = f(x – v t) implies the propagation along the
positive x-axis without change of shape with constant velocity v :

y  y  y  2y  2y  2y  2y  2y 

 v     v    v 2  2   v2  2 
(A) (B) t  x 2  (C)  x  (D)  x 
t  x    t 2   t 2  

A 15*. Transverse mechanical waves can travel in :

(A) Iron rod (B) Hydrogen gas (C) Inside Water (D) Stretched strings

A 16. A wave pulse is generated in a string that lies along x-axis. At the points A and B, as shown in figure,
if RA and RB are ratio of wave speed to the particle speed respectively then :

(A) RA > RB (B) RB > RA

(C) R A = RB (D) Information is not sufficient to decide.

A 17. Wave pulse on a string shown in figure is moving to the

right without changing shape. Consider two particles at
positions x 1 = 1.5 m and x 2 = 2.5 m. Their transverse
velocities at the moment shown in figure are along directions
(A) positive y–axis and positive y–axis respectively
(B) negative y–axis and positive y–axis respectively
(C) positive y–axis and negative y–axis respectively
(D) negative y–axis and negative y–axis respectively

A 18*. The displacement of particles in a string stretched in xdirection is represented by y . Among the
following expressions for y , those describing wave motion are : [JEE - 87, 2]
(A) cos (kx) sin(t) (B) k²x²  ²t² (C) cos² (kx + t) (D) cos (k²x²  ²t²)

A 19*. A wave equation which gives the displacement along the Y direction is given by
y = 10–4 sin (60t + 2x)
where x and y are in metres and t is time in seconds. This represents a wave [JEE - 82]
(A) travelling with a velocity of 30 m/s in the negative x direction
(B) of wavelength  metre
(C) of frequency 30/ hertz
(D) of amplitude 10–4 metre travelling along the negative x direction.
A 20*. The displacement of a particle in a medium due to a wave travelling in the x-direction through the
medium is given by y = A sin(t –  x), where t = time, and  and  are constants :
(A) the frequency of the wave is  (B) the frequency of the wave is /2
(C) the wavelength is 2/ (D) the velocity of the wave is /

SECTION (B) : POWER TRANSMITTED ALONG THE STRING

B 1. A wave moving with constant speed on a uniform string passes the point x = 0 with amplitude A0,
angular frequency 0 and average rate of energy transfer P0. As the wave travels down the string it
P0
gradually loses energy and at the point x = , the average rate of energy transfer becomes . At the
2
point x = , angular frequency and amplitude are respectively :
(A) 0 and A0/ 2 (B) 0/ 2 and A0 (C) less than 0 and A0 (D) 0/ 2 and A0 / 2

PSC SIR
 STRING WAVE
B 2. A sinusoidal wave with amplitude ym is travelling with speed V on a string with linear density . The
angular frequency of the wave is . The following conclusions are drawn. Mark the one which is correct.
(A) doubling the frequency doubles the rate at which energy is carried along the string
(B) if the amplitude were doubled, the rate at which energy is carried would be halved
(C) if the amplitude were doubled, the rate at which energy is carried would be doubled
(D) the rate at which energy is carried is directly proportional to the velocity of the wave.
B 3. Sinusoidal waves 5.00 cm in amplitude are to be transmitted along a string having a linear mass
density equal to 4.00 × 10–2 kg/m. If the source can deliver a maximum power of 90 W and the string is
under a tension of 100 N, then the highest frequency at which the source can operate is (take 2 = 10):
(A) 45.3 Hz (B) 50 Hz (C) 30 Hz (D) 62.3 Hz

B 4. For a wave displacement amplitude is 10–8 m, density of air 1.3 kg m –3, velocity in air 340 ms–1 and
frequency is 2000 Hz. The intensity of wave is
(A) 5.3 × 10–4 Wm –2 (B) 5.3 × 10–6 Wm –2 (C) 3.5 × 10–8 Wm –2 (D) 3.5 × 10–6 Wm –2

SECTION (C) : INTERFERENCE, REFLECTION, TRANSMISSION

C 1. When two waves of the same amplitude and frequency but having a phase difference of , travelling with
the same speed in the same direction (positive x), interfere, then
(A) their resultant amplitude will be twice that of a single wave but the frequency will be same
(B) their resultant amplitude and frequency will both be twice that of a single wave
(C) their resultant amplitude will depend on the phase angle while the frequency will be the same
(D) the frequency and amplitude of the resultant wave will depend upon the phase angle.
C 2. The rate of transfer of energy in a wave depends
(A) directly on the square of the wave amplitude and square of the wave frequency
(B) directly on the square of the wave amplitude and square root of the wave frequency
(C) directly on the wave frequency and square of the wave amplitude
(D) directly on the wave amplitude and square of the wave frequency
C 3. Two waves of equal amplitude A, and equal frequency travels in the same direction in a medium. The amplitude
of the resultant wave is
(A) 0 (B) A (C) 2A (D) between 0 and 2A
C 4. A wave pulse, travelling on a two piece string, gets partially reflected and partially transmitted at the junction.
The reflected wave is inverted in shape as compared to the incident one. If the incident wave has wavelength
 and the transmitted wave ,
(A)  >  (B)  = 
(C)  <  (D) nothing can be said about the relation of  and .
C 5. The effects are produced at a given point in space by two waves described by the equations,
y1 = ym sin t and y2 = ym sin (t + ) where ym is the same for both the waves and  is a phase angle.
Tick the incorrect statement among the following.
(A) the maximum intensity that can be achieved at a point is twice the intensity of either wave and
occurs if  = 0
(B) the maximum intensity that can be achieved at a point is four times the intensity of either wave and
occurs if  = 0
(C) the maximum amplitude that can be achieved at the point its twice the amplitude of either wave and
occurs at  = 0
(D) When the intensity is zero, the net amplitude is zero, and at this point  = .
C 6. The following figure depicts a wave travelling in a medium.
Which pair of particles are in phase.
(A) A and D (B) B and F
(C) C and E (D) B and G

PSC SIR
 STRING WAVE
SECTION (D) : STANDING WAVES AND RESONANCE
D 1. A wave represented by the equation y = a cos(kx  t) is superposed with another wave to form a
stationary wave such that the point x = 0 is a node . The equation for other wave is : [JEE - 88]
(A) a sin (kx + t) (B) a cos(kx + t) (C) a cos(kx  t) (D) a sin(kx  t)
D 2. A stretched sonometer wire resonates at a frequency of 350 Hz and at the next higher frequency of 420
Hz. The fundamental frequency of this wire is
(A) 350 Hz (B) 5 Hz (C) 70 Hz (D) 170 Hz

D 3. Equations of a stationary wave and a travelling wave are y1 = a sinkx cos t and y2 = a sin (t – kx). The phase
 3 
difference between two points x1 = and x2 = are 1 and 2 respectively for the two waves. The ratio 1 is:
3k 2k 2
(A) 1 (B) 5/6 (C) 3/4 (D*) 6/7

PSC SIR
 STRING WAVE

PART - I E 2. (a) 144 m/s; (b) 60.0 cm; (c) 241 Hz

SECTION (A) : E 3. 70 Hz
A 1. (a) amplitude A = 5 mm E 4. (a) 0.25 cm (b) 1.2 × 102 cm/s; (c) 3.0 cm; (d) 0
(b) wave number k = 1 cm–1 E 5. (a) 100 Hz (b) 700 Hz
2
(c) wavelength = = 2 cm E 6. (i) y1 = 1.5 cos {(/20)x  72t} ,
k
y2 =1.5 cos {(/20)x + 72 t}
 60 (ii) 10 , 30 , 50 cm and 0 , 20 , 40 , 60 cm
(d) frequency v= = Hz
2 2 (iii) 0
PART - II
1 
(e) time period T= = s
v 30 SECTION (A) :
(f) wave velocity u = v = 60 cm/s. A 1. (A) A 2. (C) A 3. (B) A 4. (B)
A 5. (B) A 6. (D) A 7. (A) A 8. (B)
A 2. (a) 10 rad/s (b) /2 rad/m (c) y = (0.120 m) sin A 9. (C) A 10. (B) A 11. (C) A 12. (D)
(1.57 x – 31.4t) (d) 1.2 m/s (e) 118 m/s2 A 13. (B) A 14. (D) A 15. (A)(D)
A 16. (A) A 17. (B) A 18. (A)(C)
  7  A 19. (A)(B)(C)(D) A 20. (B)(C)(D)
A 3. 0.5 sin  t  x 
3 18 9  SECTION (B) :
A 4. (a) D, E, F (b) A, B, H (c) C, G (d) A, E B 1. (A) B 2. (D) B 3. (C) B 4. (D)
SECTION (C) :
SECTION (B) : C 1. (C) C 2. (A) C 3. (D) C 4. (C)
B 1. 520 m/s C 5. (A) C 6. (D)
B 2. 0.02 s SECTION (D) :
B 3. 0.12 m. D 1. (B) D 2. (C) D 3. (D)
B 4. 50 Hz, 4.0 cm, 2.0 m/s
B 5. 0.25
B 6. 50 m/s

SECTION (C) :
C 1. (a) y = (7.50 cm) sin (4.19x – 314t + ) (b) 625 W

C 2. 49 mW

C 3. (a) 70 m/s, 16 cm (b)`0.67 W

SECTION (D) :

2  
D 1. y’ = 0.8 a sin  t  x   .
  2
D 2. (a) Zero (b) 0.300 m.

D 3. (a) 3 (b) 4 (c) zero, 4.0 mm

SECTION (E) :

5 10 15 10
E 1. (a) Hz; (b) 5 10 Hz; (c) Hz.
2 2

PSC SIR