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**Question 15. 1** A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched
string is 20.0 m. If the transverse jerk is struck at one end of the string, how long
does the disturbance take to reach the other end?

**Question 15. 2** A stone dropped from the top of a tower of height 300 m high splashes into the
water of a pond near the base of the tower. When is the splash heard at the top
given that the speed of sound in air is 340 m s–1 ? (g = 9.8 m s–2)

**Question 15. 3** A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the
tension in the wire so that speed of a transverse wave on the wire equals the speed
of sound in dry air at 20 °C = 343 m s–1.

**Question 15. 4 **Use the formula v
P
=
γ
ρ
to explain why the speed of sound in air
(a) is independent of pressure,
(b) increases with temperature,
(c) increases with humidity.

**Question 15. 5** 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 – v t or x + v t, i.e.
y = f (x ± v t). 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)

**Question 15. 6** A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a
water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted
sound? Speed of sound in air is 340 m s –1 and in water 1486 m s–1.

**Question 15. 7** A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the
wavelength of sound in the tissue in which the speed of sound is 1.7 km s–1 ? The
operating frequency of the scanner is 4.2 MHz.

**Question 15. 8** A transverse harmonic wave on a string is described by
y(x, t) = 3.0 sin (36 t + 0.018 x + π/4)
where x and y are in cm and t in s. The positive direction of x is from left to right.

(a) Is this a travelling wave or a stationary wave ?
If it is travelling, what are the speed and direction of its propagation ?

(b) What are its amplitude and frequency ?

(c) What is the initial phase at the origin ?

(d) What is the least distance between two successive crests in the wave ?

**Question 15. 9** For the wave described in Exercise 15. 8, plot the displacement (y) versus (t) graphs
for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does
the oscillatory motion in travelling wave differ from one point to another: amplitude,
frequency or phase ?

**Question 15. 10 **For the travelling harmonic wave
y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35)
where x and y are in cm and t in s. Calculate the phase difference between oscillatory
motion of two points separated by a distance of
(a) 4 m,
(b) 0.5 m,
(c) λ/2,
(d) 3λ/4
WAVES 389

**Question 15. 11** The transverse displacement of a string (clamped at its both ends) is given by
y(x, t) = 0.06 sin
2
3
x
cos (120 πt)
where x and y are in m and t in s. The length of the string is 1.5 m and its mass is
3.0 ×10–2 kg.
Answer the following :

(a) Does the function represent a travelling wave or a stationary wave?

(b) Interpret the wave as a superposition of two waves travelling in opposite
directions. What is the wavelength, frequency, and speed of each wave ?

(c) Determine the tension in the string.

**Question 15. 12** (i) For the wave on a string described in Exercise 15.11, do all the points on the
string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain
your answers. (ii) What is the amplitude of a point 0.375 m away from one end?

**Question 15. 13** Given below are some functions of x and t to represent the displacement (transverse
or longitudinal) of an elastic wave. State which of these represent (i) a travelling
wave, (ii) a stationary wave or (iii) none at all:

(a) y = 2 cos (3x) sin (10t)

(b) y = 2 x − vt

(c) y = 3 sin (5x – 0.5t) + 4 cos (5x – 0.5t)

(d) y = cos x sin t + cos 2x sin 2t

**Question 15. 14 **A wire stretched between two rigid supports vibrates in its fundamental mode with
a frequency of 45 Hz. The mass of the wire is 3.5 × 10–2 kg and its linear mass density
is 4.0 × 10–2 kg m–1. What is (a) the speed of a transverse wave on the string, and
(b) the tension in the string?

**Question 15. 15** A metre-long tube open at one end, with a movable piston at the other end, shows
resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when
the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the
temperature of the experiment. The edge effects may be neglected.

**Question 15. 16** A steel rod 100 cm long is clamped at its middle. The fundamental frequency of
longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of
sound in steel?

**Question 15. 17** A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is
resonantly excited by a 430 Hz source ? Will the same source be in resonance with
the pipe if both ends are open? (speed of sound in air is 340 m s–1).

**Question 15. 18** Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce
beats of frequency 6 Hz. The tension in the string A is slightly reduced and the
beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz,
what is the frequency of B?

**Question 15. 19** Explain why (or how):

(a) in a sound wave, a displacement node is a pressure antinode and vice versa

(b) bats can ascertain distances, directions, nature, and sizes of the obstacles
without any “eyes”

(c) a violin note and sitar note may have the same frequency, yet we can
distinguish between the two notes

(d) solids can support both longitudinal and transverse waves, but only
longitudinal waves can propagate in gases

(e) the shape of a pulse gets distorted during propagation in a dispersive medium.
390

**Question 15. 20** A train, standing at the outer signal of a railway station blows a whistle of frequency 400
Hz in still air. (i) What is the frequency of the whistle for a platform observer when the train
(a) approaches the platform with a speed of 10 m s–1, (b) recedes from the platform with a
speed of 10 m s–1? (ii) What is the speed of sound in each case ? The speed of sound in still
air can be taken as 340 m s–1.

**Question 15. 21** A train, standing in a station-yard, blows a whistle of frequency 400 Hz in still air. The wind
starts blowing in the direction from the yard to the station with at a speed of 10 m s–1. What
are the frequency, wavelength, and speed of sound for an observer standing on the station’s
platform? Is the situation exactly identical to the case when the air is still and the observer
runs towards the yard at a speed of 10 m s–1? The speed of sound in still air can be taken as
340 m s–1
Additional Exercises

**Question 15. 22** A travelling harmonic wave on a string is described by
y(x, t) = 7.5 sin (0.0050x +12t + π/4)
(a)what are the displacement and velocity of oscillation of a point at
x = 1 cm, and t = 1 s? Is this velocity equal to the velocity of wave propagation?
(b)Locate the points of the string which have the same transverse displacements and velocity
as the x = 1 cm point at t = 2 s, 5 s and 11 s.

**Question 15. 23** 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 ?

**Question 15. 24** One end of a long string of linear mass density 8.0 × 10–3 kg m–1 is connected to an electrically
driven tuning fork of frequency 256 Hz. The other end passes over a pulley and is tied to a
pan containing a mass of 90 kg. The pulley end absorbs all the incoming energy so that
reflected waves at this end have negligible amplitude. At t = 0, the left end (fork end) of the
string x = 0 has zero transverse displacement (y = 0) and is moving along positive y-direction.
The amplitude of the wave is 5.0 cm. Write down the transverse displacement y as function
of x and t that describes the wave on the string.

**Question 15. 25** A SONAR system fixed in a submarine operates at a frequency 40.0 kHz. An enemy submarine
moves towards the SONAR with a speed of 360 km h–1. What is the frequency of sound
reflected by the submarine ? Take the speed of sound in water to be 1450 m s–1.

**Question 15. 26** Earthquakes generate sound waves 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 ?

**Question 15. 27** A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound
emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall
surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the
bat hear reflected off the wall ?

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- Chapter 3: Motion in a Straight Line
**[Ques wise Ans]** - Chapter 1 Physical World
- Chapter 2 Units and Measurement
- Chapter 3 Motion in a Straight Line
- Chapter 4 Motion in a Plane
- Chapter 5 Laws of Motion
- Chapter 7 Systems of Particles and Rotational Motion
- Chapter 6 Work Energy and Power
- Chapter 8 Gravitation
- Chapter 9 Mechanical Properties of Solids
- Chapter 10 Mechanical Properties of Fluids
- Chapter 11 Thermal Properties of Matter
- Chapter 12 Thermodynamics
- Chapter 13 Kinetic Theory
- Chapter 14 Oscillations
- Chapter 15 Waves

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