Chapter 15 - WAVES

In the given progressive wave
y = 5 sin (100πt – 0.4πx )
where y and x are in m, t is in s. What is the
(a) amplitude
(b) wave length
(c) frequency
(d) wave velocity
(e) particle velocity amplitude.

displacement of an elastic wave.
(a) y = 5 cos (4x ) sin (20t )
(b) y = 4 sin (5x – t/2) + 3 cos (5x – t/2)
(c) y = 10 cos [(252 – 250) πt ] cos [(252+250)πt ]
(d) y = 100 cos (100πt + 0.5x )
State which of these represent
(a) a travelling wave along –x direction
(b) a stationary wave
(c) beats
(d) a travelling wave along +x direction.
Given reasons for your answers.

If c is r.m.s. speed of molecules in a gas and v is the speed of
sound waves in the gas, show that c/v is constant and independent
of temperature for all diatomic gases.

The earth has a radius of 6400 km. The inner core of 1000 km
radius is solid. Outside it, there is a region from 1000 km to a
radius of 3500 km which is in molten state. Then again from
3500 km to 6400 km the earth is solid. Only longitudinal (P)
waves can travel inside a liquid. Assume that the P wave has a
speed of 8 km s–1 in solid parts and of 5 km s–1 in liquid parts of
the earth. An earthquake occurs at some place close to the surface
of the earth. Calculate the time after which it will be recorded in a
seismometer at a diametrically opposite point on the earth if wave
travels along diameter?

Show that when a string fixed at its two ends vibrates in 1 loop,
2 loops, 3 loops and 4 loops, the frequencies are in the ratio
1:2:3:4.

A train standing at the outer signal of a railway station blows a
whistle of frequency 400 Hz still air. The train begins to move
with a speed of 10 m s-1 towards the platform. What is the
frequency of the sound for an observer standing on the platform?

A pipe 20 cm long is closed at one end. Which harmonic mode of
the pipe is resonantly excited by a source of 1237.5 Hz ?

When two waves of almost equal frequencies n 1 and n 2 reach at
a point simultaneously, what is the time interval between
successive maxima?

At what temperatures (in oC) will the speed of sound in air be
3 times its value at OoC?

A sitar wire is replaced by another wire of same length and
material but of three times the earlier radius. If the tension in the
wire remains the same, by what factor will the frequency change?

The displacement of an elastic wave is given by the function
y = 3 sin ωt + 4 cos ωt.
where y is in cm and t is in second. Calculate the resultant
amplitude.

A tuning fork A, marked 512 Hz, produces 5 beats per second,
where sounded with another unmarked tuning fork B. If B is
loaded with wax the number of beats is again 5 per second. What
is the frequency of the tuning fork B when not loaded?

An organ pipe of length L open at both ends is found to vibrate
in its first harmonic when sounded with a tuning fork of 480 Hz.
What should be the length of a pipe closed at one end, so that it
also vibrates in its first harmonic with the same tuning fork?

The length of a sonometer wire between two fixed ends is 110cm. Where the
two bridges should be placed so as to divide the wire into three segments whose
fundamental frequencies are in the ratio of 1:2:3?

Give two cases in which there is no Doppler effect in sound?

Explain briefly the analytical method of formation of beats?

How roar of a lion can be differentiated from bucking of a mosquito?

If the splash is heard 4.23 seconds after a stone is dropped into a well. 78.4
metes deep, find the velocity of sound in air?

A person deep inside water cannot hear sound waves produces in air. Why?

Can beats be produced in two light sources of nearly equal frequencies?

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 this source be in resonance with the
pipe if both the ends are open?

Why do tuning forks have two prongs?

Why longitudinal waves are called pressure waves?

Explosions on other planets are not heard on earth. Why?

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?

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 Swave is about 4.0 km s^–1, and that of P wave is 8.0 km s^–1. A seismograph records P and Swaves 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?

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.

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.

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? 

A travelling harmonic wave on a string is described by

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.

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.

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, and
(e) The shape of a pulse gets distorted during propagation in a dispersive medium.

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?

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).

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

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.

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?

(d) y = cos x sin t + cos 2x sin 2t

(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?

The transverse displacement of a string (clamped at its both ends) is given by

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

 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?

A transverse harmonic wave on a string is described by

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.

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.

(a) (x – vt)²
(b) log [(x + vt) / x₀]
(c) 1 / (x + vt)

Use the formula v = √γP/ρ to explain why the speed of sound in air
(a) is independent of pressure,
(b) increases with temperature,
(c) increases with humidity.

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.

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)

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?