Chapter 14 Solutions — Waves

Exercises

1. 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?
2. 2.
   A stone dropped from the top of a tower of height 300 m 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)
3. 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.
4. 4.
   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.
5. 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 – 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)
6. 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.
7. 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.
8. 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?…
9. 9.
   For the wave described in Exercise 14.8, 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?
10. 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
11. 11.
    The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin(2πx/3) cos(120πt) 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…
12. 12.
    For the wave on a string described in Exercise 14.11, 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?
13. 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
14. 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?
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.
16. 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?
17. 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).
18. 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?
19. 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…