352
8 Waves
Which one of these would be preferable to use in determining the surface
tension by means of ripples?
8.45 Waves in deep water travel with phase velocity given by v p
2 = g/k, where g
is the acceleration due to gravity and k is the wave number, 2π/λ. Obtain an
expression for the group velocity and show that it is equal to v p /2.
[University of Manchester 2006]
8.46 The dispersion relation for sound waves in air is ω =
γ RT
M k. Find the phase
velocity and the group velocity.
8.47 The phase velocity v p for deep water waves is given by v p
2 = (g/k + Sk/ρ).
Show that the phase velocity is minimum at λ = 2π
s
ρg .
8.2.4 Sound Waves
8.48 Let both displacement and pressure of a plane wave vary harmonically. Obtain
a relation between pressure amplitude and displacement amplitude. Also show
that the displacement is 90 ◦ out of phase with the pressure wave.
8.49 Assuming ρ = 1.29 kg/m 3 for the density of air and v = 331 m/s for the
speed of sound, find the pressure amplitude corresponding to the threshold of
hearing intensity of 10 −12 W/m 2 .
8.50 For ordinary conversation, the intensity level is given as 60 dB. What is the
intensity of the wave?
8.51 A small source of sound radiates energy uniformly at a rate of 4 W. Calculate
the intensity and the intensity level at a point 25 cm from the source if there is
no absorption.
8.52 The maximum pressure variation that the ear can tolerate is about 29 N/m 2 .
Find the corresponding maximum displacement for a sound wave in air having
a frequency of 2000 Hz. Assume the density of air as 1.22 kg/m
3 and the speed
of sound as 331 m/s.
8.53 If two sound waves, one in air and the other in water, have equal pressure
amplitudes, what is the ratio of the intensities of the waves? Assume that the
density of air is 1.293 kg/m
3 , and the speed of sound in air and water is 330
and 1450 m/s, respectively.
8.54 The pressure in a progressive sound wave is given by the equation P =
2.4 sin π(x − 330t), where x is in metres, t in seconds and P in N/m 2 . Find
(a) the pressure amplitude, (b) frequency, (c) wavelength and (d) speed of the
wave.
8.55 A note of frequency 1200 vibrations/s has an intensity of 2.0 μ W/m 2 . What
is the amplitude of the air vibrations caused by this sound?
8 Waves
Which one of these would be preferable to use in determining the surface
tension by means of ripples?
8.45 Waves in deep water travel with phase velocity given by v p
2 = g/k, where g
is the acceleration due to gravity and k is the wave number, 2π/λ. Obtain an
expression for the group velocity and show that it is equal to v p /2.
[University of Manchester 2006]
8.46 The dispersion relation for sound waves in air is ω =
γ RT
M k. Find the phase
velocity and the group velocity.
8.47 The phase velocity v p for deep water waves is given by v p
2 = (g/k + Sk/ρ).
Show that the phase velocity is minimum at λ = 2π
s
ρg .
8.2.4 Sound Waves
8.48 Let both displacement and pressure of a plane wave vary harmonically. Obtain
a relation between pressure amplitude and displacement amplitude. Also show
that the displacement is 90 ◦ out of phase with the pressure wave.
8.49 Assuming ρ = 1.29 kg/m 3 for the density of air and v = 331 m/s for the
speed of sound, find the pressure amplitude corresponding to the threshold of
hearing intensity of 10 −12 W/m 2 .
8.50 For ordinary conversation, the intensity level is given as 60 dB. What is the
intensity of the wave?
8.51 A small source of sound radiates energy uniformly at a rate of 4 W. Calculate
the intensity and the intensity level at a point 25 cm from the source if there is
no absorption.
8.52 The maximum pressure variation that the ear can tolerate is about 29 N/m 2 .
Find the corresponding maximum displacement for a sound wave in air having
a frequency of 2000 Hz. Assume the density of air as 1.22 kg/m
3 and the speed
of sound as 331 m/s.
8.53 If two sound waves, one in air and the other in water, have equal pressure
amplitudes, what is the ratio of the intensities of the waves? Assume that the
density of air is 1.293 kg/m
3 , and the speed of sound in air and water is 330
and 1450 m/s, respectively.
8.54 The pressure in a progressive sound wave is given by the equation P =
2.4 sin π(x − 330t), where x is in metres, t in seconds and P in N/m 2 . Find
(a) the pressure amplitude, (b) frequency, (c) wavelength and (d) speed of the
wave.
8.55 A note of frequency 1200 vibrations/s has an intensity of 2.0 μ W/m 2 . What
is the amplitude of the air vibrations caused by this sound?