354
8 Waves
acoustic particle displacement amplitude and (e) the condensation amplitude.
Assume that the velocity of sound in water is 1450 m/s.
8.69 Derive Laplace formula for the sound velocity in a gas.
8.70 An empirical formula giving the velocity of sound in distilled water as a function of temperature at a pressure of one atmosphere in the range 0 − 60 ◦ C is
v = 1403 + 5t − 0.06t 2 + 0.0003t 3 where t is the temperature of water in
◦ C and v is in m/s. (a) Determine the velocity of sound in distilled water at
20 ◦ C and (b) find the change of velocity of sound per degree Celsius at this
temperature.
8.2.5 Doppler Effect
8.71 A railway engine whistles as it approaches a tunnel, and the sound is reflected
back by the wall of the rock at the opening. If the train is proceeding at a
speed of 72 km/h and if the effect of the wind be neglected, find the ratio
of the relative frequencies of the reflected and direct sounds as heard by the
driver of the engine.
[University of Aberystwyth, Wales]
8.72 Two trains move towards each other at a speed of 90 km/h relative to the
earth’s surface. One gives a 520 Hz signal. Find the frequency heard by the
observer on the other train.
8.73 Two trains move away from each other at a speed of 25 m/s relative to the
earth’s surface. One gives a 520 Hz signal. Find the frequency heard by the
observer on the other train (sound velocity = 330 m/s).
8.74 A whistle of frequency 540 Hz rotates in a circle of radius 2 m at an angular
speed of 15.0 rad/s. What are the maximum and minimum frequencies heard
by a listener, standing at a long distance away at rest from the centre of the
circle (sound velocity = 330 m/s).
8.75 In the Kundt’s tube experiment, the length of the steel rod which is stroked is
120 cm long and the distance between heaps of cork dust is 8 cm when the rod
is caused to vibrate longitudinally in air. If the ends of the tube are sealed and
the air replaced by a gas and the experiment repeated, the distance between
heaps is observed to be 10 cm. (a) What is the velocity of sound in the gas if
the velocity in air is 340 m/s. (b) What is the velocity of sound in the rod?
8.76 A sound source from a motionless train emits a sinusoidal wave with a source
frequency of f s = 514 Hz. Given that the speed of sound in air is 340 m/s
and that you are a stationary observer. Find the wavelength of the wave you
observe
(i) When the train is at rest
(ii) When the train is moving towards you at 15 m/s
8 Waves
acoustic particle displacement amplitude and (e) the condensation amplitude.
Assume that the velocity of sound in water is 1450 m/s.
8.69 Derive Laplace formula for the sound velocity in a gas.
8.70 An empirical formula giving the velocity of sound in distilled water as a function of temperature at a pressure of one atmosphere in the range 0 − 60 ◦ C is
v = 1403 + 5t − 0.06t 2 + 0.0003t 3 where t is the temperature of water in
◦ C and v is in m/s. (a) Determine the velocity of sound in distilled water at
20 ◦ C and (b) find the change of velocity of sound per degree Celsius at this
temperature.
8.2.5 Doppler Effect
8.71 A railway engine whistles as it approaches a tunnel, and the sound is reflected
back by the wall of the rock at the opening. If the train is proceeding at a
speed of 72 km/h and if the effect of the wind be neglected, find the ratio
of the relative frequencies of the reflected and direct sounds as heard by the
driver of the engine.
[University of Aberystwyth, Wales]
8.72 Two trains move towards each other at a speed of 90 km/h relative to the
earth’s surface. One gives a 520 Hz signal. Find the frequency heard by the
observer on the other train.
8.73 Two trains move away from each other at a speed of 25 m/s relative to the
earth’s surface. One gives a 520 Hz signal. Find the frequency heard by the
observer on the other train (sound velocity = 330 m/s).
8.74 A whistle of frequency 540 Hz rotates in a circle of radius 2 m at an angular
speed of 15.0 rad/s. What are the maximum and minimum frequencies heard
by a listener, standing at a long distance away at rest from the centre of the
circle (sound velocity = 330 m/s).
8.75 In the Kundt’s tube experiment, the length of the steel rod which is stroked is
120 cm long and the distance between heaps of cork dust is 8 cm when the rod
is caused to vibrate longitudinally in air. If the ends of the tube are sealed and
the air replaced by a gas and the experiment repeated, the distance between
heaps is observed to be 10 cm. (a) What is the velocity of sound in the gas if
the velocity in air is 340 m/s. (b) What is the velocity of sound in the rod?
8.76 A sound source from a motionless train emits a sinusoidal wave with a source
frequency of f s = 514 Hz. Given that the speed of sound in air is 340 m/s
and that you are a stationary observer. Find the wavelength of the wave you
observe
(i) When the train is at rest
(ii) When the train is moving towards you at 15 m/s