Question

A loudspeaker sits in a large room (of length L) against a wall and faces towards the opposite wall. The speaker emits a resonant frequency of the room, thus setting up a standing wave. Assume that the wavelength is much smaller than the size of the room, so that the mode number n is high. Use vsnd for the speed of sound.

a.) Your friend starts at the speaker and runs toward the opposite wall at constant speed v. As she does, she hears a loud-soft-loud modulation of the sound intensity. From your perspective, sitting at rest, she is running through nodes and antinodes of the standing wave. Find an expression for the number of sound maxima she hears per second, only in terms of the variables given above.

b.) From your friend’s perspective, the two sound waves are not the same frequency, so they don’t create a standing wave. Find an expression for the beat frequency that she hears, again, only in terms of the variables given above.

c.) Are your answers in (a) and (b) the same, or different? Should they be?

Answer 1

Answer:

Given that:

(a) Let us setting up standing wave distance and no.of sound maxima hears in whole journey , in a time interval t, the distance covered by

and the speed of friend

Use the relation among velocity, distance and time, we get the expression for the number of sound maxima friends heart per second as

(b) Use the Doppler Effect, we can write the higher apparent frequency of source. Friend is moving towards one loudspeakers

Use the Doppler Effect, we can write the lower apparent frequency of source. Friend is moving away from other loudspeakers as

where ,

The number of sound maxima friends hears per second due to the above two frequencies and is given by

Use the relation " n " among frequency. Velocity of sound and wave length if sound the original frequency of sound produced by loudspeakers is given by

(c) In part (a) and (b) we get the same answer. They should be equal, because in both cases, we measure the no.of sound maxima friend hears per second.