Question

A sound source located at x=29 m produces a sound pulse with a frequency of 926 Hz which you (who are standing at the origin) note has a loudness of 64 dB. At the very moment that you hear the sound, there is an object located at x=94 m moving with a velocity of 52 m/s. This object reflects the sound back to you. What is the frequency and loudness of the reflected sound when the echo is first heard? (You may assume that the object perfectly reflects all sound and that the intensity is unchanged upon reflection. For a point of extra credit, determine in what way the intensity would be changed upon such a reflection!)

Answer 1

Given observer at x=0, source at x=29m and reflector at x=94m and moving with speed v=52m/s.

fs= 926 Hz

speed of sound v_s = 343m/s in dry air at temperatur = 20⁰C

When the sound was reflected from a moving away object form observer then the observed wave frequency f is

$\fn_phv f_o = \frac{v_s-v}{v_s+v} f_s = \frac{343-52}{343+52}x 926 = 682.1924 Hz$

loudness of sound = 10 log(I/I₀)

Here, I= Intensity of sound

$\fn_phv I \alpha \frac{f^2}{r^2}$

f= frequency of sound wave

r = distance from the object.

Direct wave distance r_d = 29m freuency f_d = 926 Hz

Reflected wave distance r_r = 159m frequency f_r = 682.1924 Hz

Assumption is sound wave reflected fully.

$\fn_phv 10log\left(\frac{I_r}{I_d}\right)= 20log\left(\frac{f_r r_d}{f_dr_r} \right ) \\ \\ 10log (I_r) = 10log(I_d) + 20log\left(\frac{f_r r_d}{f_dr_r} \right ) = 64 -17.43= 46.566 dB$

Loudness of echo is 46.566 dB.

The change in intensity due to change in frequency of reflected wave and the distance travelled by the reflected wave. The change in frequency depends on the velocity of moving object.