Question

Bats use the Doppler effect to determine their location. A bat flies toward a wall at a speed of 15 m/s while emitting sound of frequency 35 kHz. What is the frequency received by the bat? (hint: 2 doppler effects occur)

Answer 1

the bat is the first source; its movement through the air has some effect on the sound which hits the wall; the wall bounces the sound back unchanged (even the phase of the sound is unchanged, mostly, because of the wall's stiffness compared to that of air); and the bat picks up the bounced sound packet.

If the source moving away from the observer is emitting waves through a medium with an actual frequency f0, then an observer stationary relative to the medium detects waves with a frequency f given by

f = (v/(v+vs))* f_0

Here, the wall is the "stationary observer", v ~ 343 m/s (for sound in air), vs = -5 m/s, and f_0 = 40 kHz. Thus, the wall "hears" the bat chirp at about 40.592 kHz. This is the same frequency that is bounced back at the moving bat.

Now, the bat is the receiver, with the wall being the source. Here,

f = (v+vr)/v * f_0 ,

with v = 343 m/s, vr = +15 m/s, and f_0 = 36.53 kHz. The result is f ~ 36.6 kHz. The bat hears the returned chirp at this frequency,