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a bat is flitting about in a cave navigating via...

Question:
A bat is flitting about in a cave, navigating via ultrasonic beeps. Assume that the sound emission frequency of the bat is 40 kHz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.03 times the speed of sound in air. What frequency does the bat hear reflected off the wall?

Answer:

Ultrasonic beep frequency emitted by the bat, v = 40 kHz

Velocity of the bat, v_b = 0.03 v

Where, v = velocity of sound in air

The apparent frequency of the sound striking the wall is given as:

v' = [v / v - v_b] v

= [v / v - 0.03v] x 40

= 40/0.97 kHz

This frequency is reflected by the stationary wall (v_s) toward the bat.

The frequency (v'') of the received sound is given by the relation:

v'' = [ v + v_b / v] v'

= [ v + 0.03v / v] x 40/0.97

= 1.03 x 40 / 0.97 = 42.47 kHz

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