Question

An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer's voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notesa half-step apart have a frequency ratio of 2 1/12 1.059.

With what speed must the bat fly toward the singer?

Answer 1

The apparent frequency heard by the bat is

$f_1=\frac{c+v_b}{c}f$

The bat reflects this frequency. So it acts like a source for the stationary audience

$f_2=\frac{c}{c-v_b}f_1=\frac{c+v_b}{c-v_b}f$

$\frac{c+v_b}{c-v_b}=1.059$

$2.059v_b=0.059c$

$v_b=\frac{59}{2059}c\approx9.74\textup{ m/s}$

using

$c=340\textup{ m/s}$