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 notes a half-step apart have a frequency ratio of 2⁽¹/¹²⁾ = 1.059.
With what speed must the bat fly toward the singer?

     v’= 9.74 m / s

Explanation:

The Doppler effect is due to the relative movement of the sound source and the receiver of the sound, in this case we must perform the exercise in two steps, the first to find the frequency that the bat hears and then the frequency that the audience hears that also It is sitting.

Frequency shift heard by the murciela, in case the source is still and the observer (bat) moves closer

        f₁ ’= f₀ (v + v₀)/v

Frequency shift emitted by the speaker in the bat, in this case the source is moving away from the observer (public sitting) that is at rest

         f₂’= f₁’ v/(v - vs)

Note that in this case the bat is observant in one case and emitter in the other, called its velocity v’

            v’= vo = vs

Let's replace

           f₂’= f₀   (v + v’)/v   v/(v -v ’)

           f₂’= f₀   (v + v’) / (v -v ’)

           (v –v’ ) f₂’ / f₀ = v + v ’

           v’ (1+ f₂’ /f₀) = v (f₂’/fo - 1)

           v’ (1 + 1.059) = 340 (1.059 - 1)

           v’= 20.06 / 2.059

           v’= 9.74 m / s

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