An avant-garde composer wants to use the Doppler effect in his new opera. As the

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 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?Express your answer with the appropriate units.

Explanation / Answer

In this case the bat recieves the sound from the source and re-trasmits the recieved frequency

1) the bat moving a speed of u (observer) and the singer (source) is stationary

the oberved frequency fb by the bat due to doppler effect is

fb = f (v+u)/v , v - speed of sound 340 m/s and u speed of bat

Now when the bat re-transmits the frequency, the source is moving and observer (audience) is stationary, doppler frequency recieves by the obeserver (audience) is

fo = v/(v-u) *fb

     = v/(v-u) * f (v+u)/v = f (v+u)/(v-u)

f'/f = 1.059 = (340+u)/(340-u)

speed of bat u = 340*0.059/2.059 = 9.74 m/s

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