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Chapter 16: Q. 75 (page 454)

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鈥檚 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 21/12 = 1.059. With what speed must the bat fly toward the singer?

Short Answer

Expert verified

The speed of the bat is 9.74 m/s

Step by step solution

01

Write the given information

The frequency ratio of two notes is 21/12= 1.059

The speed of the sound is vs= 340 m/s

Let the speed of the bat is vb

02

To determine the speed of the bat

The frequency reflected by the bat is given by

f'=vs+vbvs-vbf

Here, f= frequency of the sound source

vs = speed of the sound
vb= speed of the bat

According to the statement given,

f'f=vs+vbvs-vb21/12=vs+vbvs-vb1.059=vs+vbvs-vb

Substitute the known values,

1.059=340+vb340-vb1.059(340-vb)=340+vb2.059vb=20.06vb=20.062.059=9.74m/s

Thus, the speed of the bat is 9.74 m/s

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