A speaker at the front of a room and an identical speaker at the rear of the room are being driven at 456 Hz by the same sound source. A student walks at a uniform rate of 1.12 m/s away from one speaker and toward the other. How many beats does the student hear per second? (Take the speed of sound to be 343 m/s.) Hint: The Doppler effect causes both frequencies to be shifted. The difference between those two frequencies is what causes the beats.

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boffeemadrid boffeemadrid · Answer 1 · 2019-12-16T11:15:11+01:00

Answer:

2.97795 Hz

Explanation:

v = Speed of sound in air = 343 m/s

[tex]v_r[/tex] = Relative speed between the speakers and the student = 1.12 m/s

[tex]f'[/tex] = Actual frequency of sound = 456 Hz

Frequency of sound heard as the student moves away from one speaker

[tex]f_1=f'\dfrac{v-v_r}{v}\\\Rightarrow f_1=456\dfrac{343-1.12}{343}\\\Rightarrow f_1=454.51102\ Hz[/tex]

Frequency of sound heard as the student moves closer to the other speaker

[tex]f_2=f'\dfrac{v+v_r}{v}\\\Rightarrow f_2=456\dfrac{343+1.12}{343}\\\Rightarrow f_2=457.48897\ Hz[/tex]

The difference in the frequencies is

[tex]f_2-f_1=457.48897-454.51102=2.97795\ Hz[/tex]

The student hears 2.97795 Hz