Sound Waves A microphone attached to a vertical spring oscillates up and down in

Question

Sound Waves

A microphone attached to a vertical spring oscillates up and down in simple harmonic motion, as shown below. Directly below the microphone is a sound source, which emits waves at sound level 106 decibels. When the sound source is stationary, the highest and lowest frequencies detected by the microphone are 675 Hz and 625 Hz. respectively. (a) Find the maximum speed at which the microphone is moving, as well as the true frequency and pressure amplitude of sound waves emitted by the source. (The density of air is 1.29 kg/m^3, the speed of sound in air is 343 m/s, and the reference intensity at the threshold of hearing is 1 times 10^-12 W/m^2.) (b) Now suppose the sound source moves vertically downward at a constant speed of 25 m/s. What is the highest frequency detected by the microphone?

Explanation / Answer

a) heighest and lowest freqinecies, detected by microphane correspond to, maximum speed of microphone in the downward direction and upward direction respectively. If this maximum speed is V
f(max) = fo ( Vo +V) / Vo .....1
f(min) = fo ( Vo - V) / Vo ......2
Vo is speed of sound = 343 m/s
fo is true freuency of wave
Dividing these two equations we get
675/ 625 = (343 + V)/(343-V)
V = 13.2 m/s
using this in equation 1 we get, fo = 650

For pressure amplitude
I = Io e(dB/10) = Po2 / 2 rho Vo
given Io = 10-12 , dB = 106, rho = 1.29, Vo = 343 m/s
Pressure amplitude Po = 6x10-3 Pa

b) For constant speed of 25 m/s downward, heighest frequency
f = fo (Vo + V ) / Vo
V= 25 and f0 = 650
f = 697.4 Hz

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