Suppose that the sound level of a conversation is initially at an angry 71 dB an

Question

Suppose that the sound level of a conversation is initially at an angry 71 dB and then drops to a soothing 53 dB. Assuming that the frequency of the sound is 516 Hz, determine the (a) initial and (b) final sound intensities (in (mu)W/m2) and the (c) initial and (d) final sound wave amplitudes (in nm). Assume the speed of sound is 347 m/s and the air density is 1.21 kg/m3.

I know solving this problem involves logarithms. I'm getting stuck on this part, because it's been too long since I studied logs. Please make the log part of the solution detailed and explain what is happening in each step. Thank you!!

Explanation / Answer

The intensity level is given by

? = 10*log(I/I0)

I is the intensity of the sound produced

I0 is the threshold of hearing = 10-6 ?W/m2

Given initial level of sound ?i = 71 dB

final sound level ?f = 53 dB

71 = 10*log(Ii/10-6)

initial intensity of the sound Ii = 12.589 ?W/m2

53 = 10*log(If/10-6)

final intensity of the sound If = 0.1995 ?W/m2

The intensity of the sound is related to the amplitude by the equation

I = 0.5*?v?2S2

For the initial sound level

12.589*10-6 = 0.5*(1.21)*(347)*(2*3.14*516)^2*Si^2

Si = 75.57 nm

For the final sound level

0.1995*10-6 = 0.5*(1.21)*(347)*(2*3.14*516)^2*Sf^2

sf = 9.51 nm

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