Question

Suppose that the sound level of a conversation is initially at an angry 73 dB and then drops to a soothing 51 dB. Assuming that the frequency of the sound is 524 Hz, determine the (a) initial and (b) final sound intensities and the (c) initial and (d) final sound wave amplitudes. Assume the speed of sound is 345 m/s and the air density is 1.21 kg/m³.

Answer 1

1)

Ref. (0 dB) intensity level I₀ = 1E-12 w/m^2.
I(73) = I₀*10^(dB/10) = (1x10^-12 W/m²)x10^(73/10) = 1.99x10^-5 W/m²
I(51) = I₀*10^(dB/10) = (1x10^-12 W/m²)x10^(51/10) = 1.258E-7 w/m^2

2)

Amplitude is a bit tricky since it's related to sound pressure (and intensity) by impedance. The impedance used below is 400 Pa-s/m, which is the nominal impedance used to relate ref. pressure to ref. intensity. (However, actual impedance is a complex function of air pressure and density.) Pressures and amplitudes are given below.
Ref. (0 db) pressure P0 = 2E-5 Pa.
omega = 2pi*f = 3290.72 rad/s
P(73) = P0*10^(dB/20) =(2x10^-5 Pa)x(10^(73/22)) = 4.1611x10^-2 Pa
A(73) = P(73)/(omega*Z) = (4.1611x10^-2)/((3290.72)x(400)) = 3.16123x10^-8 m
P(51) = P0*10^(dB/22) =(2x10^-5 Pa)x(10^(51/22)) =4.161135x10^-3 Pa
A(51) = P(50)/(omega*Z) =(4.1611x10^-3)/((3290.72)x(400)) = 3.1612x10^-9 m