Question

A rather typical sound wave has a pressure amplitude of 2×10-2 Pa and a frequency of 103Hz. Assuming that the wave is a plane wave propagating to the right in air at 22°C and 1.01×105 Pa:a) Write the function describing this wave. Give numerical values for the circular frequency ω, the wave number k, and the amplitude. b) What is the amplitude of the particle velocity in this wave?c) What is the maximum percentage change in the air density due to this wave? d) What is the amplitude of the temperature fluctuations in the wave? (Hint: Substitute the relations0pp p′= +, 0TT T′= +, and 0ρρ ρ′= + into the gas law and remember that0 00

Answer 1

a )

using equation for ω = 2 f = 2 X 10³ rad/s

v = 334.6 m/s

we have v = ω / k

k = 2 X 3.14 X 10³ / 334.6

k = 18.76

and k = 2 /

= 2 X 3.14 / 18.76

= 0.334 m

amplitude A = 20 log ( 2 X 10^-2 / 20 X 10^-6 )

A = 60 dB

b )

( x,t ) = 60 sin( 18.76x - 2000t )

particle velocity v = A ω

= 60 X 2 X 3.14 X 10³

= 376800 m/s

c )

ρ = ρ_o T_o R

ρ_o = 1.1929 kg/m³

P = ρv _A ω

ρ= 1.58 X 10^-10 kg/m³

change in air density = 1.58 X10^-10 / 1.1929

= 1.32 X 10^-10

d )

P = ρR T

( P_o + ρ ' ) = ( ρ_o + ρ' ) R X ( T_o + T ' )

T ' = 9.17 X 10^-3 K