A long straight rigid horizontal duct is filled with oxygen at rest at Pa and kg m

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Acoustics of fluids: generation, propagation and attenuation · Acoustics of fluids: generation, propagation and attenuation

Fluids and Continuum Physics, Ch Acoustics: sound propagation, radiation and attenuation

Problem

A long straight rigid horizontal duct is filled with oxygen at rest at

p_0 = 1.013\times10^{5}

Pa and

\rho_0 = 1.33

kg m

^{-3}

, with ratio of specific heats

\kappa = 1.40

. A loudspeaker at one end drives a plane sound wave along the duct axis

x

, so that every field depends on

x

and

t

only. Take the gas as inviscid and treat the compressions as adiabatic (no heat conduction); gravity plays no part because the duct is horizontal. Write

p = p_0 + p'

\rho = \rho_0 + \rho'

u = u'

, and close the system with

p' = c^{2}\rho'

, where

c^{2} = \kappa p_0/\rho_0

. The loudest tone used has a pressure amplitude of

0.20

Pa. ((a)) Starting from the one-dimensional continuity equation

\partial\rho/\partial t + \partial(\rho u)/\partial x = 0

and the one-dimensional Euler equation

\rho\left(\partial u/\partial t + u\,\partial u/\partial x\right) = -\partial p/\partial x

, obtain the two linearised equations. Name the term you drop in each and the ratio that makes it small, and estimate that ratio for the loudest tone. ((b)) Eliminate

u'

between them to obtain

\partial^{2}p'/\partial t^{2} = c^{2}\,\partial^{2}p'/\partial x^{2}

, and say which step exchanges the order of two partial derivatives. ((c)) Evaluate

c

for this oxygen, and the wavelength of a

500

Hz tone in the duct.

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