2.7. Solution of wave equation for air in a tube filled with absorbent material

Ducts and tubes are often filled with absorbent material in order to minimize standing waves, such as in transmission-line loudspeaker enclosures or exhaust-pipe mufflers, for example. Let us now modify the one-dimensional wave equation in rectangular coordinates, Eq. (2.34), taking into account the thermal and viscous losses in the material
(P0∂∂x2−jωRf+ω2ρ0)p˜=0
image (2.114)
where in the steady state we have let ∂2/∂t2=−ω2 image and R f is the specific flow resistance per unit length of the absorptive material in rayls/m. For simplicity we are assuming that the resistance is constant for all frequencies. A more comprehensive treatment of sound in absorbent materials will be given in Section 7.6. Notice too that we have omitted the specific heat ratio γ because we are assuming that the heat conduction within the material is such that the pressure fluctuations are isothermal. We define a complex density by
ρ=ρ0+Rfjω,
image (2.115)
so that the wave equation simplifies to
(∂2∂x2+ω2c2)p˜=0,
image (2.116)
where
c=P0ρ.
image (2.117)
Hence the solution is
p˜(x)=p˜+e−jkx+p−ejkx,
image (2.118)
where the complex wave number is given by
k=ωc=ωρP0,
image (2.119)
and the characteristic impedance of the tube is
Zs=ρP0.
image (2.120)
In general, viscous or flow losses are dynamic and therefore associated with a change in the density of the medium whereas thermal conduction is static and therefore associated with a change in the bulk modulus. Viscous and thermal losses also occur in narrow unfilled tubes and these will be treated in some detail in Sections 4.22 to 4.24.
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