Introduction to Acoustics

50 Part A Propagation of Sound
Part A 3.7
and, from (3.157), one derives
αsh = csh(ρ22 + ρ12) 2
ω
2GBb
2 . (3.205)
3.7.2 Relaxation Processes
For many substances, including air, sea water, biological
tissues, marine sediments, and rocks, the variation of
the absorption coefficient with frequency is not quadratic
[3.48–50], and the classical model is insufficient to predict
the magnitude of the absorption coefficient. The
substitution of a different value of the bulk viscosity
is insufficient to remove the discrepancy, because this
would still yield the quadratic frequency dependence.
The successful theory to account for such discrepancies
in air and sea water and other fluids is in terms of relaxation
processes. The physical nature of the relaxation
processes vary from fluid to fluid, but a general theory in
terms of irreversible thermodynamics [3.51–53] yields
appropriate equations.
The equation of state for the instantaneous (rather
than equilibrium) entropy is written as
s = s(u,ρ −1 ,ξ) , (3.206)
where ξ represents one or more internal variables. The
differential relation of (3.13) is replaced by
T ds = du + pdρ −1 + �
Aν dξν , (3.207)
ν
where the affinities Aν are defined by this equation.
These vanish when the fluid is in equilibrium with
a given specified internal energy and density. The pressure
p here is the same as enters into the expression
(3.15) for the average normal stress, and the T is the same
as enters into the Fourier law (3.16) of heat conduction.
The mass conservation law (3.2) and the Navier–Stokes
equation (3.24) remain unchanged, but the energy equation,
expressed in (3.25) in terms of entropy, is replaced
by the entropy balance equation
ρ Ds
Dt +∇·q
T = σs . (3.208)
Here the quantity σs, which indicates the rate of irreversible
entropy production per unit volume, is given
by
Tσs =µB(∇·v) 2 + 1 �
2 µ φ 2 κ
ij +
T (∇T)2
+ ρ �
ν
ij
Dξν
Aν . (3.209)
Dt
One needs in addition relations that specify how the
internal variables ξν relax to their equilibrium values.
The simplest assumption, and one which is substantiated
for air and sea water, is that these relax independently
according to the rule [3.54]
Dξν
Dt =−1
� �
ξν − ξν,eq . (3.210)
τν
The relaxation times τν that appear here are positive and
independent of the internal variables.
When the linearization process is applied to the nonlinear
equations for the model just described of a fluid
with internal relaxation, one obtains the set of equations
∂ρ ′
∂t + ρ0∇·v ′ = 0 ,
∂v
ρ0
(3.211)
′
∂t =−∇p′ + (1/3µ + µB)
× ∇(∇·v ′ ) + µ∇ 2 v ′ , (3.212)
∂s
ρ0T0
′
∂t = κ∇2 T ′ , (3.213)
ρ ′ = 1
c2 p′ � �
ρβT
− s
cp 0
′
+ � �
ξ
ν
′ ν − ξ′ �
ν,eq aν , (3.214)
T ′ � �
Tβ
= p
ρcp 0
′ � �
T
+ s
cp 0
′
+ � �
ξ ′ ν − ξ′ �
ν,eq bν , (3.215)
∂ξ ′ ν
ν
∂t =−1
τν
�
ξ ′ ν − ξ′ �
ν,eq , (3.216)
ξ ′ ν,eq = mνs ′ + nν p ′ . (3.217)
Here aν, bν, mν, andnν are constants whose values
depend on the ambient equilibrium state.
For absorption of sound, the interest is in the acoustic
mode, and so approximations corresponding to those
discussed in the context of (3.93) through (3.101) can
also be made here. The quantities aν and bν are treated
as small, so that one obtains, to first order in ɛ and in
these quantities, a wave equation of the form
∇ 2 pac − 1
c2 ∂2 ∂t2 �
pac − 2δcl
c2 ∂pac
∂t
−2 �
ν
(∆c)ν
τν
c
�
∂pν
= 0 .
∂t
(3.218)