Introduction to Acoustics

The auxiliary internal variables that enter here satisfy
the supplemental relaxation equations
∂pν
∂t =−1 (pν − pac) . (3.219)
τν
Here the notation is such that the pν are given in terms
of previously introduced quantities by
pν = ξ ′ ν /nν . (3.220)
The sound speed increments that appear in (3.218)
represent the combinations
(∆c)ν = 1/2aνnνc 2 . (3.221)
Equations (3.218) and(3.219) are independent of
the explicit physical interpretation of the relaxation processes.
Insofar as acoustic propagation is concerned, any
relaxation process is characterized by two parameters,
the relaxation time τν and the sound speed increment
(∆c)ν. The various parameters that enter into the irreversible
thermodynamics formulation of (3.214) through
(3.216) affect the propagation of sound only as they enter
into the values of the relaxation times and of the sound
speed increments. The replacement of internal variables
by quantities pν with the units of pressure implies no
assumption as to the precise nature of the relaxation
process. [An alternate formulation for a parallel class
of relaxation processes concerns structural relaxation
[3.55]. The substance can locally have more than one
state, each of which has a different compressibility. The
internal variables are associated with the deviations of
the probabilities of the system being in each of the states
from the probabilities that would exist were the system
in quasistatic equilibrium. The equations that result are
mathematically the same as (3.218)and(3.219).]
The attenuation of plane waves governed by (3.218)
and (3.219) is determined when one inserts substitutions
of the form of (3.200)forpac and the pν. The relaxation
equations yield the relations
1
ˆpν = ˆpac . (3.222)
1 − iωτν
These, when inserted into the complex amplitude
version of (3.218), yield the dispersion relation
k 2 = ω2
c2 �
1 + i 2ωδcl �
�
2(∆c)ν iωτν
+ .
c2 c 1 − iωτν
ν
(3.223)
To first order in the small parameters, δcl and (∆c)ν,this
yields the complex wavenumber
k = ω
�
1 + i
c
ωδcl �
�
(∆c)ν iωτν
+ , (3.224)
c2 c 1 − iωτν
ν
1
0.8
0.6
0.4
0.2
(αv λ)/(αv λ)m
Basic Linear Acoustics 3.7 Attenuation of Sound 51
0
0.06 0.1 0.2 0.4 0.6 1 2 4 6 10 20 40
f/fv
Fig. 3.11 Attenuation per wavelength (in terms of characteristic
parameters) for propagation in a medium with a single
relaxation process. Here the wavelength λ is 2πc/ω, and
the ratio f/ fν of frequency to relaxation frequency is ωτν.
The curve as constructed is independent of c and (∆c)ν
which corresponds to waves propagating in the +xdirection.
The attenuation coefficient, determined by the
imaginary part of (3.224), can be written
α = αcl + �
αν . (3.225)
ν
The first term is the classical attenuation determined by
(3.203). The remaining terms correspond to the incremental
attenuations produced by the separate relaxation
processes, these being
αν = (∆c)ν
c 2
ω2τν . (3.226)
1 + (ωτν) 2
Any such term increases quadratically with frequency
at low frequencies, as would be the case for classical
absorption (with an increased bulk viscosity), but
it approaches a constant value at high frequencies.
The labeling of the quantities (∆c)ν as sound speed
increments follows from an examination of the real part
of the complex wavenumber, given by
kR = ω
c
�
1 − �
ν
(∆c)ν
c
(ωτν) 2
1 + (ωτν) 2
�
. (3.227)
The ratio, of ω to kR, is identified as the phase velocity
vph of the wave. In the limit of low frequencies, the phase
velocity predicted by (3.227) is the quantity c, whichis
the sound speed for a quasi-equilibrium process. In the
limit of high frequencies, however, to first order in the
(∆c)ν, the phase velocity approaches the limit
vph → c + �
(∆c)ν . (3.228)
ν
Consequently, each (∆c)ν corresponds to the net increase
in phase velocity of the sound wave that occurs
Part A 3.7