Introduction to Acoustics

ω ′ = 0, and this is consistent with kR being odd in ω and
representable locally as a power series. Validity also requires
that kI/ω2 be finite at ω = 0, which is consistent
with the relaxation models mentioned in the section below.
The existence of the integral also places a restriction
on the asymptotic dependence of kR as ω →∞.
A possible consequence of the last relation is that
�
kI(ω)
lim
ω→∞ ω2 �
� �
kI(ω)
=
+ 2
π
ω 2
�
0
∞
0
[kR(ω ′ )/ω ′ ]−[kR(ω ′ )/ω ′ ]0
ω ′2
dω ′ . (3.253)
The validity of this requires that the integral exists, which
is so if the phase velocity approaches a constant at high
frequency. An implication is that the attenuation must
approach a constant multiplied by ω 2 at high frequency,
where the constant is smaller than that at low frequency
if the phase velocity at high frequency is higher than that
at low frequency.
Imaginary Part Within the Integrand
Analogous results, only with the integration over the
imaginary part of k, result when k/ω is replaced by k in
(3.247). Doing so yields
�∞
kI(ω)
2Pr � �� � dω
ω2 − ω2 1 ω2 − ω2 2
0
π
=
ω2 1 − ω2 �
kR(ω1)
−
ω1 2
kR(ω2)
�
, (3.254)
ω2
which in turn yields
kR(ω)
ω
= kR(ω1)
−
ω1
2(ω2 1 − ω2 )
π
�∞
kI(ω
×Pr
′ )
(ω ′2 − ω2 1 )(ω′2 − ω2 ) dω′ . (3.255)
0
Then, taking of the limit ω1 → 0, one obtains
kR(ω)
ω
� �
kR(ω)
=
ω
0
+ 2ω2
π Pr
�∞
0
kI(ω ′ )
ω ′2 (ω ′2 − ω 2 ) dω′ .
(3.256)
Basic Linear Acoustics 3.7 Attenuation of Sound 55
This latter expression requires that kI/ω 2 be integrable
near ω = 0.
Attenuation Proportional to Frequency
Some experimental data for various materials suggest
that, for those materials, kI is directly proportional to ω
over a wide range of frequencies. The relation (3.256)is
inapplicable if one seeks the corresponding expression
for phase velocity. Instead, one uses (3.255), treating ω1
as a parameter. If one inserts
kI(ω) = Kω (3.257)
into (3.255), the resulting integral can be performed
analytically, with the result
kR(ω) kR(ω1)
= −
ω ω1
2
� �
ω
K ln . (3.258)
π ω1
(The simplest procedure for deriving this is to replace
the infinite upper limit by a large finite number and
then separate the integrand using the method of partial
fractions. After evaluation of the individual terms, one
takes the limit as the upper limit goes to infinity, and
discovers appropriate cancelations.) The properties of
the logarithm are such that the above indicates that the
quantity
kR(ω) 2
+ K ln(ω) = constant (3.259)
ω π
is independent of ω. This deduction is independent of
the choice of ω1, but the analysis does not tell one what
the constant should be. A concise restating of the result
is that there is some positive number ω0, such that
kR(ω) 2
� �
ω0
= K ln . (3.260)
ω π ω
The result is presumably valid at best only over the range
of frequencies for which (3.257) is valid. Since negative
phase velocities are unlikely, it must also be such that
the parameter ω0 is above this range. This approximate
result also predicts a zero phase velocity in the limit of
zero frequency, and this is also likely to be unrealistic.
But there may nevertheless be some range of frequencies
for which both (3.257) and(3.260) would give a good
fit to experimental data.
3.7.5 Attenuation of Sound in Air
In air, the relaxation processes that affect sound attenuation
are those associated with the (quantized) internal
vibrations of the diatomic molecules O2 and N2. The
ratio of the numbers of molecules in the ground and
Part A 3.7