Introduction to Acoustics

as well as symmetry considerations, require for a dilatational
wave propagating in the x-direction, that the
off-diagonal elements of the stress tensor vanish. The
diagonal elements are given by
σxx = ρc 2 1F′ (x − c1t) , (3.195)
σyy = σzz = ρ � c 2 1 − 2c2 � ′
2 F (x − c1t) . (3.196)
Here the primes denote derivatives with respect to the
total argument.
The divergence of the displacement field in a shear
wave is zero, so a plane shear wave must cause a displacement
perpendicular to the direction of propagation.
3.7 Attenuation of Sound
Plane waves of constant frequency propagating through
bulk materials have amplitudes that typically decrease
exponentially with increasing propagation distance,
such that the magnitude of the complex pressure amplitude
varies as
| ˆp(x)|=|ˆp(0)|e −αx . (3.199)
The quantity α is the plane wave attenuation coefficient
and has units of nepers per meter (Np/m); it is an intrinsic
frequency-dependent property of the material. This
exponential decrease of amplitude is called attenuation
or absorption of sound and is associated with the transfer
of acoustic energy to the internal energy of the material.
(If | ˆp| 2 decreases to a tenth of its original value, it
is said to have decreased by 10 decibels (dB), so an attenuation
constant of α nepers per meter is equivalent
to an attenuation constant of [20/(ln 10)]α decibels per
meter, or 8.6859α decibels per meter.)
3.7.1 Classical Absorption
The attenuation of sound due to the classical processes
of viscous energy absorption and thermal conduction
is derivable [3.33] from the dissipative wave equation
(3.101) given previously for the acoustics mode.
Dissipative processes enter into this equation through
a parameter δcl, which is defined by (3.102). To determine
the attenuation of waves governed by such
a dissipative wave equation, one sets the perturbation
pressure equal to
pac = P e −iωt e ikx , (3.200)
Basic Linear Acoustics 3.7 Attenuation of Sound 49
Shear waves are therefore transverse waves. Equation
(3.83), when considered in a manner similar to that
described above for the wave equation for waves in fluids,
leads to the conclusion that plane shear waves must
propagate with a speed c2. A plane shear wave polarized
in the y-direction and propagating in the x-direction will
have only a y-component of displacement, given by
ξy = F(x − c2t) . (3.197)
The only nonzero stress components are the shear
stresses
σyx = ρc 2 2 F′ (x − c2t) = σxy . (3.198)
where P is independent of position and k is a complex
number. Such a substitution, with reasoning such
as that which leads from (3.171)to(3.172), yields an algebraic
equation that can be nontrivially (amplitude not
identically zero) satisfied only if k satisfies the relation
k 2 = ω2 2δclω3 + i
c2 c3 . (3.201)
The root that corresponds to waves propagating in the
+x-direction is that which evolves to (3.189) in the limit
of no absorption, and for which the real part of k is
positive. Thus to first order in δcl, one has the complex
wavenumber
k = ω δclω2 + i
c c3 . (3.202)
The attenuation coefficient is the imaginary part of this,
in accordance with (3.199), so
δclω2 αcl =
c3 . (3.203)
This is termed the classical attenuation coefficient for
acoustic waves in fluids and is designated by the subscript
‘cl’. The distinguishing characteristic of this
classical attenuation coefficient is its quadratic increase
with increasing frequency.
The same type of frequency dependence is obeyed
by the acoustic and shear wave modes for the Biot model
of porous media in the limit of low frequencies. From
(3.149) one derives
αac = τB
ω
2cac
2 , (3.204)
Part A 3.7