Introduction to Acoustics

Because any vector field may be decomposed into
a sum of two fields, one with zero curl and the other with
zero divergence, one can set the displacement vector ξ
to
ξ =∇Φ +∇× Ψ (3.81)
in terms of a scalar and a vector potential. This expression
will satisfy (3.77) identically, provided the two
potentials separately satisfy
∇ 2 Φ − 1
c 2 1
∇ 2 Ψ − 1
c 2 2
∂2Φ = 0 , (3.82)
∂t2 ∂2Ψ = 0 . (3.83)
∂t2 Both of these equations are of the form of the simple
wave equation of (3.74). The first corresponds to longitudinal
wave propagation, and the second corresponds
to shear wave propagation. The quantities c1 and c2 are
referred to as the dilatational and shear wave speeds,
respectively.
If the displacement field is irrotational, so that the
curl of the displacement vector is zero (as is so for sound
in fluids), then each component of the displacement field
satisfies (3.82). If the displacement field is solenoidal,
so that its divergence is zero, then each component of
the displacement satisfies (3.83).
3.3.5 Linearized Equations
for a Viscous Fluid
For the restricted but relatively widely applicable case
when the Navier–Stokes–Fourier equations are presumed
to hold and the characteristic scales of the
ambient medium and the disturbance are such that
the ambient flow is negligible, and the coefficients are
idealizable as constants, the linear equations have the
form appropriate for a disturbance in a homogeneous,
time-independent, non-moving medium, these equations
being
∂ρ ′
∂t + ρ0∇·v ′ = 0 , (3.84)
∂v
ρ0
′
∂t =−∇p′ � �
1
+ µ + µB ∇
3 � ∇·v ′�
+ µ∇ 2 v ′ , (3.85)
∂s
ρ0T0
′
∂t = κ∇2 T ′ , (3.86)
ρ ′ = 1
c 2 p′ −
� �
ρβT
cp
s
0
′ , (3.87)
Basic Linear Acoustics 3.3 Equations of Linear Acoustics 37
T ′ � �
Tβ
=
ρcp
0
p ′ � �
T
+ s
cp 0
′ . (3.88)
The primes on the perturbation field variables are needed
here to distinguish them from the corresponding ambient
quantities.
3.3.6 Acoustic, Entropy,
and Vorticity Modes
In general, any solution of the linearized equations
for a fluid with viscosity and thermal conductivity
can be regarded [3.30, 32–34] asasumofthreebasic
types of solutions. The common terminology for
these is: (i) the acoustic mode, (ii) the entropy mode,
and (iii) the vorticity mode. Thus, with appropriate
subscripts denoting the various fundamental mode
types, one writes the fluid velocity perturbation in the
form,
v ′ = vac + vent + vvor , (3.89)
with similar decompositions for the other field variables.
The decomposition is intended to be such that, for waves
of constant frequency, each field variable’s contribution
from any given mode satisfies a partial differential equation
that is second order [rather than of some higher
order as might be derived from (3.84)–(3.88)] in the
spatial derivatives. That such a decomposition is possible
follows from a theorem [3.35, 36] that any solution
of a partial differential equation of, for example, the
form,
� �� �� � 2 2 2
∇ + λ1 ∇ + λ2 ∇ + λ3 ψ = 0 , (3.90)
can be written as a sum,
ψ = ψ1 + ψ2 + ψ3 , (3.91)
where the individual terms each satisfy second-order
differential equations of the form,
� � 2
∇ + λi ψi = 0 . (3.92)
This decomposition is possible providing no two of
the λi are equal. [In regard to (3.90), one should
note that each of the three operator factors is
a second-order differential operator, so the overall
equation is a sixth-order partial differential equation.
One significance of the theorem is that one
has replaced the seemingly formidable problem of
solving a sixth-order partial differential equation by
that of solving three second-order partial differential
equations.]
Part A 3.3