Introduction to Acoustics

Because the friction associated with the motion of
the fluid with respect to the solid matrix is inherently
nonconservative (i. e., energy is lost), the original version
(3.119) of Hamilton’s principle does not apply. The
applicable extension includes a term that represents the
time integral of the virtual work done by nonconservative
forces during a variation. If such nonconservative
forces are taken to be distributed over the volume, with
the forces per unit volume on the fluid denoted by �
and those on the solid denoted by �, then the modified
version of Hamilton’s principle is
����
(δÄ +�·δU + �·δu) dx dy dz dt = 0.
Moreover, one infers that
(3.131)
� =−� , (3.132)
since the virtual work associated with friction between
the solid matrix and the fluid must vanish if the two are
moved together.
The Lagrange–Euler equations that result from the
above variational principle are
� �
∂ ∂Ä
+
∂t ∂ (∂Ui/∂t)
�
�
∂ ∂Ä
∂x
j
j ∂ � �
� = �i ,
∂Ui/∂x j
(3.133)
� �
∂ ∂Ä
+
∂t ∂ (∂ui/∂t)
�
�
∂ ∂Ä
∂x
j
j ∂ � �
� = �i .
∂ui/∂x j
(3.134)
Biot, in his original exposition, took the internal
distributed forces to be proportional to the relative velocities,
reminiscent of dashpots, so that
� �
∂Ui ∂ui
�i =−�i =−b − . (3.135)
∂t ∂t
Here the quantity b can be regarded as the apparent
dashpot constant per unit volume. This form is such that
the derived equations, in the limit of vanishingly small
frequencies, are consistent with Darcy’s law [3.44] for
steady fluid flow through a porous medium.
The equations that result from this formulation,
when written out explicitly, are
∂2 ∂t2 (ρ11u + ρ12U) −∇(A ′ ∇·u + Q∇·U)
+∇× [N (∇×u)] = b ∂
(U − u) , (3.136)
∂t
Basic Linear Acoustics 3.4 Variational Formulations 43
∂2 ∂t2 (ρ12u + ρ22U) −∇(Q∇·u + R∇·U)
=−b ∂
(U − u) , (3.137)
∂t
with the abbreviation A ′ = A + 2N. Here, and in what
follows, it is assumed that the various material constants
are independent of position, although some of the derived
equations may be valid to a good approximation
even when this is not the case.
3.4.3 Disturbance Modes in a Biot Medium
Disturbances that satisfy the equations derived in the
previous section can be represented as a superposition of
three basic modal disturbances. These are here denoted
as the acoustic mode, the Darcy mode, and the shear
mode, and one writes
u = uac + uD + ush , (3.138)
U = Uac +UD +Ush . (3.139)
At low frequencies, the motion in the acoustic mode and
in the shear mode is nearly such that the fluid and solid
displacements are the same,
Uac ≈ uac ; Ush ≈ ush . (3.140)
The lowest-order (in frequency divided by the dashpot
parameter b) approximation results from taking the sum
of (3.136)and(3.137) and then setting u = U, yielding
∂
ρeq
2
∂t 2 U −∇(BV,B∇·U) +∇× [GB (∇×U)] = 0 ,
with the abbreviations
(3.141)
ρeq = ρ11 + 2ρ12 + ρ22 , (3.142)
BV,B = A + 2N + 2Q + R , (3.143)
GB = N , (3.144)
for the apparent density, bulk modulus, and shear modulus
of the Biot medium. The same equation results for
the solid displacement field u in this approximation.
Acoustic Mode
For the acoustic mode, the curl of each displacement
field is zero, so
∂
ρeq
2
∂t2 U −∇�BV,B∇·U � = 0 . (3.145)
One can identify an apparent pressure disturbance associated
with this mode, so that
pac ≈−BV,B∇·U , (3.146)
Part A 3.4