We will first take a Fourier transform of (finaleom) in the time domain, equivalent to assuming a time dependence of the form . (Strictly speaking we should now introduce new notation for the variables that follow to account for the differences between the time-dependent coefficients and the Fourier coefficients. But we will not refer further to the time-dependent coefficients in this paper, so no confusion should arise if we use the same notation from now on for the Fourier coefficients.) Then, (finaleom) becomes

-^2q_11 & q_12 & q_13 q_12 & q_22 & q_23 q_13 & q_23 & q_33 u_i U^(1)_i U^(2)_i = _ij,j - p_,i^(1) - p_,i^(2) , where

q_11 &=& _11 + i(b_12+b_13), q_12 &=& _12 - ib_12, etc. It is also convenient to notice that

x_iu_i U^(1)_i U^(2)_i = e U^(1)_i,i U^(2)_i,i = 1 & & 1 & 1(1-v^(2))^(1) & 1 & & 1v^(2)^(2) e - ^(1) - ^(2)

e
- ^(1)
- ^(2) ,
which will permit us to write the final equation in terms of the
macroscopic strain and fluid contents *e*, , and
.The final equality in (variablechange) defines the matrix
, which we need again later in the analysis.

Taking the divergence of (fouriertime), then substituting
(variablechange) and (constitutiverelation), and finally taking
the spatial Fourier transform (having wavenumber *k*)
gives the complex eigenvalue problem
associated with wave propagation:

K_u + 43& B^(1)K_u & B^(2)K_u -a_12a_33/D & (a_11a_33-a_13^2)/D & a_12a_13/D

-a_13a_22/D & a_12a_13/D & (a_11a_22-a_12^2)/D e -^(1) -^(2) = v^2()1 & & & 1(1-v^(2))^(1) & & & 1v^(2)^(2) q_11 & q_12 & q_13 q_12 & q_22 & q_23 q_13 & q_23 & q_33 1 & & 1 & 1(1-v^(2))^(1) & 1 & & 1v^(2)^(2) e -^(1) -^(2) , where the eigenvalue has the physical significance of being the square of the complex wave velocity. With obvious definitions for the matrices , ,and , while was previously defined in (variablechange), we rewrite (eigenvalueproblem) as

e -^(1) -^(2) = v^2() e -^(1) -^(2) , and then, in terms of these matrices, the dispersion relation determining at all angular frequencies is

(- v^2()) = 0.
This is a determinant of complex numbers that must
be solved for *v ^{2}*. A method for finding the three solutions is
discussed in the next subsection.

4/20/1999