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 v2. A method for finding the three solutions is
discussed in the next subsection.