Derivation of the dispersion relation

We will first take a Fourier transform of (finaleom) in the time domain, equivalent to assuming a time dependence of the form $\exp{(-i\omega t)}$. (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, $\zeta^{(1)}$, and $\zeta^{(2)}$.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 $v^2(\omega) = \omega^2/k^2$ 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 $v^2(\omega)$ at all angular frequencies $\omega$ is

(- v^2()) = 0.   This is a $3\times 3$ determinant of complex numbers that must be solved for v². A method for finding the three solutions is discussed in the next subsection.