THE COUPLED SYSTEM

Since Equations 8, 13 and 17 cannot be solved independently, we can, by combining them, derive a coupled system of partial differential equations. In order to investigate the nature of its solutions we can set up an analog to the Christoffel equation (Auld, 1981) by looking at plane wave solutions in a homogeneous medium. To do so, we can Fourier transform the system of equations in space and time (${\partial \over{\partial t}} = -i~\omega ; {\partial \over{\partial x}} =i~k_x~;~etc.$). This operation results in an algebraic system of equations for the unknown displacements

 

$$\pmatrix{- \rho^{-1} ~ \displaystyle \mathop{\mbox{${\bf K}$... ...ix{ {\bf u_1} \cr {\bf \epsilon_2 } \cr {\bf \epsilon_3 }\cr}$$ (18)

The spatial derivative operators in the form of matrices $$\mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}}$$ and $$\mathop{\mbox{${\bf R}$}}_{\mbox{$\sim$}}$$ are given in the Appendix. Dividing Equation 18 by the square magnitude of the spatial wave number vector $\vert{\bf k}\vert^2$, we get the eigenvalue equation  

$$( \displaystyle \mathop{\mbox{${\bf L}$}}_{\mbox{$\sim$}} ~ ... ...\bf C}$}}_{\mbox{$\sim$}} )~{\bf \epsilon}~=~v^2~{\bf \epsilon}$$ (19)

which is the ``generalized'' Christoffel equation. The eigenvalues ($v^2~=~{{\omega^2}\over{{\bf k}^2}}$) of this equation are propagation velocities of plane waves in the medium. The eigenvectors are directly related to the ``particle motions''. Since $$\mathop{\mbox{${\bf L}$}}_{\mbox{$\sim$}} ~ \displaystyle \mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}}$$ is dependent on direction cosines, we can expect that propagation velocity and particle motion are dependent on the propagation direction. The dispersion relations become frequency dependent when we include heat effects or if the medium is electrically conducting. At the zero frequency limit, the contributions from heat and conductivity vanish. This result is the same as if the medium were now purely piezoelectric.