The coupled system in terms of strains

Combining Equations 21, 23 and 25, we end up with a coupled system of partial differential equations. In contrast to Equation 18, the wave equation corresponds to the linear relationship between ``generalized stresses'' and ``generalized strains'', as given in Equation 7.  

$$\begin{array} {c c c c c c c} - \displaystyle \mathop{\mbox{... ... \epsilon_3} ) & ~= & ~\omega^2~{\bf \epsilon_3} \\ \end{array}$$ (26)

However in this case, the spatial derivative operator in matrix form K has to be in uncompacted notation

$$\displaystyle \mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}} ~ = ... ... 0 & 0& {k_y \over 2} & {k_x \over 2} & {k_y \over 2} & z \cr}$$ (27)

in order to cast the problem in eigenvalue form

 

$$\displaystyle \mathop{\mbox{${\bf C}$}}_{\mbox{$\sim$}} ~{\bf \epsilon}~=~v^2~{\bf \epsilon} ,$$ (28)

which is the generalized Christoffel equation in elastic strain notation. The eigenvalues ($v^2~=~{{\omega^2}\over{{\bf k}^2}}$) of this equation are again propagation velocities of plane waves in the medium. The eigenvectors are directly related to the particle motion. With this formulation, in contrast to the displacement formulation, we have to carry out an additional computational step in order to obtain physically meaningful displacements, given the strains for a wave traveling in a particular direction.