Expressing waves in terms of wave amplitudes

For a plane wave with horizontal slownesses p₁ and p₂ and frequency $\omega$ the displacement vector ${\bf u}$ at some point in 3-space ${\bf x}$ is given by,

$${\bf u} = \sum_{n=1}^6 f(n) {\bf a}(n) e^{i \omega ( t - {\bf x} \cdot {\bf p}(n) )}$$

$${\bf p}(n) = ( p_1, p_2, p_3(n) )^T$$

The summation is over the six different wavetypes in the medium. These correspond to the six different roots of the sextic polynomial in p₃. They have slownesses vectors ${\bf p}(n)$ , polarization vectors ${\bf a}(n)$and amplitudes f(n). The vertical slownesses and displacement vectors are calculated by solving the Christoffel equation.

The strains in the medium are given by,

$$\epsilon_{jk} = \frac{1}{2} \left( \frac{\partial u_j}{\partial x_k} + \frac{\partial u_k}{\partial x_j} \right)$$

So the strains due to the n^th wavetype are,

$$\epsilon^n_{jk} = \frac{1}{2} f(n) e^{i \omega ( t - {\bf x} \cdot {\bf p}(n)) } [ a_j(n) p_k(n) + a_k(n) p_j(n) ]$$

And the stresses are given by,

$$\sigma^n_{ef} = c_{efjk} \epsilon^n_{jk}$$