ELASTIC WAVES

It is now presumed that the reader has a general knowledge of classical elasticity theory. Few textbooks, if any, develop the special subject of stratified media which is so important in seismology. Many papers on that subject may be found in the Bulletin of the Seismological Society of America (BSSA). For those readers unfamiliar with the BSSA, we now present the results of applying the general methods of this chapter to the equations of isotropic elasticity.

The conventions in elasticity are (u, w) displacements in x and z directions, $\tau$ is the stress matrix, $\lambda$ and $\mu$ are Lame's constants and $\rho$ is density. Hooke's law and Newton's law with $e^{-i\omega t}$ time dependence leads to

$$\frac{\partial}{\partial z}\, \left[ \begin{array} {l} U \\ ... ...array} {l} U\\ \tau_{zz} \\ W \\ \tau_{zx}\end{array}\right]$$ (1)

where

$$\gamma \eq \partial_{x} \frac{4\mu (\lambda + \mu)}{(\lambda + 2\mu)} \partial_{x}$$ (2)

Define also

$$\begin{array} {l} \alpha^{2} \eq \frac{\lambda + 2\mu}{\rho... ... \eq \frac{-\omega^{2}}{\beta^{2}} - 2\partial_{xx} \end{array}$$ (3)

If material properties do not vary in the x direction, we have the row eigenvector transformation $\mbox{\bf R}$ to up- and downgoing wave variables.

$$\left[ \begin{array} {l} p^{+} \\ s^{+} \\ p^{-} \\ s^{-}... ...ray} {l} u \\ \tau_{zz} \\ w \\ \tau_{zx} \end{array}\right]$$ (4)

and the column eigenvector inverse transform $\mbox{\bf C}$

$$\left[ \begin{array} {l} u \\ \tau_{zz} \\ w \\ \tau_{zx}... ...ray} {l} p^{+} \\ s^{+} \\ p^{-} \\ s^{-} \end{array}\right]$$ (5)

where

$$\Lambda = \left[ \begin{array} {llll} m & & & \\ & n & & \\ & & -m & \\ & & & -n \end{array} \right]$$

The matrices partition nicely into $2 \times 2$ blocks. The reader may verify that $\mbox{\bf CR} = \mbox{\bf RC} = \mbox{\bf I}$ and $\mbox{\bf C}\mbox{\boldmath$\Lambda$}\mbox{\bf R} = \mbox{\bf A}$.