Equations of motion

Here I follow the derivation procedure in Slawinski (2003). For anisotropic inhomogeneous media, we start with Cauchy's equations of motion,

$$\rho(\mathbf{x})\frac{\partial^2u_i}{\partial t^2}= \sum_{j=1}^3\frac{\partial\sigma_{ij}}{\partial x_j} , \qquad i\in\{1,2,3\} .$$ (1)

These equations are combined with the stress-strain equations,

$$\sigma_{ij}(\mathbf{x})=\sum_{k=1}^3\sum_{l=1}^3 c_{ijkl}(\mathbf{x}) \varepsilon_{kl}(\mathbf{x}) ,\qquad i,j\in\{1,2,3\} .$$ (2)

Then we proceed to use the definition of the strain tensor,

$$\varepsilon_{kl}=\frac12\left(\frac{\partial u_k}{\partial x_l} +\frac{\partial u_l}{\partial x_k}\right) ,$$ (3)

to obtain equations of motion in inhomogeneous anisotropic media, which are

$$\rho(\mathbf{x})\frac{\partial^2 u_i}{\partial t^2} = \sum_{j=1}^3\frac{\partial}{\partial x_j}\left[ \frac12\sum_{k=1}... ...ac{\partial u_k}{\partial x_l}+ \frac{\partial u_l}{\partial x_k}\right)\right]$$

$$= \frac12\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3 \frac{\partial c_{ijk... ...eft( \frac{\partial u_k}{\partial x_l}+\frac{\partial u_l}{\partial x_k}\right)$$

$$\hspace*{0.3in}+\frac12\sum_{j=1}^3\sum_{k=1}^3\sum_{l=1}^3 c_{ij... ...rac{\partial^2 u_l}{\partial x_j\partial x_k} \right) .\qquad i\in\{1,2,3\} .$$ (4)

These equations of motion, due to symmetries in the elasticity tensor in $k$ and $l$, reduce to

$$\rho(\mathbf{x})\frac{\partial^2 u_i}{\partial t^2} = \sum_{k=1}^3\sum_{l=1}^3 D_{ikl} \frac{\partial u_k}{\partial x_l... ...{x})\frac{\partial^2 u_k}{\partial x_j \partial x_l} ,\qquad i\in\{1,2,3\} ,$$ (5)

$$\quad D_{ikl}=\sum_{j=1}^3 \frac{\partial c_{ijkl}(\mathbf{x})}{\partial x_j} .$$

This is equivalent to the elastic wave equation studied by Stolk and de Hoop (2002), but without body forces included in the equation.