Elastic medium

The wave equation for elastic medium is a vector equation. It belongs to a class of equations of the type:

$${\bf L}{\bf u}=b{\partial^{2}{\bf u} \over \partial t^{2}}$$

where ${\bf u}$ is a vector and ${\bf L}$ is a matrix operator:

$${\bf L} = \sum^{3}_{i,j=1} {\bf A}_{ij}{\partial ^{2} \over \partial x_{i} \partial x_{j}}$$

${\bf A}_{ij}({\bf r})$ - a matrix.

If we develop the same considerations as with the scalar case, we shall obtain instead of equation (19) the vector equation

$${\bf N}{\bf u}_{{x_{1}^{\prime}}{x_{1}^{\prime}}}+ \ldots = 0$$

with a matrix

$${\bf N}= \sum {\partial \omega_{1} \over \partial x_{i}} {\p... ...ik} - b {({\partial \omega_{1} \over \partial t})}^{2} {\bf E}.$$

It is obvious that characteristics satisfy the equation  

$${\rm det}\ {\bf N}=0.$$ (23)

In general, equation (23) has three solutions. This is due to the fact that in a continuous one-phase medium where wave phenomena are described by second order equations, only three types of body waves (not more, but eventually less!) may exist.

Let us check the Lame's equation (isotropic elasticity):
 

$$(\lambda + \mu) {\rm grad}({\rm div}) {\bf u} + \mu \Delta {... ...{{\bf e}_{i}}= \rho {\partial^{2}{\bf u} \over \partial t^{2}}.$$ (24)

We may omit the terms with first derivatives of u . So we have the system:

$$(\lambda + \mu) {\partial \over \partial x_{j}} \left( \sum_... ...ts = \rho {\partial^{2}u_{j} \over \partial t^{2}},{\:}j=1,2,3.$$

It is easy to derive the correspondent system of characteristic equations. The first line is:

$$\left[ (\lambda + \mu) {\left( {\partial \tau \over \partial... ... x_{1}} {\partial \tau \over \partial x_{3}} v_{3} + \ldots = 0$$

where $v_{i}= {\partial^{2}u_{i} \over {\partial x_{1}^{`}}^{2}}$.

The matrix of the equation system may be written in the following form:  

$${\bf N} = (\lambda+ \mu) {\bf P} {\bf P}^T+(\mu \vert\nabla \tau\vert^2- \rho){\bf E}$$ (25)

where ${\bf P}$ is a column vector with the components $\partial \tau \over \partial {x_i}$.

Let us prove that the vector ${\bf P}$ is an eigenvector of ${\bf N}$:

$${\bf NP} = (\lambda + \mu) {\bf PP}^T {\bf P}+(\mu \vert\nabla \tau\vert^2- \rho) {\bf EP},$$

where ${\bf P}^T {\bf P}=\vert\nabla \tau\vert^2$, so  

$$\begin{array} {lll} {\bf NP} & = & [(\lambda + \mu)\vert\nab... ...mbda+2 \mu) \vert\nabla \tau\vert^2- \rho] {\bf P} .\end{array}$$ (26)

Therefore, the first eigenvalue is:

$$(\lambda + 2 \mu)\vert\nabla \tau \vert^2- \rho .$$

Let us consider a vector ${\bf Q} \perp {\bf P}$:

$${\bf P}^T {\bf Q}=0$$

 

$$\begin{array} {lll} {\bf NQ} & = & (\lambda + \mu) {\bf PP}^... ... & = & (\mu \vert\nabla \tau\vert^2- \rho) {\bf Q}. \end{array}$$ (27)

The latter is valid for any vector ${\bf Q} \perp {\bf P}$.Consequently, this eigenvalue is degenerated and it follows that

$$\vert{\bf N}\vert = {((\lambda+ 2 \mu) \vert\nabla \tau\vert^2- \rho) (\mu \vert\nabla \tau\vert^2- \rho)}^{2}.$$

So we may satisfy the equation (19) in both cases:

$$\vert\nabla \tau\vert^2={1\over {v_{p}}^2} ,{\:}v_{p}=\sqrt{{\lambda + 2 \mu} \over \rho}$$

$$\vert\nabla \tau\vert^2={1\over {v_{s}}^2} ,{\:}v_{s}=\sqrt{\mu \over \rho}.$$

This means that in isotropic elastic media, there are only two types of body waves which move with velocities v_p and v_s, respectively. In the general case of anisotropic elastic media, there are three families of characteristics and correspondently three body waves: quasi-P, quasi-SV, and quasi-SH. In Maxwell's equations of electrodynamics, there is only one characteristic which propagates with the velocity of light.

We may say that geometrical seismics (as well as geometrical optics and geometrical acoustics) describes the propagation of discontinuities. We have studied here only the pure geometrical aspect of wave propagation, but it will be shown that our results are also true for the amplitudes of waves.