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Equation 26 is a wave equation that combines elastic strains,
electric displacements, and entropy. It describes how, in a given medium,
a traveling wavetype is typically neither purely elastic, nor purely
electromagnetic, nor purely thermal. Instead, one wave includes all such
attributes. However it is useful to term them quasi-elastic when
most of the energy is transferred by strain changes, or
quasi-electric when most of the energy propagates in form of electromagnetic
waves.
Solving the eigenvalue equation 28, we obtain as eigenvalues the
propagation velocities for the elastic waves, the electromagnetic and
the thermal wave. Although is a 9 component tensor there are
only three non-zero eigenvalues, corresponding to three propagating elastic wavetypes.
The electromagnetic part will typically have two non-zero eigenvalues,
corresponding to two propagating wavetypes. The eigenvalue for the thermal wave
will be complex.
Consequently we can also calculate propagation velocities and particle motion for
the electromagnetic wavetypes and the heat wave.

In order to calculate the elastic displacements we have to extract the three
non-zero
eigenvalues (*v*_{1},*v*_{2},*v*_{3}) and eigenvectors () from the coupled system.
We can combine them as follows:

| |
(29) |

to produce the
purely elastic wave equation which uniquely describes the propagation
of strains with the correct (coupled) velocity.
This purely elastic equation can
be turned into a generalized eigenvalue problem for the elastic displacements
*u*, as follows:
| |
(30) |

The eigenvalues and eigenvectors represent now propagation velocity and particle
motion respectively and are physically meaningful.

**APPENDIX B: SYMBOLS**

The following definitions translate the symbols used in the notation into commonly used terms:

- = elastic stress (second rank tensor)
- = electric stress (``electric field'', first rank tensor, vector)
- = heat stress (``Temperature'', zeroth rank tensor, scalar)
- = elastic strain (second rank tensor)
- = electric strain (``electric displacement'', first rank tensor, vector)
- = heat strain (``Entropy'', zeroth rank tensor, scalar)
- = generalized stiffness matrix
- = symmetric spatial derivative matrix

The matrix notation for is
| |
(31) |

The matrix formulation for is

| |
(32) |

and are denoted as
| |
(33) |

| |
(34) |

**APPENDIX C**

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Stanford Exploration Project

1/13/1998