Next: Elastic'' slowness surfaces Up: Karrenbach: coupled wave propagation Previous: THE COUPLED SYSTEM

# CHARACTERISTIC SURFACES

A wave traveling in a homogeneous medium described by Equation 18 propagates with a velocity corresponding to one of the eigenvalues of the system. The propagation velocity depends on the prescribed direction. Such a wave generally has three elastic displacement components, three electric displacement components, and the scalar entropy component. We can only classify the waves into pure elastic, electric, and thermal wavetypes if there is no coupling between the fields. This means that the matrix only consists of the diagonal block matrices and is otherwise zero. If the offdiagonal terms are non-zero, field coupling is introduced. For any propagation direction we obtain three waves carrying energy mainly in their elastic displacement components; secondly we get two wavetypes which consist mostly of electric displacements and which cannot have purely longitudinal components; thirdly we obtain a wave type which is primarily thermal (a thermal diffusion wave). Plotting inverse propagation velocity versus direction results in a slowness surface which characterizes the physical properties of the medium. Take for example an electrically non-conductive but piezoelectric medium; in this case the propagation velocity does not depend on frequency and one 3D surface characterizes the propagation of one wavetype completely. If the medium is allowed to be electrically conductive, we need a 3D slowness surface for each frequency in order to describe propagation completely.

In the uncoupled case, elastic and electromagnetic particle motions have an indeterminate relationship; there is no interaction between them. As soon as coupling is in effect, electric particle motion'' exhibits a 90 degree phase shift, which can be explained by the fact that a propagating electromagnetic field is generated only by a temporal change in the electromagnetic field variables.

In the literature it is very hard to find complete sets of generalized stiffness constants that are measured consistently with the correct boundary conditions applied. Thus in the following paragraphs I show results mainly for piezoelectric materials. Two materials are especially attractive to investigate: quartz and Lead-Titanate-Zirconate (PZT2). Firstly quartz since it is a naturally occurring rock and known for its wave coupling effects. Secondly Lead-Titanate-Zirconate (PZT2), that resembles shales in its manufacturing'' process and composition. The generalized stiffness constants for these two materials are listed in the appendix.

Next: Elastic'' slowness surfaces Up: Karrenbach: coupled wave propagation Previous: THE COUPLED SYSTEM
Stanford Exploration Project
1/13/1998