``Elastic'' slowness surfaces

The computation of characteristic surfaces as shown by Dellinger (1989) can be extended to the coupled wave propagation case. Figure shows slowness surfaces for the uncoupled purely elastic wavetypes in the medium PZT2. It is transversely isotropic with the z-axis as symmetry axis. In appearance it resembles Greenhorn shale, which is shown in Figure 1. For the PZT2 medium the piezoelectric constants are given in the literature; and we get a very interesting result, Figure 2, when we include these effects in our calculations. Instead of showing a well-behaved picture similar to that of shale's behaviour, the two slowest wavetypes change shape radically. The slowness increases by a factor of 4 in the direction of the z-axis. This behavior directly reflects the magnitude of the piezoelectric stress coefficients, which show a strong coupling of the electric field, especially into strain components in the z-direction.

Consequently we can ask ourselves what the slowness surfaces for Greenhorn shale would look like, if we measure the piezoelectric stress constants.