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Character of Wave Propagation in Anisotropic Media

David VonSeggern
Organization: U. Nevada
Department: Nevada Seismological Laboratory

The classic relation between stress and strain in an elastic medium involves a rank-4 tensor of 81 coefficients. Invoking various symmetry relations can lead to as few as two coefficients, as in the limiting case of isotropy. The equation for wave propagation in anisotropic media was sufficiently complex that pre-computer investigators had to make simplifying assumptions on the tensor in order to evaluate, by hand calculation, even a few points for direction-dependent velocity, slowness, and wavefront. Mathematica enables one to fully plot these quantities even in the most general case. This article demonstrates the power of Mathematica to handle previously daunting hand calculations and proceeds to visualize the transverse anisotropic case (5 coefficients), equivalent to hexagonal symmetry in crystals, in some detail. It also contains a novel approach to investigating arbitrary transverse anisotropy through parameters controlled in the Manipulate function of Mathematica.

*Applied Mathematics > Visualization > Ray Tracing
*Science > Physics > Mechanics
*Science > Physics > Wave Motion

elastic media, anisotropy, hexagonal symmetry, cubic symmetry, stress and strain, elastic wave propagation
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archive_submission_anisotropy_VonSeggern.nb (1.8 MB) - Mathematica Notebook