The present study derives a formulation of the finite-difference time-domain (FDTD) method appropriate for analyzing the transient behavior of elastic wave fields in the Y - Z plane of Quartz. It is shown that this formulation may easily be adapted for use on arbitrary anisotropic solids. The staggered lattice network of the present study differs from that of an isotropic solid only in the requirement of two variables on a particle velocity node and three variables on a stress node, in contrast to the isotropic case of one variable on a particle velocity node, and two or one variables on a stress node. Two model problems are considered which allow results obtained by the present numerical method to be compared with analytical predictions. The first model is constructed to test whether the FDTD formulation correctly captures the direction of energy flow for longitudinal and shear plane waves. This direction differs from that of the wave number vector. The second model is constructed in order to test whether the FDTD formulation correctly captures the dependence of phase velocity on the wave number vector. The results obtained from the numerical method are shown to be in excellent agreement with the analytical predictions.
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Masahiro Sato (2005) studied this question.