Energy vs. group velocity for elastic waves in homogeneous anisotropic solid media

Abstract

As for electromagnetic waves the energy velocity for time harmonic elastic waves is defined by the ratio of the (real part) of the elastodynamic complex Poynting vector and their time-averaged elastic energy density. For time harmonic plane waves in homogeneous anisotropic materials an explicit analytic expression can be derived involving the phase propagation vector and the eigenvectors (polarizations) of the respective wave modes (quasi-pressure, quasi-shear). Numerical evaluation yields plane wave energy velocity diagrams for these wave modes that depend on the direction of the phase vector. On the other hand, for impulsive elastic wave packets a group velocity can be defined under certain assumptions as the gradient in phase space of the dispersion relation. As will be shown in the paper, if definable, the group velocity can be alternatively computed by the expression for the energy velocity. This is particularly interesting because frequency band-limited impulsive elastic waves emanating from point sources (band-limited time domain Green functions, Huygens elementary elastic wave packets) exhibit a spatial structure identical to plane wave energy velocity diagrams giving rise to their calculation without the explicit knowledge of the pertinent Green functions. Therefore, time harmonic plane waves serve as a computational tool for impulsive point source excited waves that has consequences for elastic wave diffraction tomographic imaging.