Dispersive body waves

Although the attenuation of body waves in an infinite medium has often been considered, little emphasis is usually given to the attendant dispersion, which is a necessary consequence of the medium absorption and is determined unambiguously by it. Dispersion equations are obtained from the application of an integral transform in the frequency domain. Such relations are of the Kramers-Krönig type, relating the dispersive part of the index of refraction of the medium to the absorptive part by means of an integral over the entire frequency range. They are a consequence of the principle of causality and follow without recourse to a specific wave equation. In particular, our results are consistent with and presume linearity for the wave motion. Three forms of the dispersion equation are exhibited. In all cases, the absorption coefficient has a linear dependence upon frequency for frequencies above a certain characteristic value—chosen low enough to be outside the range of measurement. However, it is necessary in all cases to introduce a low-frequency, nonzero cutoff for the absorption, which can in principle be taken arbitrarily small. Group and phase velocities are determined as functions of the medium Q and a characteristic velocity, which for small absorption can be related to the measured signal velocity. Seismic phenomena occur in materials where the absorption is small, Q≥30, so that the corresponding dispersion presumably will be difficult to measure in the earth. For appropriate laboratory materials the effect can be significant.