Abstract

Several papers in recent years have dealt with the causality-imposed relation between attenuation and dispersion for waves in lossy solids, with emphasis on seismic waves. While the published formulas for dispersion within a particular frequency band are supported by experimental evidence within that band, the mathematical behavior of these expressions outside the band, particularly at low frequencies, is physically unacceptable. In the present paper, one-dimensional seismic waves are modeled as propagation along a simple lumped-element transmission line, leading to expressions for attenuation and velocity as functions of frequency which not only satisfy the experimental data available, but exhibit no objectionable behavior outside the range of available data. This is achieved by introducing a resistive element whose value is inversely proportional to frequency. Numerical application of the Hilbert transform shows the condition of causality to be satisfied by this model quite accurately.

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