Time domain wave equations for lossy media obeying a frequency power law

Abstract

For attenuation described by a slowly varying power law function of frequency, α=α0‖ω‖y, classical lossy time domain wave equations exist only for the restricted cases where y=0 or y=2. For the frequently occurring practical situation in which attenuation is much smaller than the wave number, a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence. In order to obtain the corresponding time domain lossy wave equation, time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions. Three forms of lossy wave equations are found, depending on whether y is an even or odd integer or a noninteger. A time domain expression of causality analogous in function to the Kramers–Kronig relations in the frequency domain is used to derive the causal wave equations. Final causal versions of the time domain wave equations are obtained even for the cases where y≥1, which, according to the Paley–Wiener theorem, are unobtainable from the Kramers–Kronig relations. Different forms of the wave equation are derived including normal time, retarded time, and parabolic (one and three dimensional). These equations compare favorably with those from the literature corresponding to y=0, 0.5, 1, and 2.