Time‐domain three‐dimensional Green’s functions for power law media.

Abstract

Frequency‐dependent loss and dispersion are typically modeled with a power law attenuation coefficient, where the power law exponent ranges from 0 to 2. Typically, these effects are modeled in the frequency domain and then time‐domain results are obtained via inverse fast Fourier transform because analytical solutions in the time domain were previously not available. To address this problem, analytical three‐dimensional Green’s functions are derived in power law media for exponents between 0 and 2 by utilizing stable law probability distributions. For exponents equal to 0, 1/3, 1/2, 2/3, 3/2, and 2, Green’s function is expressed in terms of Dirac delta, exponential, Airy, and hypergeometric functions. For exponents strictly less than 1, Green’s functions are causal and expressed in terms of the Fox function. For exponents strictly greater than 1, Green’s functions are also expressed in terms of Fox and Wright functions and are noncausal. For exponents equal to 1, Green’s function is expressed as a stable ...