Exact and Approximate Causal Time-Domain Green’s functions for Linear with Frequency Attenuation

Exact and approximate analytical time-domain Green's functions that describe linear with frequency attenuation are presented for the Chen-Holm space-fractional wave equation. These exact and approximate time-domain Green's functions contain a Cauchy density multiplied by a Heaviside function, where the Heaviside function guarantees that the response is causal. To demonstrate the distinguishing features of these analytical expressions, exact and approximate time-domain Green's functions are numerically evaluated and compared for linear with frequency attenuation. Numerical results are evaluated for an attenuation of 0.5 dB/cm/MHz, and the results demonstrate excellent agreement between the exact and approximate solutions. At this and at all other distances, the exact and approximate solutions overlap almost perfectly. Comparisons of the peak values evaluated and the full width at half maximum (FWHM) evaluated as a function of time at different distances for the exact and approximate expressions also agree very closely. Thus, the approximate expressions are excellent approximations to the exact time-domain Green's functions for the Chen-Holm space-fractional wave equation, which is the only known fractional wave equation that yields causal analytical time-domain Green's functions describing linear with frequency attenuation.