Levy stable distribution and [0,2] power law dependence of acoustic absorption on frequency

Abstract

The absorption of acoustic wave propagation in a broad variety of lossy media is characterized by an empirical power law function of frequency, w^y. It has long been noted that exponent y ranges from 0 to 2 for diverse media. Recently, the present author developed a fractional Laplacian wave equation to accurately model the power law dissipation, which can be further reduced to the fractional Laplacian diffusion equation. The latter is known underlying the Levy stable distribution theory. Consequently, the parameters y is found to be the Levy stability index, which is known bounded within 0<y\le2. This finding first provides a theoretical explanation of empirical observations 0<y<=2. Statistically, the frequency-dependent absorption can thus be understood a Levy stable process, where the parameter y describes the fractal nature of attenuative media.