Stable Distribution and [0;2] Power Law Dependence of Acoustic Absorption on Frequency in Various Lossy Media

Absorption of acoustic wave propagation in a large variety of lossy media is characterized by an empirical power law function of frequency, 0j!j y . It has long been noted that the exponent y ranges from 0 to 2 for diverse media. Recently, the present author [J. Acoust. Soc. Am. 115 (2004) 1424] developed a fractional Laplacian wave equation to accurately model the power law dissipation, which can be further reduced to the fractional Laplacian di usion equation. The latter is known underlying the L evy stable distribution theory. Consequently, the parameters y is found to be the L evy stability index, which is known to be bounded within 0 < y 2. This nding rst provides a theoretical explanation of empirical observations y 2 [0; 2]. Statistically, the frequencydependent absorption can thus be understood a L evy stable process, where the parameter y describes the fractal nature of attenuative media.