A unifying fractional wave equation for compressional and shear waves

作者：Holm Sverre^*; Sinkus Ralph

来源：Journal of the Acoustical Society of America, 2010, 127(1): 542-548.

DOI：10.1121/1.3268508

This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable properties such as causality that describe such wave propagation. Starting with a constitutive equation which is a generalized Hooke's law with a loss term containing a fractional derivative, one can derive a causal fractional wave equation previously given by Caputo [Geophys J.R. Astron. Soc. 13, 529-539 (1967)] and Wismer [J. Acoust. Soc. Am. 120, 3493-3502 (2006)]. In the low omega tau (low-frequency) case, this equation has an attenuation with a power-law in the range from 1 to 2. This is consistent with, e.g., attenuation in tissue. In the often neglected high omega tau (high-frequency) case, it describes attenuation with a power-law between 0 and 1, consistent with what is observed in, e. g., dynamic elastography. Thus a unifying wave equation derived properly from constitutive equations can describe both cases.

• 出版日期2010-1