Nonlinear acoustic pulse propagation in dispersive sediments using fractional loss operators

Abstract

The nonlinear progressive wave equation (NPE) is a time-domain formulation of Euler’s fluid equations designed to model low-angle wave propagation using a wave-following computational domain [ McDonald et al., J. Acoust. Soc. Am. 81]. The wave-following frame of reference permits the simulation of long-range propagation that is useful in modeling the effects of blast waves in the ocean waveguide. However, the current model does not take into account sediment attenuation, a feature necessary for accurately describing sound propagation into and out of the ocean sediment. These attenuating, dispersive sediments are naturally captured with linear, frequency-domain solutions through use of complex wavespeeds, but a comparable treatment is nontrivial in the time-domain. Recent developments in fractional loss operator methods allow for frequency-dependent loss mechanisms to be applied in the time-domain providing physically realistic results [Prieur et al., J. Acoust. Soc. Am. 130]. Using these approaches, the governing equations used to describe the NPE are modified to use fractional derivatives in order to develop a fractional NPE. The updated model is then benchmarked against a Fourier-transformed parabolic equation solution for the linear case using various sediment attenuation factors.

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional order wave equations

“…Also, fractional derivatives have lately been applied to model frequency dependent power-law wave attenuation in sediments. 23 Second, we have recently demonstrated that just as non-Hookean materials lead to nonlinear acoustics, time-dependent non-Newtonian materials tend to power law behavior. 24 Interestingly, power-law characteristics are common in materials modeled using fractional derivatives, such as biological media 25− 29 and earth materials.…”