Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency

Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzer...

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Bibliographic Details
Published in:The Journal of the Acoustical Society of America 2004-04, Vol.115 (4), p.1424-1430
Main Authors: Chen, W, Holm, S
Format: Article
Language:English
Subjects:
Acoustics
Computer Simulation
Fourier Analysis
Fractals
Hot Temperature
Humans
Linear Models
Models, Theoretical
Nonlinear Dynamics
Space-Time Clustering
Ultrasonic Therapy
Ultrasonics
Viscosity
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Summary:Frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency-independent and frequency-squared dependent attenuations. The otherwise nonzero and nonsquare frequency dependency occurring in many cases of practical interest is thus often called the anomalous attenuation. In this study, a linear integro-differential equation wave model was developed for the anomalous attenuation by using the space-fractional Laplacian operation, and the strategy is then extended to the nonlinear Burgers equation. A new definition of the fractional Laplacian is also introduced which naturally includes the boundary conditions and has inherent regularization to ease the hypersingularity in the conventional fractional Laplacian. Under the Szabo's smallness approximation, where attenuation is assumed to be much smaller than the wave number, the linear model is found consistent with arbitrary frequency power-law dependency.
ISSN:0001-4966
1520-8524
DOI:10.1121/1.1646399