Fractional biharmonic operator equation model for arbitrary frequency-dependent scattering attenuation in acoustic wave propagation.
This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power law with an exponent ranging from 0 to 4. In stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature, little has been reported on frequency-dependent scattering attenuation. This is largely because the order of the fractional Laplacian is from 0 to 2 and is infeasible for scattering attenuation. In this study, the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed, as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power-law dependency and obeys the causal relation under the smallness approximation. Finally, the correlation between the fractional order and the ratio of wavelength to the diameter of the scattering heterogeneity is investigated and an expression on exponential form is also provided.
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Fuchengxi Road 8, Jiangning District, Nanjing, Jiangsu Province 211100, People's Republic of China.
Wen
W
Chen
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Fuchengxi Road 8, Jiangning District, Nanjing, Jiangsu Province 211100, People's Republic of China.
Jun
J
Fang
Beijing Computational Science Research Center, No. 10 East Xibeiwang Road, Haidian District, Beijing 100193, People's Republic of China.
Guofei
G
Pang
Department of Informatics, University of Oslo, P. O. Box 1080, NO-0316 Oslo, Norway.
Sverre
S
Holm