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Volume 29, Issue 2
Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

J. J. Liu, P. A. Krutitskii & M. Sini

DOI: 10.4208/jcm.1006-m3131

J. Comp. Math., 29 (2011), pp. 141-166.

Published online: 2011-04

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  • Abstract

The wave scattering problem by a crack $\Gamma$ in $\mathbb{R}^2$ with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.

  • Keywords

Wave scattering, Impedance boundary, Integral equations, Singularity analysis, Numerics.

  • AMS Subject Headings

35P25, 35R30, 78A45.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-29-141, author = {J. J. Liu, P. A. Krutitskii and M. Sini}, title = {Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {2}, pages = {141--166}, abstract = {

The wave scattering problem by a crack $\Gamma$ in $\mathbb{R}^2$ with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1006-m3131}, url = {http://global-sci.org/intro/article_detail/jcm/8470.html} }
TY - JOUR T1 - Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space AU - J. J. Liu, P. A. Krutitskii & M. Sini JO - Journal of Computational Mathematics VL - 2 SP - 141 EP - 166 PY - 2011 DA - 2011/04 SN - 29 DO - http://doi.org/10.4208/jcm.1006-m3131 UR - https://global-sci.org/intro/article_detail/jcm/8470.html KW - Wave scattering, Impedance boundary, Integral equations, Singularity analysis, Numerics. AB -

The wave scattering problem by a crack $\Gamma$ in $\mathbb{R}^2$ with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.

J. J. Liu, P. A. Krutitskii and M. Sini. (2011). Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space. Journal of Computational Mathematics. 29 (2). 141-166. doi:10.4208/jcm.1006-m3131
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