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

Abstract

The wave scattering problem by a crack Γ in 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.Mathematics subject classification: 35P25, 35R30, 78A45.

Scattering of plane waves by an open arc with different boundary conditions on its sides

“…The solution to the Dirichlet, Neumann and impedance problems has only a square root singularity at the arc tips. This singularity can be eliminated from consideration using the cosine substitution [6] - [11] or with the introduction of the additional condition [12]. The purpose of this paper is to give an efficient computing method for the determination of the scattered waves for the open arc with different boundary conditions on its sides, using the collocation method without considering the different structure of the solution singularities at tips.…”

Boundary Galerkin Method of a Skew-Derivative Problem in the Exterior of an Open Arc Based on Chebyshev Polynomials

“…In [5], the author constructs the solution by singlelayer potential and nonclassical angular potential [6][7][8][9][10] and proves the existence and uniqueness of the solution. An angular potential can be seen as a double layer potential under certain conditions.…”