Boundary integral equation methods for the scattering problem by an unbounded sound soft rough surface with tapered wave incidence

• A rigorous derivation of BIE is presented for studying the scattering problem.
• Some properties of the integral equation in an energy space with weights are proved.
• The convergence of the numerical method is also obtained.

In this paper, we consider the scattering problem of tapered acoustic wave by an unbounded sound soft surface. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with Dirichlet boundary condition. Although the tapered wave is often introduced to realize asymptotic truncation for unbounded rough surface, the standard Helmholtz integral equations which derive for the scattering of plane waves by an arbitrary bounded obstacle are often used to generate benchmark numerical solutions. Different from the scattering of plane waves by an arbitrary bounded obstacle, we use the angular spectrum representation radiation condition to replace the Sommerfeld radiation condition, and derive a boundary integral equation for studying the scattering problem. Then we study the integral equation by the truncation method, whereby the integral equation posed on an unbounded region is approximated by an integral equation on a bounded region. Some properties of the integral equation in an energy space with weights are proved. Then the collocation method is used to solve the integral equation on a bounded region, and its convergence is also obtained.