Boundary Integral Equations in Time-harmonic Acoustic Scattering

We first review the basic existence results for exterior boundary value problems for the Helmholtz equation via boundary integral equations. Then we describe the numerical solution of these integral equations in two dimensions for a smooth boundary curve using trigonometric polynomials on an equidistant mesh. We provide a comparison of the NystrGm method, the collocation method and the Gale&in method. In each case we take proper care of the logarithmic singularity of the kernel of the integral equation by choosing appropriate quadrature rules. ln the case of analytic data the convergence order is exponential. The NystrGm method is the most efficient since it requires the least computational effort. Finally, we consider boundary curves with corners. Here, we use a graded mesh based on the idea of transforming the nonsmooth case to a smooth periodic case via an approprrate substitution. Then, the application of Nystram’s method again yields rapid convergence. 1. BOUNDARY INTEGRAL EQUATIONS We begin with a brief review on the theory of boundary integral equations in time-harmonic acoustic scattering. For a comprehensive study we refer to [l]. Consider acoustic wave propagation in a homogeneous isotropic medium in lR2 or lR3 with speed of sound c. The wave motion can be determined from a velocity potential CT = U(z, t) from which the velocity field w is obtained by w = grad U, and the pressure p by DU P-Po=-~’ where p. denotes the pressure of the undisturbed medium. In the linearized theory, the velocity potential U satisfies the wave equation d2U -w c”AU = 0. Hence, for time-harmonic acoustic waves of the form 1/(x, t) = u(x) eeiwf with frequency w, we deduce that the space dependent part u satisfies the reduced wave equation or Helmholtr equation Au + ti2u = 0, where the wave number is given by h: = w/c. The mathematical description of the scattering of time-harmonic waves by an obstacle leads to exterior boundary value problems for the IIelmholtz equation. For example, prescribing the values of u on the boundary of the obstacle physically corresponds to prescribing the pressure of the acoustic wave on the boundary. Therefore, scattering from sound-soft obstacles leads to a Dirichlet boundary value problem. Similarly, prescribing the normal derivative on the boundary corresponds to prescribing the normal velocity of the acoustic wave. Thus, scattering from sound-hard obstacles leads to a Neumann boundary value problem. For the sake of brevity we confine our presentation to the Dirichlet, boundary condition.