Boundary integral formulation of the standard eigenvalue problem for the 2-D Helmholtz equation

Abstract

In this paper, a boundary integral formulation is presented for obtaining the standard eigenvalue problem for the two-dimensional (2-D) Helmholtz equation. The formulation is derived by using the series expansions of zero-order Bessel functions for the fundamental solution to the Helmholtz equation. The proposed approach leads to a series of new fundamental functions which are independent of the wave number k of the Helmholtz equation. The coefficient matrix of the resulting homogeneous system of boundary element equations is of the form of a polynomial matrix in k which allows a much faster search for the eigenvalues by scanning k over an interval of interest or the standard eigenvalue problem to be formulated for directly solving for the eigenvalues without resort to iterative methods. The proposed technique was used to solve some known problems with available analytical solutions: 2-D domains with circular and rectangular geometries under Dirichlet and/or Neumann boundary conditions. The outcomes demonstrate that the proposed approach is computationally efficient and highly accurate.