The Reflection Principle in Two-Dimensional Boundary-Value Problems for the Helmholtz Equation

The possibility of employing the principle of reflection in constructing solutions and internal and external boundary-value problems for the Helmholtz equation in two-dimensional domains whose boundaries contain rectilinear segments is analyzed. The principal idea of the approach consists in extending the desired solution in a canonical domain such as a circle by employing the reflection principle for solving the Helmholtz equation through the rectilinear segments of the boundary (at homogeneous boundary conditions). In this case the solution of the boundary-value problem is expressed in terms of series in particular solutions of the Helmholtz equation in polar coordinates; the unknown coefficients of this series can be found from an infinite set of linear algebraic equations. The closure equations at the segments of the circle that do not serve as physical boundaries of the original domain are formulated here by reflection of the desired equation. Various examples of boundary-value problems for the Helmholtz equation for a rectilinear-circular lune (internal and external problems) are analyzed. The manner in which allowance can be made for local singularities of the wave field associated with corner points of the domain under study and the mixed nature of the boundary conditions is shown. Numerical computations that verify the suggested method are performed for one of the problems.