On a uniquely solvable integral equation in a mixed Dirichlet-Neumann problem of acoustic scattering

The mixed Dirichlet—Neumann problem for the Helmholtz equation in the exterior of several bodies (obstacles) is studied in 2 and 3 dimensions. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called the “method of interior boundaries", because additional boundaries are introduced inside scattering bodies, where the Neumann boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers. The Neumann and Dirichlet problems are particular cases of our problem.