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| In this work we present a novel integral equation formulation for the transmission problem for the Helmholtz equation. We start introducing the admittance operators, which map the transmission boundary conditions to the exterior and respectively interior Cauchy data on the interface between the media. With such mapping, the solution of the transmission problem is straightforward since the densities needed to construct the acoustic fields are the result of applying these admittance operators to the jumps of the trace and normal derivative. However, since these mappings are usually difficult to compute, we propose instead to construct suitable approximations of such operators based on approximating linear combinations of Dirichlet-to-Neumann mappings, assuming that they exist, by boundary layer operators with pure imaginary wave-numbers. We then prove that the interior and exterior acoustic fields can be evaluated in terms of layer potentials whose densities are the solution of a system of integral equations, that the suggested approximations of the admittance operator make this system a compact perturbation of the identity in Sobolev spaces (property which hold without needing to assume that the above mentioned combination of the Dirichlet-to-Neumann operators exist), and that this system is uniquely solvable. |
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