On Single Integral Equations for the Transmission Problem of Acoustics

The transmission problem, namely scattering of time-harmonic waves in a compressible fluid by a fluid inclusion with different material properties, is usually formulated as a pair of coupled boundary integral equations over the interface S between the inclusion and the exterior fluid. In this paper, however,we consider methods for solving the transmission problem using a single integral equation over S for a single unknown function. In fact, we derive four different integral equations, using a hybrid of the direct (Green’s theorem) and indirect (layer ansatz) methods, and give conditions for the unique-solvability of each and for the subsequent construction of the solution to the transmission problem. Some of our single integral equations are Fredholm integral equations of the second kind with weakly-singular kernels. Thus, these equations, all of which appear to be new, are attractive computationally.