The Detection of the Source of Acoustical Noise in Two Dimensions

We consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin. Mathematically this field satisfies the Helmholtz equation. In this paper we consider the model two-dimensional case. We show that any regular solution of this equation admits a unique representation by a single layer potential, so that the problem is reduced to the solution of a linear integral equation of the first kind. We prove uniqueness of reconstruction and obtain a sharp stability estimate. Finally, for two geometries and sources of noise simulating the cabin of the aircraft and two engines, we give results of the numerical solution of this integral equation, comparing regularization by the truncated singular value decomposition and the conjugate gradient method.