The detection of surface vibrations from interior acoustical pressure

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 three-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 equivalent to the solution of a linear integral equation of the first kind. We study uniqueness of reconstruction and obtain a sharp stability estimate and convergence rates for some regularization algorithms when the domain is a sphere. We have developed a boundary element code to solve the integral equation. We report numerical results with this code applied to three geometries: a sphere, a cylinder with spherical endcaps and a cylinder with a floor modelling the interior of an aircraft cabin. The exact test solution is given by a point source exterior to the surfaces with about 1% random noise added. Regularization methods using the truncated singular value decomposition with generalized cross validation and the conjugate gradient (cg) method with a stopping rule due to Hanke and Raus are compared. An interesting feature of the three-dimensional problem is the relative insensitivity of the optimal regularization parameter (number of iterations) for the cg method to the wavenumber and the multiplicity of the singular values of the integral operator.