The detection of surface vibrations from interior acoustical pressure

Abstract

We consider the problem of detecting the source of acoustical noise inside thecabin of a midsize aircraft from measurements of the acoustical pressure fieldinside the cabin. Mathematically this field satisfies the Helmholtz equation. Inthis paper we consider the three-dimensional case. We show that anyregular solution of this equation admits a unique representation by asingle-layer potential, so that the problem is equivalent to the solutionof a linear integral equation of the first kind. We study uniqueness ofreconstruction and obtain a sharp stability estimate and convergence rates forsome regularization algorithms when the domain is a sphere. We havedeveloped a boundary element code to solve the integral equation. We reportnumerical results with this code applied to three geometries: a sphere, acylinder with spherical endcaps and a cylinder with a floor modellingthe interior of an aircraft cabin. The exact test solution is given by apoint source exterior to the surfaces with about 1% random noise added.Regularization methods using the truncated singular value decomposition withgeneralized cross validation and the conjugate gradient (cg) method with astopping rule due to Hanke and Raus are compared. An interesting feature ofthe three-dimensional problem is the relative insensitivity of the optimalregularization parameter (number of iterations) for the cg method to thewavenumber and the multiplicity of the singular values of the integral operator.