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.
Computational methods for the inverse problem of detecting the source of acoustical noise in an interior region from pressure measurements in the nearfield are discussed. The methods are based on a single layer potential representation of solutions to the Helmholtz equation. Regularization is peformed using the singular value decomposition and the conjugate gradient method.
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