Reconstruction of a general mass density in a rectangular membrane from finite eigenvalue data

This paper is concerned with the estimation of the mass density in a rectangular taut membrane from finite eigenvalue data of the small undamped transverse vibration. It is known that the first lower eigenvalues of the membrane supported at the boundary may allow the reconstruction of small perturbations of the uniform density, provided that the mass density is symmetric with respect to both the midlines of the domain. Here we show how the addition of suitable sets of eigenvalues corresponding to different boundary conditions can be useful for the determination of mass densities either with one symmetry axis only or without symmetry. The reconstruction procedure is based on a sequence of linearizations of the inverse problem in a neighborhood of the uniform membrane, under the assumption that the eigenvalues of the initial membrane used in identification are all simple. An extended series of numerical simulations performed for various mass densities allowed to test the effectiveness of the reconstruction, and also to highlight some indeterminacy inherent in this inverse eigenvalue problem with finite data.