Identification of a symmetric mass density in a rectangular membrane from finite eigenvalue data

In this paper we seek to recover the mass density of a rectangular taut membrane from the first eigenvalues of the small transverse vibration under supported boundary conditions, provided the mass is a small perturbation of the constant density and is symmetric with respect to both midlines. The proposed method allows for the determination of the generalized Fourier coefficients of the mass density change evaluated on a suitable basis of functions, which naturally arises from the linearized Taylor expansion of the eigenvalues. The reconstruction is based on a sequence of successive linearizations of the inverse problem in a neighborhood of the uniform membrane, and numerical results indicate improvements over other techniques available in the literature. The method has been tested also on mass densities that fall outside the range of small perturbations of the constant density, showing some potential for the reconstruction.