Abstract Summary
The determination of the mass density of a taut vibrating membrane from measurements of resonant frequency values (eigenvalues) is a problem of practical interest in various areas of applied science, from structural mechanics to biology. The inverse problem that arises is of great mathematical difficulty and it is still open in many respects. Typically, for general domains and for arbitrary mass densities, uniqueness requires the knowledge of all the infinite eigenvalues and the trace of the normal derivative of the corresponding eigenfunctions at the boundary. It is expected that the requests on the data can be weakened for simple domains, such as rectangular ones. Although these latter conditions are of interest in applications, few general results are available in the literature for this simplified version of the problem. Things get even more complicated when as is the case considered in this work the available data are a finite number of resonant frequencies, a situation that always occurs in engineering applications. Despite the lack of well established theoretical framework, some encouraging results have been obtained in the reconstruction of small perturbations of a uniform mass density in a rectangular membrane with fixed boundary. All previous works, however, consider mass perturbations symmetric with respect to both the midlines of the domain [1]-[3]. The main objective of this work is to deal with general mass perturbations. Our reconstruction method is based on the fact that if the unknown mass density ρ= (ρo+r) is a small smooth perturbation of the uniform mass density ρo, then the differences between unperturbed and perturbed resonant frequencies are correlated with certain generalized Fourier coefficients of the unknown perturbation r. This property was recently applied in [4] for the determination of doubly symmetrical mass distributions from information on Dirichlet spectrum data only. Here we show that a detailed study of the linearized inverse problem around the uniform membrane is useful in selecting appropriate spectral data to be added to the Dirichlet spectrum to ensure a proper formulation and solution of the inverse problem. It turns out that, besides the Dirichlet spectrum, it is necessary to resort to three additional spectra. The method leads to an iterative algorithm based on successive linearizations of the inverse problem in a neighbourhood of the unperturbed membrane. The reconstruction technique has been validated on an extended series of numerical simulations. The results show a notable agreement between target and identified densities, even for perturbations with disconnected support. Moreover, a certain ability of the method also emerged in approximating mass distributions that do not necessarily fall within the small perturbations or that are less regular. The results obtained are valid under the hypothesis that the eigenvalues of the initial membrane are all simple. The analysis of cases with multiple eigenvalues is significantly more complex and is currently under study. [1] R. Knobel et al., Z. Angew. Math. Phys. (1994). [2] C. McCarthy, Appl. Anal. (2001). [3] Q. Gao et al., Comput. Math. Appl. (2015). [4] A. Kawano et al., Comput. Math. Appl. (2022).
