Reconstruction of One-Dimensional Inhomogeneities in Elastic Modulus and Density from Measurements of Acoustic Reasonances

Abstract

We present here a simple procedure for reconstructing an inhomogeneity in density and elastic modulus in a one-dimensional structure using only its measured resonant frequencies (fundamental and overtones). The shifts in the resonant frequencies of the structure (e. g., a rod) due to a material inhomogeneity (computed relative to the frequencies of the homogeneous state) are shown to bear a simple relation to the coefficients in a Fourier expansion of the inhomogeneity which is exact to first order in the material perturbation. As an example, consider extensional waves excited in a one dimensional rod. We show that the fractional frequency shift of the n^th normal mode gives, to within a constant and to first order in the material perturbation, the 2n^th Fourier coefficient of the inhomogeneity for the case of free-free (or fixed-fixed) boundary conditions at the rod ends.¹