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Reconstructing an even damping from a single spectrum

We consider the wave equation on a finite interval with fixed ends and nonuniform viscous damping. We prove there can be at most one even damping associated with a given spectrum. We then develop a refined asymptotic formula for the high frequencies. When the damping is even about the domain's midpoint, terms in this expansion are Fourier coefficients for functions of the damping. Furthermore, the expansion is often quite accurate even at low frequencies, thus suggesting a simple numerical procedure for reconstructing even damping coefficients from measured eigenvalues. Computational examples illustrate the efficacy of this procedure, even in the presence of noise.