Suppose a musician wants a guitar string that has a certain specific sound. This specification can be translated into eigenvalue locations in the complex plane: persistent tones correspond to eigenvalues near the imaginary axis; highly damped tones give eigenvalues farther in the left half of the complex plane. Can we design a string that has the desired response?

In this talk, we investigate the construction of a spatially-varying damping coefficient that induces a wave operator having specified eigenvalues. This problem is distinguished from the more familiar inverse Sturm-Liouville problem by the fact that the operator in question in non-self-adjoint. Still, many of the same theoretical devices, based on eigenvalue asymptotics, prove useful. The constructed damping functions are even about the midpoint and should vary mildly about their mean. Numerical results, in the absence and presence of noise, demonstrate the efficacy of this technique.