The Spectrum of the Damped Wave Operator for a Bounded Domain in { $\boldsymbol{R^2}$}

Abstract

The spectrum of the damped wave operator for a bounded domain in {$R^2$} is shown to be related to the asymptotic average of the damping function by the geodesic flow. This allows the calculation of an asymptotic expression for the distribution of the imaginary parts of the eigenvalues for a radially symmetric geometry. Numerical simulations confirm the theoretical model. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the rays of geometrical optics, and the geometry of the damping region. The MATLAB code used in this paper is provided.