An Asymptotic Average Decay Rate for the Wave Equation with Variable Coefficient Viscous Damping

The analysis of the damping rate for eigenmodes of a vibrating system with variable coefficient viscous damping is usually difficult because no explicit formulas are available in general. In this paper, using the one-dimensional wave equation as a model, it is shown that for such a system there is an asymptotic average decay rate of eigenmodes that is equal to the uniform damping rate of high frequencies of a wave equation with homogenized constant damping coefficient. The proof is obtained by asymptotic estimation of eigenfrequencies based on an earlier work of Birkhoff and Langer [Proc. Amer. Acad. Arts Sci., 58 (1923), pp. 51–128]. Numerical confirmation is also included, with details of computations given in [C. Qi, Computing the Spectrum of a Vibrating String with Positive and Negative Locally Distributed Viscous Damping, Masters Thesis, Pennsylvania State University, 1987].