THE WAVE METHOD FOR DETERMINING THE ASYMPTOTIC DAMPING RATES OF EIGENMODES .1. THE WAVE-EQUATION ON A RECTANGULAR OR CIRCULAR DOMAIN

The uniform exponential decay property of the wave equation with viscous boundary damping has been studied by several people. The mathematical proofs used therein are commonly based on energy identities, which cannot determine the actual decay rates of the solution. In Quinn and Russell [Proc. Roy. Soc. Edinburgh Sect. A, 77 (1977), pp. 97-127] and Chen [Ph.D. thesis, University of Wisconsin, Madison, WI, May 1977], such damping rates have been calculated for rectangular and circular domains, respectively, using separation of variables and perturbation approach good for small viscous damping parameters. In this paper, we extend some earlier geometrical optics and diffraction methods of Keller and Rubinow [Ann. Phys., 9 (1960), pp. 24-75] to treat the eigenvalue problems with dissipative conditions. Such methods provide strong insights into the physical properties of the solutions. Asymptotic estimates of damping and wavenumber are shown to agree favorably with the earlier results of Quinn and Russell and Chen for small damping parameters, as well as with the numerical solutions computed herein for cases even when the damping parameters are not small.