Localization of travelling and standing waves in a circular membrane coupled to a continuous viscoelastic support

We present an analytical approach for realizing the localization of travelling and standing waves in a circular membrane by means of an interior, continuous, ring-type viscoelastic support when the membrane is subjected to harmonic boundary excitation. This work is a continuation of the wave separation dynamics induced by nonclassical damping in one-dimensional elastic continua. For a two-dimensional azimuthally symmetric configuration, a necessary and sufficient condition is established to realize wave localization, leading to uniformly distributed spring stiffness and damping coefficients for the interior support. A rotating wave localization pattern can also be realized by applying a circumferentially moving boundary excitation. The desired results are shown to depend strongly on the excitation frequency, the circumferential wavenumber, and the radius of the continuous support. For a weakly eccentric configuration with uniformly distributed boundary excitation, non-uniform required distributions of the support properties can be obtained analytically. However, waveform distortion will occur for either non-uniformly distributed boundary excitation with weak support eccentricity or highly eccentric coupling, such that the parameter values of the support have zero crossings on the nodal diameters of the membrane and thus lose physical meaning. The results presented in this work are expected to be useful in the design of mechanical and acoustic systems to localize energy to a sonic bullet by means of the continuous distribution of damping.