Time-dependent wave propagation on a variable Winkler foundation with compression

Abstract

The time-dependent motion of flexural waves generated by the vibration of an elastic plate resting on a variable Winkler foundation with compression is studied. The physical problem is analysed under the assumption of small amplitude structural response in two dimensions. The dispersion equation is analysed in detail, and points of wave blocking are shown to exist. The time-dependent propagation is simulated for the case of both constant and variable properties. For the case of varying properties, it is shown that the wave amplitude near the blocking point for this class of flexural waves satisfies the hyper-Airy differential equation, which is solved analytically in terms of the fourth-order Airy function. Further, the asymptotic solution of the hyper-Airy differential equation is derived by employing the WKB method to match the far-field solution with the near-field solution. The results obtained analytically are compared with a novel time-domain simulation based on a spectral method.