Dispersive Wave Attenuation and Refocusing due to Disordered Orographic Forcing
Dispersion | disordered media | refocusing
In this work we study the interaction of weakly-nonlinear, weakly-dispersive water waves with shallow channels with highly variable depth, by using a Boussinesq-type system. We generalize the O'Doherty-Anstey approximation for the linearization of this weakly-dispersive system forced by a disordered orography. This is achieved by applying the invariant imbedding method to the equations governing the transmitted and reflected fields. We find that dispersive effects alter the medium's correlation function increasing the overall attenuation, delay and the pulse's spreading rate. Furthermore, at a fixed medium's station the wave's coherent front is found to stabilize, independent of the particular realization of the orography, when observed relative to its first arrival time. The linear theory is found to be in excellent agreement with the weakly-nonlinear simulations performed for different values of the relevant parameters by using a high-order finite difference scheme proposed by Wei and Kirby. This computing scheme is further employed to explore the refocusing property, a subject of great interest treated in recent works by Clouet and Fouque and Prada and Fink. New results for the refocusing of dispersive Gaussian-shaped disturbances and solitary waves are presented.