Water waves in shallow channels of rapidly varying depth

The linear water-wave equations for shallow channels with arbitrary rapidly varying bottoms was analysed. A theory for reflected waves was developed, based on an asymptotic analysis for stochastic differential equations when both the horizontal and vertical scales of the bottom variations are comparable to the depth but small compared to a typical wavelength so the shallow water equations cannot be used. The full, linear potential theory was used and the reflection-transmission problem for time-harmonic (monochromatic) and pulse-shaped disturbances was studied. For the monochromatic waves a formula is given for the expected value of the transmission coefficient which depends on depth and on the spectral density of the 0(1) random depth perturbations. For the pulse problem an explicit formula if given for the correlation function of the reflection process. The theory is compared with numerical results produced using the boundary-element method. Several, realisations of the bottom profile are considered and a Gaussian-shaped disturbance is propagated over each topography sampled and the reflected signal for each realisation is recorded. The numerical experiments produced reflected waves whose statistics are in good agreement with the theory.