We describe the use of iterative techniques for the solution of the integral equation that arises in an exact treatment of scalar wave scattering from randomly rough surfaces. The surfaces vary in either one or two dimensions, and the special case of a Dirichlet boundary condition is treated. It is found that these techniques, particularly when preconditioning is applied, are much more efficient than direct inversion techniques. Moreover, convergence is obtained for rms roughness of the order of 1, so the techniques have applicability over a wide parameter regime. Convergence is always to the exact solution found by direct inversion, exceptly for cases of extremely large-scaled rms surface heights in which the iterative techniques fail. In addition, by monitoring the residuals in the iteration process, it is immediately clear if the iterative techniques are failing, or performing badly in any given case. Finally, numerical results are compared with existing data in the enhanced backscattering regime.

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Using the zero divergence approximation, we calculate the response of a continental shelf to an oscillating coastal current source which acts through a gap of finite width in the coastline. It is shown that this response consists of a forced oscillation in the neighbourhood of the gap, together with shelf waves appropriate to the shelf on either side. For a shelf of exponential slope similar to that of the East Australian shelf, and a flux through a channel of the dimensions of Bass Strait, the shelf wave response is shown to be qualitaticely similar to the results obtained in the Australian Coastal Experiment. This supports the contention that Bass Strait is a dominant source of shelf waves on the East Australian continental shelf. The 'eddy' mode required to explain the observations may also be attributed to the forced response of the shelf directly off-shore to Bass Strait. This paper also investigates the effect of adopting, in this physical context, simpler boundary conditions at the shelf edge. It is shown that there is some computational simplification, and that the results are largely unaffected by this simplification.

In this article, a formulation is given for two-dimensional scattering of a plane wave incident on an infinitely long surface with a Gaussian distribution of surface height. The surface height may have any physically reasonable correlation function. This method generalizes previous work by the present authors, in which a finite length surface was treated numerically by Fourier methods. For each numerically generated random surface, an integral equation must be solved to find the pressure gradient on the surface. The reflection coefficients, describing the angular distribution of scattered energy, are then formed. It is shown how the various moments of the scattered pressure may be obtained efficiently from averages of these reflection coefficients. Using these techniques, numerical results are presented for the angular distribution of energy for a surface with a Gaussian correlation function of surface height and for a correlation function modeling a more realistic sea surface. Results are presented showing that it is not necessary to taper the incoming wave field in order to obtain accurate answers. For large values of the ratio of the standard deviation of height to the surface correlation length, it is found that the mode of scattering is very different to that found at small surface heights. For some cases, the maximum in the angular distribution is found in the backscatter direction, with the specular peak not at all apparent. It appears that this phenomenon is due to multiple scattering of the acoustic wave at the surface. Finally, it is shown that for the class of problems treated here, the ensemble average intensity is constant at all normal distances from the surface, while the normalized covariance of intensity varies with normal distance from the surface in a way that is, in some limiting cases, very similar to the scattering of a wave passing through a phase modulating screen or an extended random medium.

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