A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain into a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly-varying, smooth functions in the transformed free surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.
|Translated title of the contribution||Approximations to water wave scattering by steep topography|
|Pages (from-to)||279 - 302|
|Journal||Journal of Fluid Mechanics|
|Publication status||Published - 10 Sep 2006|
Bibliographical notePublisher: Ca\mbridge Univ Press
Other identifier: IDS Number 082MA