Differential equations for long-period gravity waves on fluid of rapidly varying depth

Abstract

The conventional long-wave equations for waves propagating over fluid of variable depth depend for their formal derivation on a Taylor series expansion of the velocity potential about the bottom. The expansion, however, is not possible if the depth is not an analytic function of the horizontal co-ordinates and it is a necessary condition for its rapid convergence that the depth is also slowly varying. We show that if in the case of two-dimensional motions the undisturbed fluid is first mapped conformally onto a uniform strip, before the Taylor expansion is made, the analytic condition is removed and the approximations implied in the lowest-order equations are much improved.In the limit of infinitesimal waves of very long period, consideration of the form of the error suggests that by modifying the coefficients of the reformulated equation we may find an equation exact for arbitrary depth profiles. We are thus able to calculate the reflexion coefficients for long-period waves incident on a step change in depth and a half-depth barrier. The forms of the coefficients of the exact equation are not simple; however, for these particular cases, comparison with the coefficients of the reformulated long-wave equation suggests that in most cases the latter may be adequate. This opens up the possibility of beginning to study finite amplitude and frequency effects on regions of rapidly varying depth.

Soliton dynamics in a strong periodic field: The Korteweg–de Vries framework

“…in particular for the Korteweg-de Vries and Boussinesq equations specially derived for the this geometry (Hamilton, 1977;Rosales and Papanicolaou, 1983;Nachbin and Papanicolaou, 1992;Nachbin, 2003). It is important to say that the same situation should be realized in the final stage of wave transformation on very long shelves, because due to the iinitial damping, the width of the soliton will become comparable with shelf width, and its damping will then be stopped.…”

On step approximations for water-wave problems

“…See also Keller & Weitz (1953). Furthermore the transformation appears to convert rapid changes in depth to smoother variations in the free-surface condition which enables a wider range of bottom topographies, even discontinuities, to be considered (see Hamilton 1977).…”

Oblique Water Wave Diffraction by a Step

“…The problems of water wave scattering by an irregular bottom have received some considerable interest in the literature on linearised theory of water waves due to their importance in finding the effects of naturally occurring bottom obstacles such as sand ripples on the wave motion (cf. Roseau [6], Kreisel [7], Fitz Gerald [8], Hamilton [9], Newman [10], Miles [11], Mandal and Gayen [12], Dolai and Dolai [13]). …”