A numerical method for calculating the interaction of steep (nonlinear) ocean waves with large coastal or offshore structures of arbitrary shape is described. The evolution of the flow, and in particular the loads on the structure and the runup around it, are obtained by a time stepping procedure in which the flow at each time step is calculated by an integral equation method based on Green's theorem.

A few comparisons are made with available solutions and results are presented for a typical design wave in shallow water. It is concluded that the method is capable of predicting forces due to steep waves quite accurately and without prohibitive computer effort.


The prediction of wave forces on large offshore structures on the basis of linear diffraction theory, which is formally valid for small amplitude sinusoidal waves, is now an established part of offshore design procedure. Reviews of the approaches generally used have been given by Hogben et al.2, Isaacson4 and in the text by Sarpkaya and Isaacson10.

In order to account more realistically for the effect of large wave heights, research has recently been directed primarily towards developing a second approximation based on the Stokes expansion procedure. However, such an approach is of practical value only under somewhat restricted conditions, as in the case of an undisturbed wave train described by Stokes second-order theory. In particular, nonlinear wave effects are expected to be of greatest importance for steep shallower waves, and these are precisely the conditions in which a Stokes second-order solution becomes invalid.

A numerical solution to the complete boundary value problem without any wave height perturbation procedure is clearly desirable. The approach outlined here is described in detail in a report by Isaacson5. In this method, the wave diffraction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident wave form approaching the structure. The development of the flow can then be obtained by a time stepping procedure, in which the velocity potential of the flow at anyone instant is obtained by an integral equation method based on Green's theorem.

Comparison with known diffraction solutions can be made only for relatively restricted situations. A few such comparisons have been carried out and are quite favorable. Results are also presented for a typical design wave in shallow water, and these are found to differ significantly from linear theory predictions.

Let x, y, z form a Cartesian coordinate system as indicated in Fig. 1, with × measured in the direction of incident wave propagation and z measured upwards from the still water level. Also let t denote time and ? the free surface elevation above the still water level. The seabed is assumed horizontal along the plane z = -d. The fluid is assumed incompressible and inviscid, and the flow irrotational. (In the usual way, flow separation effects are neglected for a large structure since water particle displacements are small relative to the structure size.)

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