Wave Prediction in Shallow Water

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Abstract
Introduction Considerable research has been concentrated on deep-water prediction in recent years. This has led to the development of rather sophisticated wave prediction schemes which have been tested successfully in field experiments under a variety of conditions. However, progress in deep-water wave prediction has not been followed by commeasurate advances in shallow water wave prediction. Since many offshore structures are located in relatively shallow water, this inability to obtain accurate predictions can have large economic consequences. There are two types of shallow water wave-prediction which this paper intends to address. First, there is the situation in which the entire generation area lies in relatively shallow water. Second, there is the situation in which waves are generated in deep water can be integrated over all propagation directions and then propagate into shallow water. The first situation can be important in selected areas such as semi-enclosed basins, shallow bays or regions where ice cover can restrict the fetch area to a shallow region along the coast. The latter situation is important in coastal area all around the world, where large storm waves oropagate from deep water in to the coast. The theory presently used by most engineers in estimating shallow water wave heights is based on published charts in the Shore Protection Manual (1973). Basically, the concept behind this theory follows from a theoretical treatment of direct atmospheric input into a monochromatic wave train with a simultaneous solution for the rate of energy loss due to bottom friction (Bretschneider and Reid, 1953). It is well known that the bottom friction factor is highly sensitive to bottom materials. Since the energy flux into the wave field from the atmosphere should be somewhat independent of bottom materials, this theory suggests that the predicted wave heights should be highly dependent on the bottom conditions. The published graphs in the Shore Protection Manual are specifically for a bottom friction factor 0.Ol. Theoretical Treatment Of Wave Generation In Shallow Water For surface gravity waves in a constant depth of water, the rate of change of a two-dimensional wave spectrum can be written asEquation (1) (Available in full paper) where E2 (f, ?) is the two-dimensional frequency direction spectrum, S2(f, ?)) is the net energy flux into (or out of) a spectral component, v is the group velocity vector and v is a horizontal gradient operator .In simple, idealized situations, equation 1 can be integrated over all propagation directions to obtain a representation of the rate of change - of the one-dimensional spectrum (Hasselmann et al., 1976).Equation (2) (Available in full paper) where x is the mean propagation direction and v xis the average group velocity along the x-axis and S1 E1 are the forms for S2 and E2 integrated around the propagation direction ?. Hasselmann (1962) developed a theory for nonlinear wave-wave interactions which predicts a transfer of energy among various components in a spectrum.

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