An equation is derived which governs the propagation of a cnoidal wave train over a gently sloping bottom. The equation is solved numerically, the solution being tabulated in terms of fH (Eq. 47) as a function of Ei = (Etr/pg) 1/3/gT2 and hi = h/gT2. Results are compared with sinusoidal wave theory. Two numerical examples are included.

This paper considers steady, slowly varying water waves propagating over a gently sloping bed on a steady current. The current varies linearly with depth, and so has constant vorticity ω. The analysis is two-dimensional and dissipation is neglected. Definitions, and expressions correct to second order in the amplitude, are given for the radiation stress, wave energy density E and total energy flux. An average La-grangian [Lscr ], obtained by heuristic arguments from Clebsch potentials, leads to the result that for this particular problem E equals the wave action [Lscr ]ω times the angular frequency ωrm relative to a frame of reference moving with the average-over-depth current velocity Um. This determines the variation of the amplitude with distance explicitly. An analytical expression for the height of the mean water surface is found by a heuristic argument which compares the conservation equations for total energy and wave action. All the results have been checked directly by substitution back into the basic equations. Graphs illustrate the effect of the vorticity ω on the wavelength, amplitude and set-down.

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Severe storm and normal tidal conditions in the North Sea were used to verify a fast-running computer model based on complete nonlinear equations for nearly horizontal flow, including wind stresses, barometric pressure gradients, Coriolis forces, and bottom shear stresses. When complemented with a short, intensive program of local measurements, this model provides detailed information on currents and surface elevations. Introduction Activities associated with production of and exploration for oil have led to increased interest in the hydrography of the North Sea. Construction of a marine pipeline. pollution fighting in case of an oil spill, and, in general, pollution fighting in case of an oil spill, and, in general, operations connected with the large structures presently used in the North Sea require detailed knowledge of hydrographic conditions.The emphasis in hydrographic conditions is often on waves. However. for several important problems, knowledge of currents is also essential. Water-particle movement is a combined effect of wave-induced motion and currents. Although wave-induced motion can be computed fairly well from wave theory, information on currents, especially currents under extreme conditions, often is more difficult to obtain in sufficient detail. This paper presents a computer model that, when complemented presents a computer model that, when complemented with a short. intensive program of local measurements, provides this detailed information. provides this detailed information. For example. currents are important in pipeline construction. During laying, crosscurrents contribute to stresses in the part of the pipe suspended between the lay barge and the sea bed. For an unburied pipe on the sea bed, knowledge of possible extreme currents is important for stability considerations. The risk of scour must be assessed for the buried pipe. Scour may uncover the pipe and a free span may result. Vortex shedding in the current across the pipe may cause the free span to develop to an unacceptable length. A few extreme current: may be enough to cause the problem to develop. About 1 year ago an argument occurred concerning the Ekofisk-Emden pipeline that had become uncovered at some points. This situation may indicate the relevance of this points. This situation may indicate the relevance of this problem. problem. As more oil is produced and transported, the risk of pollution also increases. For effective decision making pollution also increases. For effective decision making when combatting oil pollution, the migration of a slick must be predicted, which requires predicting currents in detail.The size of structures for exploration and production in the North Sea has increased dramatically. As the size increases, so do the problems. In several of these problems, knowledge of currents, generally emphasizing problems, knowledge of currents, generally emphasizing possible extreme values, is important. For example, this is possible extreme values, is important. For example, this is the case in force calculations. Although the actual size of a structure does not alter force calculations in principle, the sheer magnitude of a calamity to a large structure requires calculations to be more precise. Also, more precise calculations could lead to better pay-off by precise calculations could lead to better pay-off by avoiding equipment unnecessarily large for the job. Another structural problem related to currents is the possibility of vortex shedding in tubular frames and the vibrations associated with it.For most of these problems, the normal current situation can be determined with on-site measuring. However, for design considerations, the possible extreme current during or after a design storm must be determined.In our opinion, a computer model, supplemented with a short, intensive program of local measurements. provides this information. provides this information. JPT P. 1369