In offshore design, currents must be included in the wave force design procedure as they always are present due to the actions of the wind, tidal forces, and oceanic circulation. The neglect of these currents can result in the under-design of a structure; for instance, if the maximum velocity under a design wave is 16 fps, the inclusion of a 2 fps current component will increase the drag force by 25%. A 3 fps current component will increase the drag force by over 40%. However, most wave force design procedures neglect the current, use a constant current over depth, or, as used by Hall (1972), superimpose current profiles over the wave velocity field generated in the absence of a current.
In this paper, two wave models are provided for the computer generation of a design (symmetric) wave of given height, H, wave period, T, in a known water depth, h, which propagates on an ambient current which has a velocity profile over depth describable by two or three straight lines. These are the linear and the bilinear shear current models, respectively. These models represent an extension of the Stream Function wave theory described by Dean (1965), which generates a wave propagating on a current with a velocity profile which is constant in magnitude over the depth of the water. These extended models, which include more realistically the shape of the current profile, offer the design engineer a better capability to input accurate current data into the design procedures. As an example of the fits of the two current representations to a reasonable current velocity profile, Figure 1 is presented. One further advantage of these models is that they converge directly on a desired wave height.
To treat the problem of wave propagating on a current in the ocean mathematically, simplifying assumptions are necessary. First, the current velocity profile over the water column is well-established and thus, over a short period of time, the effect of viscosity may be neglected. Next, the directions of the waves and the current are the same. 1 The wave is assumed to be periodic in space- -and time and to travel without change in form. This last assumption, which is most valid for large swell, makes it possible to render the wave motion steady in time by translating the coordinate system with the speed, C, of the wave. The fluid is taken to be incompressible which can be expressed mathematically in two dimensions as
(available in full paper)where U is the steady current velocity, u and v are the wave-induced horizontal and vertical water particle velocities, and x and yare the horizontal and vertical coordinate directions. Incompressibility is reasonably valid for ocean waves and allows the definition of a stream function, ?(x, y), for the steady wave motion,(available in full paper)
