Stream function representation of nonlinear ocean waves

Abstract

An analytical stream function expression representing a nonlinear gravity water wave is applied both to the representation of measured wave forms and also to nonlinear theoretical waves. The stream function form is chosen so that it is a solution to the Laplace equation and the bottom boundary condition; the parameters in the stream function expression are chosen by a numerical perturbation procedure that provides a best fit to the kinematic and dynamic free surface boundary conditions. The boundary condition errors associated with the nonlinear stream function representation of four measured wave profiles are compared with estimates of the corresponding errors associated with a linear representation. The stream function method is judged more accurate than a linear method if the wave height is greater than 50% of the breaking height. The stream function method is also applied to represent theoretical waves for which only the wave height and period are available to characterize the wave profile. It is demonstrated that the method represents an improvement over other available nonlinear procedures. The method can be employed to represent wave conditions which include a prescribed uniform steady current and a specified pressure distribution on the free surface.

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

“…,"""1 .. 1>011",,, 10_"",,_ ", Furthermore, although the equations presented here are internally coherent and correspond well to existing theories (e.g. Airy, 1845;Stokes, 1847;1880;Boussinesq, 1871;Korteweg and De Vries, 1895;Dean, 1965;Cokelet, 1977;Miles, 1980;Sakkai and Battjes, 1980;Fenton, 1985;1988;Fenton and McKee, 1990) and published laboratory observations (e.g. Shore Protection Manual, 1984;Dean and Dalrymple, 1991), they have not yet been tested rigorously under field conditions, where complications may arise because of wave interference, different types of marine currents, wave reflection and refraction, as well as irregular bottom topography.…”

“…Nevertheless, although the same basic equations are used for all water depths, they appear to correspond well to most of the major theories (Airy, 1845;Boussinesq, 1871;Stokes, 1880;Cokelet, 1977;Dean, 1965;Fenton, 1985;1988;Fenton and McKee, 1990) within their applicable depth zones.…”

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“…The stream function solution is found iteratively as a solution of Laplace's equation, satisfying (numerically) the bottom BC as well as the kinematic and dynamic freesurface conditions. The fluid kinematics are then computed as the derivatives of the scalar stream function (Dean, 1965). Fig.…”

“…17b). The wave profiles were time shifted such that t' = 0 corresponds to the maximum crest elevation; all profiles being shown over one wave period T. The analytical expressions required to compute the wave profiles and water particle kinematics are excluded here for brevity, and reference is made to Fenton (1985) (5 th order Stokes), Wiegel (1960) (2 nd order cnoidal), Munk (1949) (higher order solitary) and Dean (1965) (stream function). The evaluation of the theoretical wave profiles is based upon the experimental amplitudes and up-crossing periods.…”

“…Stokes solution (Fenton, 1985), a 2 nd order Cnoidal model (Wiegel, 1960), a higher order solitary wave theory (Munk, 1949) as well as stream function theory (Dean, 1965).…”