Characteristics of gravity waves of permanent form

Abstract

In the classical problem of two-dimensional gravity waves of permanent form over a horizontal sea bed, one seeks a velocity potential and/or a stream function which satisfies Laplace's equation plus certain boundary conditions and symmetry conditions (see for example, Lamb, 1945, art. 227). Although the velocity potential satisfies a linear condition at the sea bed and in the interior of the fluid, it has a highly nonlinear boundary condition imposed at the sea surface. Part of this nonlinearity arises because the shape of the surface itself is a dependent variable of the problem. The classical procedure for the arrested wave form (i.e., the Stokes representation) is to express the complex velocity potential (φ + iψ) as a cyclic function of the complex coordinate (x + iy), the functional form being taken as a Fourier series with unknown coefficients such that the nonlinear surface boundary condition are satisfied. In contrast to the above procedure one can regard (x + iy) as the dependent and (φ + iψ) as the independent (complex) variable of the problem and seek a cyclic representation of the former in terms of the latter. This procedure has the advantage that the free-surface condition occurs as a known constant value of the coordinate ψ. Moreover, both the velocity field and the surface profile can be evaluated directly form the cyclic function F(φ + iψ). In this work the surface boundary condition is expressed in non-dimensional form in terms of the above representation. An iterative procedure which requires a simple harmonic analysis at each step is employed to obtain successive approximations of the coefficients entering into the unknown cyclic functions. This set of coefficients is calculated for 62 sets of assumed initial wave conditions through an iterative numerical process. The output of this process is compared with results obtained from Stokes' first, third, and fifth order theory by previous investigators.