Characteristics of gravity waves of permanent form

A conformal transformation of the equations that constitute the boundary value problem of permanent-form finite gravity waves from the x, y plane to the velocity-potential-stream-function plane is presented. After the resulting equations are given in a nondimensional form the Fourier coefficients of the solutions to this problem are determined for a representative sampling of theoretical waves of various heights at different depths of water through a numerical iterative process. A comparison of the results is made with existing Stokes first, third, and fifth order theories and with Chappelear's [1961] direct numerical calculations. The approach presented has the following advantages: (a) simplification of the boundary conditions through the mapping transformation, (b) one set of coefficients to define the wave characteristics, (c) easy evaluation of the particle trajectories, and (d) a wide range of depths of water and heights of waves to which it can be applied.