6.5. Stream Function Solutions

The Dean Stream Function wave theory (Dean 1965) may be applied to analyses of both symmetric (theoretical) and asymmetric (real) ocean wave profiles. Dalrymple (1974) modified the Dean Stream Function algorithm for nonlinear waves propagating on a shear current where the shear current velocity was modeled by either a linear or a bilinear steady current profile. Computational procedures were modified by adding two Lagrangian constraints to the iterative numerical algorithm given by Dean (1965) that resulted in convergence of the solution to a specified wave height H and to a zero-mean free surface displacement ? =0. For a design wave condition specified by the wave height H, the wave period T and the water depth h, the field variables are computed by a finite Fourier cosine series. The solution chosen satisfies the governing field equation, the bottom boundary condition, and the kinematic free-surface boundary condition exactly. The unknown Fourier coefficients are then computed iteratively such that the dynamic free surface boundary condition errors are minimized in a best least-squares sense.

Von Schwind and Reid (1972) develop a stream wave function wave theory that is similar to Dean (1965). The principal difference between the two theories is that Von Schwind and Reid applied a conformal transformation of the coordinates for the boundary value problem from the complex ( x+iz) plane to the complex ( +i ?) plane. The dimensionless boundary value equations that result from the conformal transformation may be...