Variations on Fourier wave theory

Abstract

SUMMARYA review of the analytical and numerical background of Fourier wave theory establishes the commonality of existing formulations and identifies a number of analytical and numerical assumptions that are unnecessary. Some formulations in particular lack flexibility in excluding the possibility of Stokes' second definition of phase speed. A generalized formulation is introduced for comparative purposes and it is shown that published solutions differ only in the approach to the limit wave. Detailed consideration of truncation order confirms that it is the crucial parameter, especially at extreme wave heights. All formulations considered are shown to provide acceptable solutions for small to moderately extreme waves.

Stokes Approximation Solutions for Steep Standing Waves

“…This was the expected result, being also the trend predicted for steady progressive waves (Sobey, 1989) of moderate height in deep water.…”

Wave momentum flux parameter: a descriptor for nearshore waves

“…Fourier approximation wave theory (Rienecker and Fenton, 1981;Fenton, 1988;Sobey, 1989) provides good characterization of steady, finiteamplitude waves of permanent form over the entire range of water depths from deepwater to nearshore and for wave heights approaching the limiting steepness. This hybrid analytical/computational methodology represents the wave stream function by a truncated Fourier series that exactly satisfies the field equation (Laplace), the kinematic bottom boundary condition, and the lateral periodicity boundary conditions.…”

“…This hybrid analytical/computational methodology represents the wave stream function by a truncated Fourier series that exactly satisfies the field equation (Laplace), the kinematic bottom boundary condition, and the lateral periodicity boundary conditions. Nonlinear optimization is used to complete the solution by determining values for the remaining unknowns that best satisfy the nonlinear kinematic and dynamic free surface boundary conditions (Sobey, 1989). Generally, more terms are needed in the truncated Fourier series to represent waves with pronounced asymmetry about the still water line, i.e., steep waves and shallow water waves.…”

Phase-resolving integral wave evolution in a coastal environment

“…Steady progressive wave evolution is a significant sub-set of those flows that must be accommodated by Boussinesq-style integral wave evolution equations. For steady progressive water waves, near-exact kinematics are provided by Stokes (or Fourier) Approximation wave theory (Rienecker and Fenton, 1981;Sobey, 1989) in deep water and by Cnoidal Approximation wave theory (Sobey, 2012) in shallow water. This provides the opportunity to explore the phase variation of the balance of terms in the exact mass and momentum conservation equations, and the evolution of this balance from shallow to deep water.…”