Characteristic Velocities for Higher-Order Stokes Waves in Deep Water

Abstract

The accuracy of asymptotic expressions for phase, wave energy, and wave action velocities for steady, plane gravity waves progressing in deep water and with no ambient currents is investigated. The calculations are to a large extent based on Fenton's fifth-order Stokes wave theory. Using the dispersion relation correct to sixth order for arbitrary water depth, the phase velocity is calculated to fourth order in wave steepness for infinitely deep water; this is in accordance with a number of previously found results. Potential and kinetic energy is calculated, which leads to the energy density correct to sixth order. Comparison with previous findings showed one discrepancy, which can however be explained. Also, the energy flux is found correct to sixth order, leading to a fourth-order expression for the wave energy velocity. From sixth-order action and action flux results, the wave action velocity is finally calculated correct to fourth order. Agreement with more accurate numerical results is excellent for all three velocities except near the wave of maximum steepness. Practical asymptotic formulas are introduced, enabling one to calculate the characteristic velocities explicitly from given wave height and period. Two new exact relations between integral properties for deep water with no current are presented.