OSCILLATORY 3RD-ORDER PERTURBATION SOLUTIONS FOR SUMS OF INTERACTING LONG-CRESTED STOKES WAVES ON DEEP-WATER

Oscillatory third-order perturbation solutions for sums of interacting long-crested Stokes waves on deep water are obtained. A third-order p erturbation expansion of the nonlinear free boundary value problem, de fined by the coupled Bernoulli equation and kinematic boundary conditi on evaluated at the free surface, is solved by replacing the exponenti al term in the potential function by its series expansion and substitu ting the equation for the free surface into it. There are second-order changes in the frequencies of the first-order terms at third order. T he waves have a Stokes-like form when they are high. The phase speeds are a function of the amplitudes and wavenumbers of all of the first-o rder terms. The solutions are illustrated. A preliminary experiment at the United States Naval Academy is described. Some applications to se akeeping are bow submergence and slamming, capsizing in following seas and bending moments.