HIGHER-ORDER WATER-WAVES IN CURRENTS OF UNIFORM VORTICITY, IN THE PRESENCE OF SURFACE-TENSION

A third order Stokes expansion is developed for periodic waves travelling on an opposing surface current whose undisturbed horizontal velocity profile is varying linearly down to some fixed depth h and zero otherwise. Deep water is assumed beneath the current. Surface tension is included. Special emphasis is laid on the case of deep currents, e-2kh much greater than 1, since this case permits useful analytical approximations. Third order Stokes theory is necessary to obtain the first non-vanishing correction to the phase velocity, which is of second order.