Nonlinear wave interaction with a vertical circular cylinder. Part I: Diffraction theory

Abstract

A closed-form solution is developed for the velocity potential resulting from the interaction of second-order Stokes waves with a large vertical circular cylinder. At first-order, the solution is the usual linear diffraction theory. At second-order, the solution consists of forced wave motions, due to nonlinear wave-wave interactions in the free surface boundary condition, plus scattered free wave motions, due to the interaction of the forced waves with the fixed cylinder. The velocity potentials are then used to determine the theoretical free surface elevations around the cylinder consistent to second-order. Second-order terms are found to significantly alter wave envelopes around the cylinder as a result of nonlinear diffraction. For example, the maximum wave crest run-up on the cylinder from the nonlinear theory is found to exceed that predicted by the linear diffraction theory by up to 50%. A brief comparison of the nonlinear theory with the measured run-up data is found to largely confirm the theoretical solution.