Nonlinear wave loads on a slender vertical cylinder

TLDR

Diffraction of water waves by a vertical circular cylinder is studied in the regime of small amplitude and radius relative to wavelength, where linear theory applies far from the cylinder but significant nonlinear effects arise near the cylinder due to the free‑surface boundary condition. The leading‑order nonlinear velocity potential contains terms proportional to A²a and A³, producing wave loads with second‑ and third‑harmonic components proportional to A²a² and A³a, and the second‑ and third‑order force components are of comparable magnitude for realistic wave slopes.

Abstract

The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a , it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A 2 a and A 3 . The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A 2 a 2 and A 3 a . In a conventional perturbation analysis, where A [Lt ] a , these components would be ordered in magnitude corresponding to the different powers of A , but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.