The run-up on a cylinder in progressive surface gravity waves: Harmonic components

Wave run-up is the vertical uprush of water that occurs when an incident wave impinges on a partially immersed body. In this present work, wave run-up is studied on a fixed vertical cylinder in plane progressive waves. These progressive waves are of a form suitable for description by Stokes' wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The limitations of canonical wave diffraction theory-whereby the free-surface boundary condition is treated by a Stokes expansion-in predicting the harmonic components of the wave run-up are discussed. An experimental campaign is described and the choice of monochromatic waves is indicative of the diffraction regime for large volume structures where the assumption of potential flow theory is applicable, or more formally A/a < O(1) (A and a being the wave amplitude and cylinder radius, respectively). The wave environment is represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The zeroth-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave nonlinearities. It is shown that the complete wave run-up is not well accounted for by second-order diffraction theory. This is, however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order diffraction theory. Experimental evidence suggests this to be caused by higher than second-order nonlinear diffraction. Moreover, these effects most noticeably operate at the first-harmonic in waves of moderate to large steepness when ka << 1. (c) 2005 Elsevier Ltd. All rights reserved.