On Wave Kinematics Approximate Methods
Authors: Zhang Jun , Randall R.E. , Spell C.A 
Published: October 2008

ABSTRACT
Three common approximate methods known as Wheeler stretching, linear and vertical extrapolation are investigated to evaluate their application for predicting wave kinematics. The measured horizontal velocity of regular and dual component waves, which conceptual1y represent a very narrow and broad bandwidth wave train respectively, is found to be satisfactorily consistent with numerical results based on the finite-amplitude wave theory. Both numerical results and measurements are compared with the predictions of the three methods, and this comparison indicates that the prediction accuracy of each approximate method depends upon the bandwidth of the wave spectrum. Wheeler stretching better predicts wave kinematics for a broad bandwidth spectrum and 1inear extrapol ation is the better predictor for a narrow bandwidth spectrum.

INTRODUCTION
The accurate prediction of the kinematics of irregular ocean surface waves is essenti a1 to offshore and coastal engineering, and it is also a challenging and sophisticated task for the study of non linear wave mechanics. Historically, particle velocities induced by irregular waves have been predicted approximately using the fast Fourier transform (FFT) spectral method, which linearly decomposes the wave elevation into regular component waves and superposes their kinematics. Under the linear wave assumption, non linear wave interactions are ignored. Consequently, large discrepancies are found between predicted and measured velocities near the surface, where the contribution from the large wavenumber (high frequency) tail of the wave spectrum becomes unrealistically large. To correct the discrepancy, semi-empirical and semi-theoretical modifications have been made to the spectral method, resulting in a variety of approximate methods for the prediction of wave kinematics. The commonly used method include Wheeler stretching1, linear extrapolation2 and vertical (uniform) extrapolation3 all of which provide fast computation time. However, recent studies have revealed that predicted wave kinematics using the three aforementioned approximate methods cause a fifty to one hundred percent difference in the predicted responses of compliant offsh re structures, such as a complianttower platform.3 The 1arge discrepancy has rai red concerns for the use of these approximate methods and stimulated interest in their reinvestigation, verification and development of new methodologies for predicting wave kinematics. The long-term objectives are to thoroughly understand the nonlinear interaction between component waves and to develop a methodology which accurately predicts wave kinematics. In this study, the objective is to examine Wheeler stretching, linear extrapolation and vertical extrapol ation methods to provide a better understanding of their prediction errors and application range. Several studies2–4 have compared predicted wave kinematics with field or laboratory measurements directly or indirectly in the past decade, and the conclusions have been inconsistent and sometimes controversial regarding the superiority of a particular approximate method. For example, Forristall4 compared the predicted results from Wheeler stretching and linear extrapolation with the measurements of Exxon Ocean Test Structure (OTS) experiments and Netherlands Marine Technology Research (MATS) program and found no consistent evidence indicating that one metho is generally superior to the other. Steel et a1. compared the predicted and measured wave response of a compliant tower and indirectly concluded that Wheeler stretching was the better overall approximation.