Error Analysis of Popular Wave Models for Long-Crested Seas
Authors: Xu Hongbo 
Published: October 2008

Abstract
Wave kinematics provide an important basis for wave load computations in design analyses of offshore structures. For simplicity, it is a standard practice in the offshore industry to use the linear wave theory and empirical stretchings to calculate wave kinematics. Popular stretching methods include the Wheeler Stretching!, Delta Stretching and Linear Extrapolation. Although these wave models are practically useful for low sea states, their errors are difficult to assess for higher sea states. Consequently, uncertainty in calculated wave kinematics remains when wave parameters vary. The same is true for calculated wave loads. To reduce the uncertainty in wave kinematics calculations, an error analysis for the forementioned three stretching models has been carried out, and compared with that of second-order Hybrid Wave Model. Extensive comparisons with lab data were made to verify the analytical results. The present results help engineers to reduce uncertainty in their wave load estimates through reducing errors in wave kinematics calculations.

Introduction
In dynamic structural analyses of offshore platforms, the Morison's equation is widely used for wave load calculations. This hydrodynamic load model has two components: theinertial force is proportional to the local fluid acceleration, and the drag force is proportional to the square of the local fluid velocities. The computed inertial force is sensitive to errors in wave acceleration estimates, whereas the calculated drag force is especially sensitive to errors of computed flow velocities near wave surface where the wave motion is larger than that at other depth. Clearly, the accuracy of wave kinematic model is essential in wave load computations.
As the first approximation, the linear random wave theory (LRWT) describes linearized irregular wave motions, wherein the wave field is expressed as the sum of linear wave components. Each component wave is a solution of the linearized wave equations with its own amplitude, frequency, direction, celerity and phase. In LRWT, the wave phase is usually considered a random variable. Once the initial phases of the component waves are given, LRWT indicates that the subsequent wave motions are deterministic. Interactions among wave components, and wind-wave-current are neglected for simplicity. This linearized wave theory provides the basis for current wave load calculations.
Comparisons between experimental data and results computed by the linear wave theory have shown that the agreement is reasonably good at elevations far below the mean water line, but degrades increasingly as elevation increases (e.g., Bosma and Vugts, 1981). Furthermore, the linear wave theory can not be directly used to compute kinematics in wave crests since the linear wave solution is only defined below the mean water line. To overcome these restrictions, considerable efforts have been devoted in the last three decades to improve the accuracy of wave modeling and to extend the capability of computing wave kinematics to the instantaneous free surface. As the results of these relentlessefforts, several extensions and modifications to the linear wave theory have been developed. The popular methods include Wheeler Stretching (Wheeler, 1970), Linear Extrapolation (Forristall, 1981), Delta Stretching (Rodenbusch and Forristall, 1986), Vertical Uniform Extrapolation (steele et al., 1988).