A Probabilistic Method for Predicting Velocities and Accelerations of Water Wave Particles
Authors: Schueller G.I. 
Published: October 2008

1. Abstract
The accurate prediction of water wave particle velocities and accelerations is needed for the calculation of wave forces on ocean structures. Most of the methods of calculating these forces until now used Airy's or Stokes' theoretical expressions. Experimental studies performed by Morison (1953), Elliot (1953) and Le Mehaute et. al. (1968) reveal that Airy's and Stokes' expressions are not adequate, even in the range of their validity. The method used here is free from theoretical assumptions about wave mechanisms and hence, explains the behavior of experimental results. Water particle velocities and accelerations are considered random variables. Morison's and Elliot's data are utilized. Multiple linear regression analysis is applied for the calculation of the mean value functions and all related statistics. The assumption of the lognormal distribution of both variables is reasonably well verified. For this purpose error and normal probability graphs are plotted. Partial correlation coefficients and regression coefficients indicate the importance of the independent variables. Horizontal particle velocity and acceleration are most sensitive to an increase in wave length normalized over the wave height.

2. Introduction
The accurate prediction of wave forces acting on submerged marine structures such as piling is a vital problem of offshore technology. Applying either of the two most commonly used semi empirical approaches described by Morison et. al.1 and Crooke2 for predicting wave induced forces one is confronted with the problem of choosing a suitable method for determining the fluid motion i.e., water particle velocities and accelerations. For this purpose various wave theories have been employed. Their relative validity has been discussed by several investigators. 3, 4 Dean3 has compared the closeness of fit of several theories to the free surface boundary conditions as a criterion of relative validity.
However the one theory providing the best fit to the boundary conditions must not necessarily provide the best prediction equations for water particle velocities and accelerations. Le Mehaute et.al.4 based their comparison upon the horizontal particle velocity under the crest, which they also determined experimentally, as the single most important feature. Since for the calculation of wave induced forces the knowledge of the water particle motion is of primary importance the obvious next step is, rather than compare the results of an analytical method with experimental values and then choose on this basis the most appropriate theory for certain wave conditions, to develop a method which is solely based on experimental data. This can be done by employing a statistical analysis such as the multiple linear regression analysis.

3. Multiple Linear Regression Analysis
By statistical analysis, an empirical description of the relationship between the variables is obtained and a discussion of the uncertainty of the relationship is found. It is essentially a method of analyzing experimental data that relates (either known mathematical variables or observed results of random variables). The method of least squares forms the theoretical basis for the estimation of the parameters in the regression analysis. The procedure is entirely empirical. Thus it eliminates the need for initial assumptions about the wave properties (i.e., low finite height in deep water, infinitesimal height, shallow water, etc.).