Statistics of Ocean Wave Groups
Authors: Nolte K.G. 
Published: December 2007

Abstract
Statistical parameters are derived that describe the occurrence probability for the number and height of consecutive waves in as group, which are large compared with the average wave height. Such wave groups can create extreme forces in mooring lines of large vessels. All the parameters that describe the statistics of wave groups can be derived from the energy spectrum representing the sea state, if the energy spectrum is assumed to contain only a narrow band of wave frequencies. Good agreement was found between the theoretical expression derived in this paper and actual wave data.

Introduction
It has been shown by Hsu and Blenkarn that be long-period sway motion and peak mooring-line force observed in a model study of a barge resulted from the combined action of a sequence of consecutive high waves in the random wave train. Such a sequence of waves will be referred to as a wave group. Independent model-tank tests conducted by Remery and Hermans substantiated the findings presented in Ref. 1. it is anticipated that the presented in Ref. 1. it is anticipated that the action of wave groups may also be important in other types of ocean engineering systems that are subjected to horizontal motions with periods much longer than the period of individual waves. Such systems could include dynamically positioned vessels. In the analysis and design of such systems it would be prudent to check whether the action of wave groups could subject the systems to motions beyond those calculated on the basis of individual waves. As a basis to perform such a check, one can use either recorded wave groups measured in the sea, or wave groups generated in a model tank. There are also available electronic and numerical methods to simulate ocean waves. However, to make proper engineering decisions, one would need to know whether the occurrence of such wave groups is a common or a rare event. More specifically, one would require a probabilistic description of the basic characteristics of wave groups in order to assess whether a system is likely to survive. The purpose of this paper is to develop tools with which one can characterize the statistics of wave groups in terms of occurrence probabilities. No attempt is made here to illustrate how such statistical tools could be used in practical engineering. Rational procedures for utilizing such tools for specific problems can best be developed by individual users.

Theoretical Derivations
In this paper a wave group will be defined in terms of the envelopes of wave crests and troughs as shown in Fig. 1. The distance between the crest envelope and the trough envelope will be called the height envelope, whereas the distance between the crest envelope and the mean water surface will be called the amplitude envelope. The height envelope indicates the change of wave heights. Similarly, the amplitude envelope indicates the change of wave amplitude. In reality, ocean waves are not symmetrical as is implied by the term "amplitude". Measured from mean water level, the distance to the crest of a wave is generally greater than the distance to the trough. However, in order to use the theoretical methods developed by Rice, we shall assume that the wave amplitude is half the wave height. As one can see from Fig. 1, the height envelope has the characteristics of a wave with a much longer period than the individual waves. period than the individual waves.