SPECTRAL ANALYSIS OF RANDOM WAVES

Fundamentals of stochastic processes

Throughout this chapter as well as others, we will evaluate various characteristics of wind-generated waves based on the stochastic process concept. The fundamentals of the stochastic process concept are outlined here.

First, the stochastic process (or random process), x(t), is defined as a family of random variables. In the strict sense, x(t) is a function of two arguments, time and sample space. To elaborate on this definition of a stochastic process, let us consider a set of n wave recorders (1x, 2x, 3x, …, nx) dispersed in a certain area in the ocean as illustrated in Figure 2.1 (a). Let us consider a set of time histories of wave records {1x(t), 2x(t), …, nx(t)} as illustrated in Figure 2.1(b). It is recognized that at any time tj, x(tj) is a random variable, and a set {1x(tj), 2x(tj), …, nx(tj)} can be considered as a random sample of size n. This simultaneous collection of wave data observed at a specified time is called an ensemble. If we construct a histogram from a set of ensembles of wave records, it may be normally distributed with zero mean and a certain variance as shown in Figure 2.1(c).