INTRODUCTION

Wave spectra discussed in earlier chapters represent wave energy at a certain location in the ocean where the wave energy is an accumulation of the energy of all waves coming from various directions. The spectrum may therefore be called the point spectrum. The spectral analyses and prediction methods of wave heights and periods presented in the preceding chapters assume that wave energy is traveling in a specific direction, commonly considered the same direction as the wind. In this respect, the wave spectrum may be considered as a uni-directional spectrum.

In reality, however, wind-generated wave energy does not necessarily propagate in the same direction as the wind; instead, the energy usually spreads over various directions, though the major part of the energy may propagate in the wind direction. Thus, for an accurate description of random seas, it is necessary to clarify the spreading status of energy. The wave spectrum representing energy in a specified direction is called the directional spectrum, denoted by S(ω, θ).

Information on wave directionality is extremely significant for the design of marine systems such as ships and ocean structures. This is because the responses of a system in a seaway computed using a unidirectional wave spectrum are not only overestimated but the associated coupled responses induced by waves from other directions are also disregarded.

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).

INTRODUCTION

The statistical analysis of ocean waves discussed in previous chapters assumes that waves are a Gaussian random process; namely, waves are a steady-state, ergodic random process and displacement from the mean obeys the normal probability law. Verification that deep ocean waves are a Gaussian process was given in Section 1.1 through the central limit theorem. It has also been verified through observations at sea as well as in laboratory tests that waves can be considered a Gaussian random process even in very severe seas if the water depth is sufficiently deep.

The above statement, however, is no longer true for waves in finite water depth. Time histories of waves in shallow water show a definite excess of high crests and shallow troughs as demonstrated in the example shown in Figure 1.1(b), and thereby the histogram of wave displacement is not symmetric with respect to its mean value, as shown in Figure 1.2(b). Thus, waves in shallow water are considered to be a non-Gaussian random process. This implies that ocean waves transform from Gaussian to non-Gaussian as they propagate from deep to shallow water.

Figure 9.1 shows a portion of wave records measured simultaneously at various water depths during the ARSLOE Project carried out by the Coastal Engineering Research Center at Duck, North Carolina.

BASIC CONCEPT OF EXTREME VALUES

This section presents the theoretical background for predicting extreme values (extreme wave height, extreme sea state, etc.) which provide invaluable information for the design and operation of marine systems. The extreme value is defined as the largest value of a random variable expected to occur in a specified number of observations. Note that the extreme value is defined as a function of the number of samples. In the naval and ocean engineering area, however, it is highly desirable to estimate the largest wave height expected to occur in one hour, or the severest sea state expected to be encountered in 50 years, for example. This information can be obtained by estimating the number of waves (or sea states) per unit time, and thereby the number of samples necessary for evaluating the extreme value is converted to time.

The concept supporting the estimation of extreme values is order statistics which is outlined in the following. Let us consider a sample set consisting of wave heights taken in the sequence of observations (x1, x2, …, xn). Each element of the random sample xi is assumed to be statistically independent having the same probability density function f(x). In the case of wave height observations, each xi is considered to obey the Rayleigh probability law.

We can only obtain analytic solutions to the equations describing nonlinear waves for a very limited set of geometries, constitutive equations, and boundary conditions. For example, the Riemann integrals yield a solution for a one-dimensional simple wave in an infinite volume of nonlinear-elastic material; however, obtaining analytic solutions for the interaction of two simple waves ranges from difficult to impossible depending upon the constitutive equations of the medium. Without the advent of the digital computer and sophisticated methods of numerical analysis, advances in the field of nonlinear wave motion and the development of constitutive theories for dynamic loading of materials would have stagnated over the past 40 years.

The study of waves is important to virtually every branch of science and engineering. Indeed, waves are also important to everyday life. Sound waves allow us to hear, and electromagnetic waves allow us to see. In this book, we restrict our study to one important class of waves often called stress waves. These waves propagate through gases, liquids, and solids. Stress waves in gases and liquids are usually called pressure waves. This nomenclature is derived from the fact that internal forces in solid bodies are represented by a stress tensor, and internal forces in inviscid liquids and gases are represented by a special form of stress called pressure. In Chapter 1, we shall learn the distinction between stress and pressure.

Sound is the most commonly experienced pressure wave. Indeed, anyone standing in a thunderstorm knows that lightening is seen before the report of its thunder is heard. This simple observation reveals the single, most important feature of any wave – its finite velocity. We know there is always a lapse of time between the cause of a sound wave and when we hear it.

Elastic Material

It is a common observation that individual materials react differently to applied loads.

We often calculate nonlinear wave phenomena with complex numerical computer codes. With such powerful capabilities you may be tempted to simply characterize the energy state of a material with large tables of data that yield values of equilibrium stress, temperature, stiffnesses, specifics heats, and thermal expansion coefficients for given values of F and η. Accurate computations of some very common materials rely heavily upon this method. The computation of nonlinear pressure waves in water is an important example. However, even with this capability, questions arise about the constitutive descriptions of these materials. For example, if water is heated by a shock wave, it often spontaneously “flashes” to steam when the pressure drops again in response to a release wave.

Index Notation

We begin by introducing a compact and convenient way to describe quantities in three spatial dimensions with a Cartesian coordinate system. We also show how to transform quantities between different Cartesian coordinate systems.

Cartesian Coordinates and Vectors

A Cartesian coordinate system has three straight coordinates that are mutually perpendicular to each other (see Figure 1.1). The coordinates are named x1, x2, and x3. The subscripts 1, 2, and 3 are called indices. When we let k represent any of these indices, we can refer to the coordinates by the compact notation xk. Each coordinate xk also has a shaded arrow that represents a unit coordinate vectorik. We use a boldface symbol to denote a vector. The length of each ik is equal to unity.

Now consider another vector v that is drawn with arbitrary angles to the three coordinates. We draw a box with sides parallel to ik and with v as a diagonal.

Why must we study thermodynamics? In Chapter 2 we studied a wide range of wave phenomena using the mechanical concepts of force and motion. Using only the constitutive model for an elastic material, we developed detailed analyses of the nonlinear structured wave, the transition of a compression wave into a shock wave, and the decay of a shock wave as it is overtaken by a rarefaction wave. We have analyzed wave motion in solids, liquids, and adiabatic gases.

During July 1994 while I was writing this book, fragments of the comet Shoemaker–Levy 9 crashed into Jupiter. As each fragment entered the Jovian atmosphere, it formed a shock wave, similar to the bow wave in front of a boat. Heat generated within these waves ignited the atmosphere of Jupiter creating fireballs the size of the earth.