Introduction

In the problems of interest to us, the solution is either a vector or a function. How can we measure the error of approximations to such solutions? One approach is to measure the error in each component of the vector or in each value of the function. Then the error is itself a vector or a function. However, such error vectors and functions provide a wealth of useless information, and may actually obscure critical evaluation. Thus, for most purposes, we prefer a single number measuring the overall size of the error. Such a measure is called a vector or functional norm.

After discussing error norms, this chapter describes the two basic categories of numerical errors: round-off errors and discretization errors. Round-off errors are any errors caused by the use of finite-precision real numbers, rather than the true infinite-precision real numbers. Discretization errors are any errors caused by using a finite-dimensional vector rather than the true infinite-dimensional vector, or a finite sequence rather than the true infinite sequence, or a finite series rather than the true infinite series, or a finite sequence or series such as a truncated Taylor series rather than the true function.

Norms and Inner Products

This section concerns vector and functional norms and inner products. Consider a scalar x. The absolute value |x| measures the size of x. Recall that the absolute value has the following properties:

The last property is sometimes called the triangle inequality.

Consider a vector x. The norm ∥x∥ measures the size of x. More specifically, ∥x∥ is any scalar function of x such that

Vector norms are the natural extensions of absolute value to vectors.

Introduction

This chapter concerns simple scalar models of the Euler equations, called scalar conservation laws. Scalar conservation laws mimic the Euler equations, to the extent that any single equation can mimic a system of equations. In order to stress the parallels between scalar conservation laws and the Euler equations, the first part of this chapter essentially repeats the last two chapters, albeit in a highly abbreviated and simplified fashion. As a result, besides its inherent usefulness, the first part of this chapter also serves as a nice review and reinforcement of the last two chapters.

Like the Euler equations, scalar conservation laws can be written in integral or differential forms. In integral forms, scalar conservation laws look exactly like Equation (2.21) except that the vector of conserved quantities u is replaced by a single scalar conserved quantity u, and the flux vector f(u) is replaced by a single scalar flux function f(u). Similarly, in differential forms, scalar conservation laws look exactly like Equations (2.27) and (2.30) except that the vector of conserved quantities u is replaced by a single scalar conserved quantity u, the flux vector f(u) is replaced by a single scalar flux function f(u), and the Jacobian matrix A(u) is replaced by a single scalar wave speed a(u), not to be confused with the speed of sound.

Replacing several interlinked conserved quantities by a single conserved quantity dramatically simplifies the governing equations while retaining much of the essential physics. In particular, scalar conservation laws can model simple compression waves, simple expansion waves, shock waves, and contact discontinuities. Like simple waves in the Euler equations, scalar conservation laws have a complete analytical characteristic solution.

INTRODUCTION

In this chapter, the stochastic properties and probability distributions applicable to wave amplitude and height will be presented under various conditions.

First, the underlying assumptions considered throughout this chapter are that random waves are considered to be weakly steady-state, ergodic, and a normal (Gaussian) random process. The normal process assumption is valid only for waves in deep water. Waves in finite water depths are commonly treated as nonlinear, and considered to be a non-Gaussian random process, as will be discussed in detail in Chapter 9. With these basic assumptions, probability distribution functions which represent the statistical characteristics of random waves are analytically derived.

The most commonly considered probability distribution for wave amplitude is developed assuming that the wave spectrum is narrow-banded. The wave profile under this condition is slowly changing with constant period, and there exists a single peak or trough during each half-cycle. Waves generated by moderate wind speeds, an example of which is shown in Figure 3.1(a), demonstrate that the narrow-banded spectrum assumption is generally acceptable. In this case, wave amplitude follows the Rayleigh probability law as will be presented in Section 3.2. The Rayleigh probability distribution is most commonly considered for the design of marine system.

This book is intended to provide uniform and concise information necessary to comprehend stochastic analyss and probabilistic prediction of wind-generated ocean waves.

Description and assessment of wind-generated ocean waves provide information vital for the design and operation of marine systems such as ships and ocean and coastal structures. Wind-generated seas continuously vary over a wide range of severity depending on geographical location, season, presence of tropical cyclones, etc. Furthermore, the wave profile in a given sea state is extremely irregular in time and space — any sense of regularity is totally absent, and thereby properties of waves cannot be readily defined on a wave-by-wave basis.

Characterization of the stochastic properties of ocean waves was first presented in the early 1950s; Neumann (1953), Pierson (1952, 1955), St Denis and Pierson (1953) introduced the stochastic approach for analysis of random seas, and Longuet-Higgins (1952) demonstrated the probabilistic estimation of random wave height. The four decades following the introduction of the stochastic prediction approach have seen phenomenal advances in the probabilistic analysis and prediction methodologies of random seas.

For the design of marine systems, information on the real world is required. Recent advances in technology permit the use of the probabilistic approach to estimate the responses of marine systems in a seaway, including extreme values, with reasonable accuracy.

STOCHASTIC CONCEPT AS APPLIED TO OCEAN WAVES

Introduction

The profile of wind-generated waves observed in the ocean changes randomly with time; it is non-repeatable in time and space. In reality, both wave height (peak-to-trough excursions) and wave period vary randomly from one cycle to another. It is often observed that waves break when the wave steepness exceeds a certain limit. Furthermore, during the process of the wind-generated waves traveling from one location to another after a storm, waves of shorter length gradually lose their energy resulting in the wave profile becoming less irregular (this situation is called swell) than that observed during a storm.

A more distinct difference in the wave profile can be observed when the water depth becomes shallow. As an example, Figure 1.1 shows portions of wave profiles recorded in severe seas; one in deep water in the North Atlantic, the other in a nearshore area of water depth 2.1 m. As seen in Figure 1 (a), positive and negative sides of the wave profile in deep water are, by and large, similar, while for waves in shallow water (Figure 1(b)), peaks are much sharper than troughs, and the order of magnitude of the peaks is different from that of the troughs.

STATISTICAL PRESENTATION OF SEA SEVERITY

Probability distribution of significant wave height

Statistical presentation of sea severity provides information vital for the design and operation of marine systems. For the design of marine systems, information is necessary not only on the severest sea condition expected to occur during the system's lifetime (50 years for example), but also on the frequency of occurrence of all sea conditions, the latter being especially necessary for evaluating fatigue loadings.

The most commonly available information on sea severity is the statistical tabulation of significant wave height constructed from data accumulated over several years. Of course, the greater the number of accumulations, the more reliable the data. As to the time interval between data sampling, it is highly desirable that data be obtained at least at 3-hour intervals so that a relatively fast change in sea condition will not be missed. During a storm, sampling at no more than one hour intervals is strongly recommended (see Section 5.2.1).

Table 5.1 is a tabulation of 5412 significant wave heights obtained over a 3-year period in the North Sea (Bouws 1978). The data indicate that the measurements were made, on average, five times per day throughout the 3 years.

INTRODUCTION

For a complete description of wind-generated random waves, it is necessary to consider wave height and period as well as the direction of travel. In particular, serious consideration must be given to the combined effect of height, period and direction, if any correlation exists. Wave data measured in the ocean show that period dispersion for very large wave height is not widely spread; as is also the case of height dispersion for large wave periods. In other words, height and period of incident waves are not statistically independent. Hence, the joint probability distribution of wave height and period plays a significant role in predicting statistical properties of waves such as the frequency of occurrence of wave breaking in a seaway.

Wave breaking takes place when wave height and period cannot maintain the equilibrium condition needed for stability. Therefore, for estimating the possibility of the occurrence of wave breaking in a given sea condition, knowledge of statistics on wave height and associated period, namely, the joint probability distribution, is necessary.

Further, the joint probability distribution of wave height and period is of the utmost importance for the design of floating marine systems. This is because one of the most important considerations for the design of a floating marine system lies in estimation of the possible occurrence of resonant motion which may occur when wave periods are close to the natural motion period of the system.

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.