Supported by:
CHAPTER 1
AN INTRODUCTION TO OCEAN WAVES
A. K. Laing: editor
| 1.1 INTRODUCTION 1.2 THE SIMPLE LINEAR WAVE | 1.3 WAVE FIELDS ON THE OCEAN
1.3.2 Wave groups and group velocity 1.3.3 Statistical description of wave records 1.3.4 Duration of wave records 1.3.5 Notes on usage of statistical parameters 1.3.6 Distribution of wave heights 1.3.7 The wave spectrum 1.3.8 Wave parameters derived from the spectrum 1.3.9 Model forms for wave spectra |
Ocean surface waves are the result of forces acting on the ocean.The predominant natural forces are pressure or stress from the atmosphere (especially through the winds), earthquakes, gravity of the Earth and celestial bodies (the Moon and Sun), the Coriolis force (due to the Earth's rotation) and surface tension. The characteristics of the waves depend on the controlling forces. Tidal waves are generated by the response to gravity of the Moon and Sun and are rather large-scale waves. Capillary waves, at the other end of the scale, are dominated by surface tension in the water. Where the Earth's gravity and the buoyancy of the water are the major determining factors we have the so-called gravity waves.
Waves may be characterized by their period. This is the time taken by successive wave crests to pass a fixed point. The type and scale of forces acting to create the wave are usually reflected in the period. Figure 1.1 illustrates such a classification of waves.
On large scales, the ordinary tides are ever present but predictable. Less predictable are tsunamis (generated by earthquakes or land movements), which can be catastrophic, and storm surges. The latter are associated with the movement of synoptic or meso-scale atmospheric features and may cause coastal flooding.
Wind-generated gravity waves are almost always present at sea. These waves are generated by winds somewhere on the ocean, be it locally or thousands of kilometres away. They affect a wide range of activities such as shipping, fishing, recreation, coastal and offshore industry, coastal management (defences) and pollution control. They are also very important in the climate processes as they play a large role in exchanges of heat, energy, gases and particles between the oceans and atmosphere. It is these waves which will be our subject in this Guide.
To analyse and predict such waves we need to have a model for them, that is we need to have a theory for how they behave. If we observe the ocean surface we note that the waves often form a rather complex pattern. To begin we will seek a simple starting model, which is consistent with the known dynamics of the ocean surface, and from this we will derive a more complete picture of the wind waves we observe.
The model of the ocean which we use to develop this picture is based on a few quite simple assumptions:
- The incompressibility of the water. This means that the density is constant and hence we can derive a continuity equation for the fluid, expressing the conservation of fluid within a small cell of water (called a water particle);
- The inviscid nature of the water. This means that the only forces acting on a water particle are gravity and pressure (which acts perpendicular to the surface of the water particle). Friction is ignored;
- The fluid flow is irrotational. This means that the individual particles do not rotate. They may move around each other, but there is no twisting action. This allows us to relate the motions of neighbouring particles by defining a scalar quantity, called the velocity potential, for the fluid. The fluid velocity is determined from spatial variations of this quantity.
From these assumptions some equations may be written to describe the motion of the fluid. This Guide will not present the derivation which can be found in most textbooks on waves or fluids (see for example Crapper, 1984).
 |
| Fig 1.1 - Classification of ocean waves by wave period (derived from Munk, 1951) |
The simplest wave motion may be represented by a sinusoidal, long-crested, progressive wave. The sinusoidal descriptor means that the wave repeats itself and has the smooth form of the sine curve as shown in Figure 1.2. The long-crested descriptor says that the wave is a series of long and parallel wave crests which are all equal in height, and equidistant from each other. The progressive nature is seen in their moving at a constant speed in a direction perpendicular to the crests and without change of form.
- The wavelength, λ is the horizontal distance (in metres) between two successive crests.
- The period, T, is the time interval (in seconds) between the passage of successive crests passed a fixed point.
- The frequency, f, is the number of crests which pass a fixed point in 1 second. It is usually measured in numbers per second (Hertz) and is the same as 1/T.
- The amplitude, a, is the magnitude of the maximum displacement from mean sea-level. This is usually indicated in metres (or feet).
- The wave height, H, is the difference in surface elevation between the wave crest and the previous wave trough. For a simple sinusoidal wave H = 2a.
- The rate of propagation, c, is the speed at which the wave profile travels, i.e. the speed at which the crest and trough of the wave advance. It is commonly referred to as wave speed or phase speed.
- The steepness of a wave is the ratio of the height to the length (H/λ).
 |
| Fig 1.2 - A simple sinusoidal wave |
For all types of truly periodic progressive waves one can write:
 | (1.1) |
i.e. the wavelength of a periodic wave is equal to the product of the wave speed (or phase speed) and the period of the wave. This formula is easy to understand. Let, at a given moment, the first of two successive crests arrive at a fixed observational point, then one period later (i.e. T seconds later) the second crest will arrive at the same point. In the meantime, the first crest has covered a distance c times T.
The wave profile has the form of a sinusoidal wave:
 | (1.2) |
In Equation 1.2, k = 2 π/λ is the wavenumber and ω= 2π/T, the angular frequency. The wavenumber is a cyclic measure of the number of crests per unit distance and the angular frequency the number of radians per second. One wave cycle is a complete revolution which is 2 radians. Equation 1.2 contains both time (t) and space (x) coordinates. It represents the view as may be seen from an aircraft, describing both the change in time and the variations from one point to another. It is the simplest solution to the equations of motion for gravity wave motion on a fluid, i.e. linear surface waves. The wave speed c in Equation 1.1 can be written as λ/T or, now that we have defined and k, as ω/k. The variation of wave speed with wavelength is called dispersion and the functional relationship is called the dispersion relation. The relation follows from the equations of motion and, for deep water, can be expressed in terms of frequency and wavelength or, as it is usually written, between ω and k:
 | (1.3) |
where g is gravitational acceleration, so that the wave speed is:
 |
If we consider a snapshot at time t = 0, the horizontal axis is then x and the wave profile is "frozen" as:
 |
However, the same profile is obtained when the wave motion is measured by means of a wave recorder placed at the position x = 0. The profile then recorded is
 | (1.4) |
Equation 1.4 describes the motion of, for instance, a moored float bobbing up and down as a wave passes by.
The important parameters when wave forecasting or carrying out measurements for stationary objects, such as offshore installations, are therefore wave height, wave period (or wave frequency) and wave direction. An observer required to give a visual estimate will not be able to fix any zero level as in Figure 1.2 and cannot therefore measure the amplitude of the wave. Instead, the vertical distance between the crest and the preceding trough, i.e. the wave height, is reported. In reality, the simple sinusoidal waves described above are never found at sea; only swell, passing through an area with no wind, may come close. The reason for starting with a description of simple waves is that they represent the basic solutions of the physical equations which govern waves on the sea surface and, as we shall see later, they are the "building blocks" of the real wave fields occurring at sea. In fact, the concept of simple sinusoidal waves is frequently used as an aid to understanding and describing waves on the sea surface. In spite of this simplified description, the definitions and formulae derived from it are extensively used in practice and have proved their worth.
1.2.3 ORBITAL MOTION OF WATER PARTICLES 
It is quite evident that water particles move up and down as waves travel through water. By carefully watching small floating objects, it can be seen that the water also moves backwards and forwards; it moves forward on the crest of a wave and backward in the trough. If the water is not too shallow relative to the wavelength, the displacements are approximately as large in the horizontal as in the vertical plane. In fact, during one cycle of a simple wave (i.e. a wave period) the particles describe a circle in a vertical plane. The vertical plane is the cross-section which we have drawn in Figure 1.2. In shallow water the motion is an ellipse. Figure 1.3 illustrates this particle motion for a simple sinusoidal wave in deep water.
Consider the speed at which a water particle completes its path. The circumference of the circle is equal to πH. This circumference is covered by a particle within a time equal to one period T. The speed of the water is therefore πH/T. This is also the greatest forward speed reached in the crests. The speed of individual water particles should not be confused with the speed at which the wave profile propagates (wave speed). The propagation rate of the wave profile is usually far greater, as it is given by λ/T, and the wavelength λ is generally much greater than πH.
 |
| Fig 1.3 - Progression of a wave motion. Thirteen snap shots each with an interval of 1/12th period (derived from Gröen and Dorrestein, 1976) |
 | Fig 1.4 - Path shift of a water particle during two wave periods) |
Figure 1.3 has been slightly simplified to show the progression of wave crests and troughs as the result of water particle motion. In reality, depending on the wave steepness, a water particle does not return exactly to the starting point of its path; it ends up at a slightly advanced position in the direction in which the waves are travelling (Figure 1.4). In other words, the return movement in the wave trough is slightly less than the forward movement in the wave crest, so that a small net forward shift remains. This difference increases in steep waves (see Section 1.2.7).
We have noted that waves are associated with motion in the water. Therefore as a wave disturbs the water there is kinetic energy present which is associated with the wave, and which moves along with the wave. Waves also displace particles in the vertical and so affect the potential energy of the water column. This energy also moves along with the wave. It is an interesting feature of the waves that the total energy is equally divided between kinetic energy and potential energy. This is called the equipartition of energy.
It is also important to note that the energy does not move at the same speed as the wave, the phase speed. It moves with the speed of groups of waves rather than individual waves. The concept of a group velocity will be discussed in Section 1.3.2 but it is worth noting here that, in deep water, group velocity is half the phase speed. The total energy of a simple linear wave can be shown to be ρwga2/2 which is the same as ρwgH2/8, where ρw is the density of water. This is the total of the potential and kinetic energies of all particles in the water column for one wavelength.
1.2.5 INFLUENCE OF WATER DEPTH
As a wave propagates, the water is disturbed so that both the surface and the deeper water under a wave are in motion. The water particles also describe vertical circles, which become progressively smaller with increasing depth (Figure 1.5). In fact the decrease is exponential.
Below a depth corresponding to half a wavelength, the displacements of the water particles in deep water are less than 4 per cent of those at the surface. The result is that, as long as the actual depth of the water is greater than the value corresponding to λ/2, the influence of the bottom on the movement of water particles can be considered negligible. Thus, the water is called deep with respect to a given surface wave when its depth is at least half the wavelength. In practice it is common to take the transition from deep to transitional depth water at h =λ/4. In deep water, the displacements at this depth are about 20 per cent of those at the surface. However, so long as the water is deeper than λ/4, the surface wave is not appreciably deformed and its speed is very close to the speed on deep water. The following terms are used to characterize the ratio between depth (h) and wavelength (λ):
- Deep water h > λ/4;
- Transitional depth λ/25 < h < λ/4;
- Shallow water h < λ/25.
Note that wave dissipation due to interactions with the bottom (friction, percolation, sediment motion) is not yet taken into account here.
When waves propagate into shallow water, for example when approaching a coast, nearly all the characteristics of the waves change as they begin to "feel" the bottom. Only the period remains constant. The wave speed decreases with decreasing depth. From the relation λ= cT we see that this means that the wavelength also decreases.
From linearized theory of wave motion, an expression relating wave speed, c, to wavenumber, k = 2π/λ, and water depth, h, can be derived:
 | (1.5) |
where g is the acceleration of gravity, and tanh x denotes the hyperbolic tangent:
 | (1.5_1) |
The dispersion relation for finite-depth water is much like Equation 1.5. In terms of the angular frequency and wavenumber, we have then the generalized form for Equation 1.3:
 | (1.3a) |
In deep water (h > λ/4), tanh kh approaches unity and c is greatest. Equation 1.5 then reduces to
 | (1.6) |
or, when using λ= cT (Equation 1.1)
 | (1.7) |
and
 | (1.8) |
and
 | (1.9) |
Expressed in units of metres and seconds (m/s), the term g/2π is about equal to 1.56 m/s2. In this case, one can write λ = 1.56 T2 m and c = 1.56 T m/s. When, on the other hand, c is given in knots, λ in feet and T in seconds, these formulae become c = 3.03 T knots and λ = 5.12 T2 feet.
When the relative water depth becomes shallow (h < λ/25), Equation 1.6 can be simplified to
 | (1.10) |
This relation is of importance when dealing with long-period, long-wavelength waves, often referred to as long waves. When such waves travel in shallow water, the wave speed depends only on water depth. This relation can be used, for example, for tsunamis for which the entire ocean can be considered as shallow.
If a wave is travelling in water with transitional depths (λ/25 < h < λ/4), approximate formulae can be used for the wave speed and wavelength in shallow water:
 | (1.11) |
 | (1.12) |
with c0 and λ0 the deep-water wave speed and wavelength according to Equations 1.6 and 1.8, and k0 the deep-water wavenumber 2π/λ0.
A further feature of changing depth is changing wave height. As a wave approaches the shore its height increases. This is a result of the changes in group velocity. The energy propagating towards the coast must be conserved, at least until friction becomes appreciable, so that if the group velocity decreases and wavelength decreases, the energy in each wavelength must increase. From the expression for energy in Section 1.2.4 we see that this means that the height of the wave must increase.
 | Fig 1.5 - Paths of the water particles at various depths in a wave on deep water. Each circle is one-ninth of a wavelength below the one immediately above it.) |
1.2.6 REFRACTION AND DIFFRACTION
As waves begin to feel the bottom, a phenomenon called refraction may occur. When waves enter water of transitional depth, if they are not travelling perpendicular to the depth contours, the part of the wave in deeper water moves more rapidly than the part in shallower water, according to Equation 1.11, causing the crest to turn parallel to the bottom contours. Some examples of refraction patterns are shown in Figures 1.6, 1.7 and 1.8.
Generally, any change in the wave speed, for instance due to gradients of surface currents, may lead to refraction, irrespective of the water depth. In Section 4.5.1 a few examples are given to illustrate refraction under simplified conditions. A more complete description of methods for the analysis of refraction and diffraction can be found in Sections 7.3 and 7.4, and in CERC (1984).
Finally, the phenomenon of wave diffraction should be mentioned. It most commonly occurs in the lee of obstructions such as breakwaters. The obstruction causes energy to be transformed along a wave crest. This transfer of energy means that waves can affect the water in the lee of the structure, although their heights are much reduced. An example is illustrated in the photograph in Figure 1.9.
 |  |
| Fig 1.6 - Refraction along a straight beach with parallel bottom contours | Fig 1.7(a) - Refraction by a submarine ridge |
 |  |
| Fig 1.8 - Refraction along an irregular shoreline | Fig 1.7(b) - Refraction by a submarine canyon |
In Section 1.2.3, it was noted that the speed of the water particles is slightly greater in the upper segment of the orbit than in the lower part. This effect is greatly magnified in very steep waves, so much so that the maximum forward speed may become not πH/T but 7H/T. Should 7H become equal to the wavelength λ (i.e. H/λ = 1/7), the forward speed of the water in the crest would then be equal to the rate of propagation which is λ/T. There can be no greater forward speed of the water because the water would then plunge forward out of the wave: in other words, the wave would break. According to a theory of Stokes, waves cannot attain a height of more than one-seventh of the wavelength without breaking. In reality, the steepness of waves is seldom greater than one-tenth. However, at values of that magnitude, the profile of the wave has long ceased to be a simple undulating line and looks more like a trochoid (Figure 1.10). According to Stokes' theory, at the limiting steepness of one-seventh, the forward and backward slopes of the wave meet in the crest under an angle of 120° (Figure 1.11).
When waves propagate into shallow water their characteristics change as they begin to feel the bottom, as we have already noted in Section 1.2.5. The wave period remains constant, but the speed decreases as does the wavelength. When the water depth becomes less than half the wavelength, there is an initial slight decrease in wave height*. The original height is regained when the ratio h/λ is about 0.06 and thereafter the height increases rapidly, as does the wave steepness, until breaking point is reached:
 | (1.13) |
in which hb is termed the breaking depth and Hb the breaker wave height.
 |
| Fig 1.9 - Wave diffraction at Channel Islands harbour breakwater (California) (from CERC, 1977) |
 |
| Fig 1.10 - Trochoidal wave profile. here the crests project farther above the mean level then the troughs sink under it |
 |
| Fig 1.6 - Ultimate form which water waves can attain according to Stokes' theory |
1.3.1 A COMPOSITION OF SIMPLE WAVES
Actual sea waves do not look as simple as the profile shown in Figure 1.2. With their irregular shapes, they appear as a confused and constantly changing water surface, since waves are continually being overtaken and crossed by others. As a result, waves at sea are often short-crested. This is particularly true for waves growing under the influence of the wind (wind sea). A more regular pattern of long-crested and nearly sinusoidal waves can be observed when the waves are no longer under the influence of their generating winds. Such waves are called swell and they can travel hundreds and thousands of kilometres after having left the area in which they were generated. Swell from distant generating areas often mixes with wind waves generated locally.
The simple waves, which have been described in Section 1.2, can be shown to combine to compose the observed patterns. To put it differently, any observed wave pattern on the ocean can be shown to comprise a number of simple waves, which differ from each other in height, wavelength and direction.
As a first step, let us consider waves with long, parallel crests but which differ in height, for example, the profile as shown in the top curve of Figure 1.12. Although this curve looks fairly regular, it is certainly no longer the profile of a simple sinusoidal wave, because the height is not everywhere the same, nor are the horizontal distances between crests. This profile, however, can be represented as the sum of two simple wave profiles of slightly different wavelength (see I and II in Figure 1.12). In adding the vertical deviations of I and II at corresponding points of the horizontal axis, we obtain the vertical deviations of the sum of wave I and wave II, represented by the top wave profile in Figure 1.12.
Thus, the top profile can be broken down, or decomposed, into two simple waves of different wavelength. The reason why the crests are of varying height in the sum of I and II is that at one place waves I and II are "in phase" and their heights therefore add up, whereas the resulting height is reduced at those places where the waves are out of phase.
Taking this idea one step further, we can see how an irregular pattern of wind waves can be thought of as a superposition of an infinite number of sinusoidal waves, propagating independently of each other. This is illustrated in Figure 1.13, which shows a great number of sinusoidal waves piled up on top of each other. Think, for example, of a sheet of corrugated iron as representing a set of simple sinusoidal waves on the surface of the ocean and caught at an instant in time. Below this, there is another set of simple sinusoidal waves travelling in a slightly different direction from the one on top. Below that again is a third one and a fourth one, etc. - all with different directions and wavelengths. Each set is a classic example of simple sinusoidal waves.
It can be shown that, as the number of different sinusoidal waves in the sum is made larger and larger and the heights are made smaller and smaller, and the periods and directions are packed closer and closer together (but never the same and always over a considerable range of values), the result is a sea surface just like the one actually observed. Even small irregularities from the sinusoidal shape can be represented by superpositions of simple waves.
1.3.2 WAVE GROUPS AND GROUP VELOCITY
We have seen how waves on the ocean are combinations of simple waves. In an irregular sea the number of differing wavelengths may be quite large. Even in regular swell, there are many different wavelengths present but they tend to be grouped together. In Figure 1.12 we see how simple waves with close wavelengths combine to form groups of waves. This phenomenon is common. Anyone who has carefully observed the waves of the sea will have noticed that in nature also the larger waves tend to come in groups. Although various crests in a group are never equidistant, one may speak of an average distance and thus of an average wavelength. Despite the fact that individual crests or wave tops advance at a speed corresponding to their wavelength, the group, as a coherent unit, advances at its own velocity - the group velocity. For deep water this has magnitude (group speed):
 | (1.14) |
A more general expression also valid in finite depth water is:
 | (1.15) |
The general form for the group speed can be shown to be:
Derivations may be found in most fluid dynamics texts (e.g. Crapper, 1984).
We can also show that the group velocity is the velocity at which wave energy moves. If we consider the energy flow (flux) due to a wave train, the kinetic energy is associated with the movement of water particles in nearly closed orbits and is not significantly propagated. The potential energy however is associated with the net displacement of water particles and this moves along with the wave at the phase speed. Hence, in deep water, the effect is as if half of the energy moves at the phase speed, which is the same as the overall energy moving at half the phase speed. The integrity of the wave is maintained by a continuous balancing act between kinetic and potential energy. As a wave moves into previously undisturbed water potential energy at the front of the wave train is converted into kinetic energy resulting in a loss of amplitude. This leads to waves dying out as they outrun their energy. At the rear of the wave train kinetic energy is left behind and is converted into potential energy with the result that new waves grow there.
One classical example of a wave group is the band of ripples which expands outwards from the disturbance created when a stone is cast into a still pond. If you fix your attention on a particular wave crest then you will notice that your wave creeps towards the outside of the band of ripples and disappears. Stating this slightly differently, if we move along with waves at the phase velocity we will stay with a wave crest, but the waves ahead of us gradually disappear. Since the band of ripples is made up of waves with components from a narrow range of wavelengths the wavelength of our wave will also increase a little (and there will be fewer waves immediately around us). However, if we travel at the group velocity, the waves ahead of us may lengthen and those behind us shorten but the total number of waves near us will be conserved.
Thus, the wave groups can be considered as carriers of the wave energy (see also Section 1.3.7), and the group velocity is also the velocity with which the wave energy is propagated. This is an important consideration in wave modelling.
1.3.3 STATISTICAL DESCRIPTION OF WAVE RECORDS
The rather confusing pattern seen in Figure 1.13 can also be viewed in terms of Equation 1.4 as the motion of the water surface at a fixed point. A typical wave record for this displacement is shown in Figure 1.14, in which the vertical scale is expressed in metres and the horizontal scale in seconds. Wave crests are indicated with dashes and all zero-downcrossings are circled. The wave period T is the time distance between two consecutive downcrossings (or upcrossings*), whereas the wave height H is the vertical distance from a trough to the next crest as it appears on the wave record. Another and more commonly used kind of wave height is the zero-crossing wave height Hz, being the vertical distance between the highest and the lowest value of the wave record between two zero-downcrossings (or upcrossings). When the wave record contains a great variety of wave periods, the number of crests becomes greater than the number of zero-downcrossings. In that case, there will be some difference between the crest-to-trough wave height and Hz. In this chapter, however, this difference will be neglected and Hz will be used implicitly. A simple and commonly used method for analysing wave records by hand is the Tucker-Draper method which gives good approximate results (see Section 8.7.2).
A measured wave record never repeats itself exactly, due to the random appearance of the sea surface. But if the sea state is "stationary", the statistical properties of the distribution of periods and heights will be similar from one record to another. The most appropriate parameters to describe the sea state from a measured wave record are therefore statistical. The following are frequently used:
H- = Average wave height; Hmax = Maximum wave height occurring in a record; T-z = Average zero-crossing wave period; the time obtained by dividing the record length by the number of downcrossings (or upcrossings) in the record; H-1/n = The average height of the 1/n highest waves (i.e. if all wave heights measured from the record are arranged in descending order, the one-nth part, containing the highest waves, should be taken and H-1/n is then computed as the average height of this part); T-H1/n = The average period of the 1/n highest waves. A commonly used value for n is 3: H-1/3 = Significant wave height (its value roughly approximates to visually observed wave height); T-H1/3 = Significant wave period (approximately equal to the wave period associated with the spectral maximum, see Section 1.3.8).
1.3.4 DURATION OF WAVE RECORDS
The optimal duration of wave records is determined by several factors. First of all, for a correct description of the sea state, conditions should be statistically stationary during the sampling period. In fact, this will never be achieved completely as wave fields are usually evolving (i.e. growing or decaying). On the other hand, to reduce statistical scatter, the wave record should contain at least 200 zero-downcrossing waves. Hence, the optimal time over which waves are usually measured is 15-35 minutes, as this reasonably accommodates both conditions.
So far, we have introduced the manual analysis of analogue "stripchart" records. Most analyses are performed by computer for which digital records are used, i.e. the vertical displacement of the ocean surface (or the position of the pen at the chart recorder) is given with a sampling rate of 1-10 times per second (1-10 Hz). For example, a record of 20 minutes duration with a sampling rate of 4 Hz contains 4 800 values.
When wave records are processed automatically, the analysis is always preceded by checks on the quality of the recorded data points to remove outliers and errors due to faulty operation of sensors, in data recording equipment, or in data transmission.
1.3.5 NOTES ON USAGE OF STATISTICAL PARAMETERS
In this Guide, the term sea state is used as a wave condition which is described by a number of statistical parameters. It is common to use the significant wave height, H-1/3, and the average zero-crossing period, T-z, or some other characteristic period, to define the sea state. The corresponding maximum wave height can be deduced as shown in Section 1.3.6.
The use of the average zero-crossing period, T-z, has its drawbacks. The distribution of individual zero-downcrossing periods of a record is usually very wide and is also somewhat sensitive to noise, in contrast with the distribution of periods of, say, the highest one-third of waves. Moreover, the average period of the highest waves of a record is usually a good approximation of the period associated with the peak of the wave spectrum (see Section 1.3.8). It has been found that average wave periods of the 1/n highest waves with n > 3 are not essentially different from T-H1/n, but exhibit a larger scatter.
In this Guide, as elsewhere, various definitions of wave steepness are used. The general form is = H/ which becomes, using Equation 1.8:
where H represents a wave height (e.g. H-1/3, Hm0, Hrms, m0) and T the wave period (e.g. T-z, T-H1/3, Tp, Tm02). Some of these parameters are introduced in Section 1.3.8.
1.3.6 DISTRIBUTION OF WAVE HEIGHTS
The elevation of the sea surface is denoted (x,t). This formulation expresses the variations of sea surface in space and time for both simple waves (see Equation 1.2) and more complicated sea states. If the range of wavelengths in a given sea state is not too broad, it has been shown (Longuet-Higgins, 1952) that the elevation, , has a statistical distribution which is Gaussian (i.e. normal).
For a normally distributed parameter, such as , the maximum values are known to be distributed with a Rayleigh distribution. For a sea state these maximum values are directly related to the wave heights. Hence, the distribution of (zero-downcrossing) wave heights can be represented by the Rayleigh distribution. This feature has been shown theoretically and verified empirically. If F(H) denotes the probability of heights not exceeding a given wave height H1 in a sea state characterized by a known value of H-1/3, F(H) is given by:
 | (1.16) |
The probability Q(H1) of heights exceeding H1 is then
 | (1.17) |
Example:
Given a sea state for which H-1/3 = 5 m,what is the probability of observing waves higher than 6 m?
Since F(H1) = 1 - exp [- 2 (6/5)2] = 0.94,the probability of heights exceeding 6 m is
Q(H1) = 1 - 0.94 = 0.06.If H-1/3 is computed from a wave record of finite length, the record length or the number of waves used for the computation should be taken into account. If, on a record containing N waves, n (n ? N) waves exceed a given height H1, the probability of heights exceeding H1 is:
 | (1.18) |
Inserting the relationships from Equations 1.16 and 1.17 into Equation 1.18 leads to:
 | (1.19) |
Equation 1.19 provides a quick method for the determination of H-1/3 from a wave record. On the other hand, if H-1/3 is known, the distribution of a wave record can be compared with the Rayleigh distribution by using:
 | (1.19a) |
For the prediction of the maximum wave height Hmax for a sequence of N waves with known H-1/3, it is common to take the mode of the distribution of maximum values:
 | (1.20) |
Alternatively, if the 50-percentile of the distribution of maximum values is taken, we get a more conservative estimate of Hmax because of the asymmetry of the distribution, i.e. about 5 per cent greater than according to Equation 1.20:
 | (1.21) |
The prediction of Hmax must be based on a realistic duration, e.g. six hours, apart from the usual confidence limits of the H-1/3 forecast. This implies N = 2000-5000 (in 6 hours there are about 2700 waves if the peak period is 8 seconds). Using Equation 1.20 we get*:
Hmax 2.0 H-1/3 Hm0.We have noted in Section 1.3.1 that a sea surface with a random appearance may be regarded as the sum of many simple wave trains. A way of formalizing this concept is to introduce the wave spectrum. A wave record may be decomposed by means of harmonic (or Fourier) analysis into a large number of sinusoidal waves of different frequencies, directions, amplitudes and phases. Each frequency and direction describes a wave component, and each component has an associated amplitude and phase. The harmonic (Fourier) analysis thus provides an approximation to the irregular but quasi-periodic form of a wave record as the sum of sinusoidal curves. For a surface elevation varying in time in a single direction:
in which:
The phase angle allows for the fact that the components are not all in phase, i.e. their maxima generally occur at different times. The high frequency components tend to become insignificant and hence there is a reasonable limit to n. Each wave component travels at its own speed (which depends on the wave frequency - or period - as expressed in Equation 1.10). Hence, the spectrum of wave components is continuously changing across the sea surface as the low frequency (large period or long wavelength) components travel faster than the high frequency components.
The expected values of the squares of the amplitudes aj are the contribution to the variance of the surface elevation ( ) from each of the wave components (i.e. the variance is E [ jaj2]). The resulting function is known as the wave-variance spectrum S(f)*. Typical spectra of wave systems have a form as shown in Figure 1.15 where the squared amplitudes for each component are plotted against their corresponding frequencies. The figure shows the spectrum from a measured wave record, along with a sample of the data from which it was calculated†. On the horizontal axis, the wave components are represented by their frequencies (i.e. 0.1 Hertz (Hz) corresponds to a period of ten seconds).
In actual practice, wave spectra can be computed by different methods. The most commonly used algorithm is the fast-Fourier transform (FFT), developed by Cooley and Tukey (1965). A much slower method, now superseded by the FFT, is the auto-correlation approach according to the Wiener-Kinchine theorem, introduced for practical use by Blackman and Tukey (1959) (see also Bendat and Piersol, 1971). Experience has shown that the difference between spectra computed by any two methods does not exceed confidence limits of each of them.
Since the wave energy E equals wgH2/8 or wga2/2 (H = 2a), wave spectra in the earlier literature were expressed in terms of E and called wave-energy spectra. However, it has become common practice to drop the term wg and to plot a2/2 or, simply, a2 along the vertical axis. The wave-energy spectrum is thus usually regarded synonymously with the "variance spectrum".
Wave spectra are usually given as a continuous curve connecting the discrete points found from the Fourier analysis and systems typically have a general form like that shown in Figure 1.16. The curve may not always be so regular. Irregular seas give rise to broad spectra which may show several peaks. These may be clearly separated from each other or merged into a very broad curve with several humps. Swell will generally give a very narrow spectrum concentrating the energy in a narrow range of frequencies (or wavelengths) around a peak value. Such a narrow spectrum is associated with the relatively "clean" appearance of the waves. Recall from Section 1.3.2 (and Figure 1.12) that this was often a condition where wave groups were clearly visible.
It is important to note that most measurements do not provide information about the wave direction and therefore we can only calculate an "energy" distribution over wave frequencies, E(f). On the vertical axis, a measure for the wave energy is plotted in units of m2/Hz. This unit is usual for "frequency spectra". We have seen earlier that, although the spectrum may be continuous in theory, in practice the variances (or energies) are computed for discrete frequencies. Even when a high-speed computer is used, it is necessary to regard the frequency domain (or the frequency-directional domain) as a set of distinct or discrete values. The value of a2 at, for instance, a frequency of 0.16 Hz is considered to be a mean value in an interval which could be 0.155 to 0.165 Hz. This value, divided by the width of the interval, is a measure for the energy density and expressed in units of m2/Hz (again omitting the factor wg). In fact the wave spectrum is often referred to as the energy-density spectrum. Thus, this method of analysing wave measurements yields a distribution of the energy of the various wave components, E(f, ). It was noted in Section 1.3.2 that wave energy travels at the group velocity cg, and from Equation 1.15 we see that this is a function of both frequency and direction (or the wavenumber vector) and possibly water depth. The energy in each spectral component therefore propagates at the associated group velocity. Hence it is possible to deduce how wave energy in the local wave field disperses across the ocean.
It is important to note that a wave record and the spectrum derived from it are only samples of the sea state (see Section 1.3.4). As with all statistical estimates, we must be interested in how good our estimate is, and how well it is likely to indicate the true state. There is a reasonably complete statistical theory to describe this. We will not give details in this Guide but refer the interested reader to a text such as Jenkins and Watts (1968). Suffice to say that the validity of a spectral estimate depends to a large extent on the length of the record, which in turn depends on the consistency of the sea state or statistical stationarity (i.e. not too rapidly evolving). The spectral estimates can be shown to have the statistical distribution called a 2 distribution for which the expected spread of estimates is measured by a number called the "degrees of freedom". The larger the degrees of freedom, the better the estimate is likely to be.
1.3.8 WAVE PARAMETERS DERIVED FROM THE SPECTRUM
A wave spectrum is the distribution of wave energy (or variance of the sea surface) over frequency (or wavelength or frequency and direction, etc.). Thus, as a statistical distribution, many of the parameters derived from the spectrum parallel similar parameters from any statistical distribution. Hence, the form of a wave spectrum is usually expressed in terms of the moments* of the distribution (spectrum). The nth-order moment, mn, of the spectrum is defined by:
 | (1.22) |
(sometimes = 2 f is preferred to f). In this formula, E(f) denotes the variance density at frequency, f, as in Figure 1.16, so that E(f) df represents the variance ai2/2 contained in the ith interval between f and f + df. In practice, the integration in Equation 1.22 is approximated by a finite sum, with fi = i df:
 | (1.22a) |
From the definition of mn it follows that the moment of zero-order, m0, represents the area under the spectral curve. In finite form this is:
which is the total variance of the wave record obtained by the sum of the variances of the individual spectral components. The area under the spectral curve therefore has a physical meaning which is used in practical applications for the definition of wave-height parameters derived from the spectrum. Recalling that for a simple wave (Section 1.2.4) the wave energy (per unit area), E, was related to the wave height by:
then, if one replaces the actual sea state by a single sinusoidal wave having the same energy, its equivalent height would be given by:
the so-called root-mean-square wave height. E now represents the total energy (per unit area) of the sea state. We would like a parameter derived from the spectrum and corresponding as closely as possible to the significant wave height H-1/3 (as derived directly from the wave record) and, equally, the characteristic wave height Hc (as observed visually). It has been shown that Hrms should be multiplied by the factor 2 in order to arrive at the required value. Thus, the spectral wave height parameter commonly used can be calculated from the measured area, m0, under the spectral curve as follows:
Note that we sometimes refer to the total variance of the sea state (m0) as the total energy, but we must be mindful here that the total energy E is really wgm0. In theory, the correspondence between Hm0 and H-1/3 is valid only for very narrow spectra which do not occur often in nature. However, the difference is relatively small in most cases, with Hm0 = 1.05 H-1/3 on average. The significant wave height is also frequently denoted by Hs. In that case, it must be indicated which quantity (4 m0 or H-1/3) is being used.
The derivation of parameters for wave period is a more complicated matter, owing to the great variety of spectral shapes related to various combinations of sea and swell. There is some similarity with the problem about defining a wave period from statistical analysis (see Section 1.3.5). Spectral wave frequency and wave period parameters commonly used are:
fp = Wave frequency corresponding to the peak of the spectrum (modal or peak frequency); Tp = Wave period corresponding to fp: i.e. Tp = fp-1; Tm01 = Wave period corresponding to the mean frequency of the spectrum: | (1.24) |
 | (1.25) |
where J, the wave power in kW/m of wave front, is computed as J = 0.49 Hm02Tm-10.
Note that the wave period Tm02 is sensitive to the high frequency cut-off in the integration (Equation 1.22) which is used in practice. Therefore this cut-off should be noted when presenting Tm02 and, in particular, when comparing different data sets. For buoy data, the cut-off frequency is typically 0.5 Hz as most buoys do not accurately measure the wave spectrum above this frequency. Fitting a high frequency tail before computing the spectral moments can be a useful convenience when high frequency information is not available (for example in a wave model hindcast).
Goda (1978) has shown that, for a variety of cases, average wave periods of the higher waves in a record, e.g. T-H1/3 (see Section 1.3.5), remain within a range of 0.87 Tp to 0.98 Tp.
Finally, the width of the spectral peak can be used as a measure of the irregularity of the sea state. The spectral width parameter is defined by:
Parameter varies between 0 (very narrow spectrum; regular waves) and 1 (very broad spectrum; many different wave periods present; irregular wave pattern). The use of is not recommended, however, because of its sensitivity to noise in the wave record due to the higher order moments, in particular m4. Rye (1977) has shown that the peakedness parameter, Qp, by Goda (1970) is a good alternative: | (1.26) |
1.3.9 MODEL FORMS FOR WAVE SPECTRA
The concept of a wave spectrum is commonly used for modelling the sea state. Models of the spectrum enable the spectrum to be expressed as some functional form, usually in terms of frequency, E(f), frequency and direction, E(f, ), or alternatively in terms of the wavenumber, E(k). Since the wavenumber and frequency are related by the dispersion relation (see Equations 1.3 and 1.3a), the frequency and wavenumber forms can be transformed from one to the other.
Models of the spectrum are used to obtain an estimate of the entire wave spectrum from known values of a limited number of parameters such as the significant wave height and wave period. These may be obtained by hindcast calculations, by direct measurement or visual observation. To give an idea of the various factors which need to be taken into account, a few models are given below as examples. In the first three models no bottom effects have been taken into account. The TMA spectrum (see p. 14) is proposed as a general form for a model spectrum in depth-limited waters. In all cases E is used to represent the variance density spectrum.
The Phillips' spectrum describes the shape particularly of the high frequency part of the spectrum, above the spectral peak. It recognizes that the logarithm of the spectrum is generally close to a straight line, with a slope that is about -5. Hence, the general form:
 | (1.27) |
The Pierson-Moskowitz spectrum (Pierson and Moskowitz, 1964) is often used as a model spectrum for a fully developed sea, an idealized equilibrium state reached when duration and fetch are unlimited. This spectrum is based on a subset of 420 selected wave measurements recorded with the shipborne wave recorder - developed by Tucker (1956) - on board British ocean weather ships during the five-year period 1955-1960. In its original form, this model spectrum is:
 | (1.28) |
in which E(f) is the variance density (in m2/s), f the wave frequency (Hz), u the wind speed (m/s) at 19.5 m above the sea surface, g the acceleration due to gravity (m/s2) and a dimensionless quantity, = 0.0081. It can be shown that the peak frequency of the Pierson-Moskowitz spectrum is:
 | (1.29) |
Equation 1.28, together with Equations 1.22 and 1.23, allows us to calculate m0 as a function of wind speed. Hence Hm0 (the significant wave height) for a fully grown sea is:
 | (1.30) |
with Hm0 in metres and u in m/s, with the wind speed now related to 10 m height*. This agrees well with limiting values of wave-growth curves in Chapter 4 (e.g. Figure 4.1). Equations 1.29 and 1.30 are valid for fully developed sea only, as is their combination:
 | (1.31) |
The JONSWAP spectrum is often used to describe waves in a growing phase. Observations made during the Joint North Sea Wave Project (JONSWAP) (Hasselmann et al., 1973) gave a description of wave spectra growing in fetch-limited conditions, i.e. where wave growth under a steady offshore wind was limited by the distance from the shore. The basic form of the spectrum is in terms of the peak frequency rather than the wind speed, i.e as in Equation 1.28 but after the substitution for g/(2 u) using Equation 1.29:
 | (1.32) |
The function is the peak enhancement factor, which modifies the interval around the spectral peak making it much sharper than in the Pierson-Moskowitz spectrum. Otherwise, the shape is similar. The general form of the JONSWAP spectrum is illustrated in Figure 1.17.
Using JONSWAP results, Hasselmann et al. (1976) proposed a relation between wave variance and peak frequency for a wide range of growth stages. Transforming their results into terms of Hm0 and fp we get:
 | (1.33) |
again with Hm0 in m, fp in Hz and u in m/s at 10 m above mean water level. This equation is connected with developing waves and so is not exactly comparable with Equation 1.30 for fully developed waves. The peak frequency can be obtained by reversing Equation 1.33:
 | &a