However, at the present day, understanding of the mechanism of wave formation and the way that waves travel across the oceans is by no means complete

11

' ... the chidden billow seems to pelt the clouds ...'

Othello,

Act II, Scene I.

Sea waves have attracted attention and comment throughout recorded history. Aristotle (384-322 BC) observed the existence of a relationship between wind and waves, and the nature of this relationship has been a subject of study ever since. However, at the present day, understanding of the mechanism of wave formation and the way that waves travel across the oceans is by no means complete. This is partly because observations of wave characteristics at sea are difficult, and partly because mathematical models of wave behaviour are based upon the dynamics of idealized fluids, and ocean waters do not conform precisely with those ideals. Nevertheless, some facts about waves are well established, at least to a first

approximation, and the purpose of this Chapter is to outline the qualitative aspects of water waves and to explore some of the simple relationships of wave dimensions and characteristics.

We start by examining the dimensions of an idealized water wave, and the terminology used for describing waves (Figure 1.1).

:igure 1.1 Vertical profile of two successive _.ealized ocean waves, showing their linear Jimensions and sinusoidal shape.

Wave height (H) refers to the overall vertical change in height between the wave crest (or peak) and the wave trough. The wave height is twice the wave amplitude (a). Wavelength (L) is the distance between two

successive peaks (or two successive troughs). Steepness is defined as wave height divided by wavelength

(H/L)

and, as can be seen in Figure 1.1, is not the same thing as the slope of the sea-surface between a wave crest and its adjacent trough. The time interval between two successive peaks (or two successive troughs) passing a fixed point is known as the period (T), and is generally measured in seconds. The number of peaks (or the number of troughs) which pass a fixed point per second is known as the frequency (f).

As the answer to Question 1.1 shows, period is the reciprocal of frequency.

We will return to this concept in Section 1.2.

12

Waves are a common occurrence in everyday life, and are manifested as, for example, sound, the motion of a plucked guitar string, tipples on a pond, or the billows on the ocean. It is not easy to define a wave. Before attempting to do so, let us consider some of the characteristics of wave motion:

1 A wave transfers a disturbance from one part of a material to another.

(The disturbance caused by dropping a stone into a pond is transmitted across the pond by tipples.)

2 The disturbance is propagated through the material without any substantial overall motion of the material itself. (A floating cork merely bobs up and down on the tipples, but experiences very tittle overall movement in the direction of travel of the ripples.)

3 The disturbance is propagated without any significant distortion of the wave form. (A ripple shows very little change in shape as it travels across a pond.)

4 The disturbance appears to be propagated with constant speed.

The answer, 'energy', provides a reasonable working definition of wave m o t i o n - a means whereby energy is transported across or through a material without any significant overall transport of the material itself.

There are two aspects to be considered: first, the progress of the waves (which we have already noted), and secondly, the movement of the water particles themselves. Superficial observation of the effect of ripples on a floating cork suggests that the water particles move 'up and down', but closer observation will reveal that, provided the water is very much deeper than the ripple height, the cork is describing a nearly circular path in a vertical plane, parallel with the direction of wave movement. In a more general sense, the particles are displaced from an equilibrium position, and a wave motion is the propagation of regular oscillations about that

equilibrium position. Thus, the particles experience a displacing force and a restoring force. The nature of these forces is often used in the descriptions of various types of waves.

1.1.1 TYPES OF WAVES

All waves can be regarded as progressive waves, in that energy is moving through, or across the surface of, the material. The so-called standing wave, of which the plucked guitar string is an example, can be considered as the sum of two progressive waves of equal dimensions, but travelling in opposite directions. We examine this in more detail in Section 1.6.4.

Waves which travel through the material are called body waves. Examples of body waves are sound waves and seismic P- and S-waves, but our main concern in this Volume is with su~.ace waves (Figure 1.2). The most

familiar surface waves are those which occur at the interface between atmosphere and ocean, caused by the wind blowing over the sea. Other external forces acting on the fluid can also generate waves. Examples range from raindrops falling into tidal pools, through diving gannets and ocean- going liners to earthquakes (see Section 1.6.3).

The tides are also waves (Figure 1.2), caused by the gravitational influence of the Sun and Moon and having periods corresponding to the causative forces. This aspect is considered in more detail in Chapter 2. Most other waves, however, result from a non-periodic disturbance of the water. The water particles are displaced from an equilibrium position, and to regain that position they require a restoring force, as mentioned above. The restoring force causes a particle to 'overshoot' on either side of the equilibrium position. Such alternate displacements and restorations establish a characteristic oscillatory 'wave motion', which in its simplest form has sinusoidal characteristics (Figures 1.1 and 1.6), and is sometimes referred to as simple harmonic motion. In the case of surface waves on water, there are two such restoring forces which maintain wave motion:

1 The gravitational force exerted by the Earth.

2 Surface tension, which is the tendency of water molecules to stick together and present the smallest possible surface to the air. So far as the effect on water waves is concerned, it is as if a weak elastic skin were stretched over the water surface.

pe of wave CAPILLARY WAVES

0.017 m

I

0.1m l m

I

wavelength 10 m 100 m 1000 m

I I I GRAVITY WAVES

wind waves long-period ordinary

waves tide waves

(fixed period)

Figure 1.2 Types of surface waves, showing the relationships between wavelength, wave frequency and period, the nature of the forces that cause them, and the relative amounts of energy in each type of wave. Unfamiliar terms will be explained later. Note: Waves caused by 'other wind waves' are waves resulting from interactions between waves of higher frequency as they move away from storm areas - see Section 1.4.2.

14

Water waves are affected by both of these forces. In the case of waves with wavelengths less than about 1.7 cm, the principal restoring force is surface tension, and such waves are known as capillary waves. They are important in the context of remote sensing of the oceans (Section 1.7.1). However, the main interest of oceanographers lies with surface waves of wavelengths greater than 1.7 cm, and the principal restoring force for such waves is gravity; hence they are known as gravity waves (Figure 1.2).

Gravity waves can also be generated at an interface between two layers of ocean water of differing densities. Because the interface is a surface, such waves are, strictly speaking, surface waves, but oceanographers usually refer to them as internal waves. These occur most commonly where there is a rapid increase of density with depth, i.e. a steep density gradient, or pycnocline. Pycnoclines themselves result from steep gradients of temperature and/or salinity, the two properties which together govern the density of seawater. Because the difference in density between two water layers is much smaller than that between water and air, less energy is required to displace the interface from its equilibrium position, and oscillations are more easily set up at an internal interface than at the sea- surface. Internal waves travel considerably more slowly than most surface waves. They have greater amplitudes than all but the largest surface waves (up to a few tens of metres), as well as longer periods (minutes or hours rather than seconds, cf. Figure 1.2) and longer wavelengths (hundreds rather than tens of metres). Internal waves are of considerable importance in the context of vertical mixing processes in the oceans, especially when they break.

Not all waves in the oceans are displaced primarily in a vertical plane. For example, because atmosphere and oceans are on a rotating Earth, variation of planetary vorticity with latitude (i.e. variation in the angular velocity of the Earth's surface and hence in the effect of the Earth's rotation on horizontal motions) causes horizontal deflection of atmospheric and oceanic currents, and provides restoring forces which establish oscillations mainly in a horizontal plane, so that easterly or westerly currents tend to swing back and forth about an equilibrium latitude. These large-scale horizontal oscillations are known as planetary (or Rossby) waves, and may occur as surface or as internal waves. They are not gravity waves (i.e.

the restoring force is not gravity) and so do not appear in Figure 1.2.

1.1.2 WIND-GENERATED WAVES IN THE OCEAN

In 1774, Benjamin Franklin said: 'Air in motion, which is wind, in passing over the smooth surface of the water, may rub, as it were, upon that surface, and raise it into wrinkles, which, if the wind continues, are the elements of future waves'.

In other words, if two fluid layers having differing speeds are in contact, there is frictional stress between them and there is a transfer of momentum and energy. The frictional stress exerted by a moving fluid is proportional to the square of the speed of the fluid, so the wind stress exerted upon a water surface is proportional to the square of the wind speed. At the sea- surface, most of the transferred energy results in waves, although a small proportion is manifest as wind-driven currents. In 1925, Harold Jeffreys suggested that waves obtain energy from the wind by virtue of pressure differences caused by the sheltering effect provided by wave crests (Figure 1.3).

15

:igure 1.3 Jeffreys' 'sheltering' model of wave 3neration. Curved grey lines indicate air flow;

horter, black arrows show water movement.

he rear face of the wave against which the wind lows experiences a higher pressure than the

"ont face, which is sheltered from the force of ]e wind. Air eddies are formed in front of each 9 ave, leading to excesses and deficiencies of ressure (shown by plus and minus signs

~spectively), and the pressure difference pushes le wave along.

Although Jeffreys' hypothesis fails to explain the formation of very small waves, it does seem to work if:

1 Wind speed exceeds wave speed.

2 Wind speed exceeds 1 m s-1.

3 The waves are steep enough to provide a sheltering effect.

Empirically, it can be shown that the sheltering effect is at a maximum when wind speed is approximately three times the wave speed. In general, the greater the amount by which wind speed exceeds wave speed, the steeper the wave. In the open oceans, most wind-generated waves have steepness

(H/L)

of about 0.03 to 0.06. However, as we shall see later, wave speed in deep water is not related to wave steepness, but to wavelength- the greater the wavelength, the faster the wave travels.

Consider the sequence of events that occurs if, after a period of calm weather, a wind starts to blow, rapidly increases to a gale, and continues to blow at constant gale force for a considerable time. No significant wave growth occurs until wind speed exceeds 1 m s -1. Then, small steep waves form as the wind speed increases. Even after the wind has reached a constant gale force, the waves continue to grow with increasing rapidity until they reach a size and wavelength appropriate to a speed which corresponds to one-third of the wind speed. Beyond this point, the waves continue to grow in size,

wavelength and speed, but at an ever-diminishing rate. On the face of it, one might expect that wave growth would continue Until wave speed was the same as wind speed. However, in practice wave growth ceases whilst Wave speed is still at some value below wind speed. This is because:

1 Some of the wind energy is transferred to the ocean surface via a tangential force, producing a surface current.

2 Some wind energy is dissipated by friction, and is converted to heat and sound.

3 Energy is lost from larger waves as a result of white-capping, i.e.

breaking of the tip of the wave crest because it is being driven forward by the wind faster than the wave itself is travelling. Much of the energy dissipated during white-capping is converted into forward momentum of the water itself, reinforcing the surface current initiated by process 1 above.

16

1.1.3 THE FULLY DEVELOPED SEA

We have already seen that the size of waves in deep water is governed not only by the actual wind speed, but also by the length of time the wind has been blowing at that speed. Wave size also depends upon the unobstructed distance of sea, known as the fetch, over which the wind blows.

Provided the fetch is extensive enough and the wind blows at constant speed for long enough, an equilibrium is eventually reached, in which energy is being dissipated by the waves at the same rate as the waves receive energy from the wind. Such an equilibrium results in a sea state called a fully developed sea, in which the size and characteristics of the waves are not changing. However, the wind speed is usually variable, so the ideal fully developed sea, with waves of uniform size, rarely occurs. Variation in wind speed produces variation in wave size, so, in practice, a fully developed sea consists of a range of wave sizes known as a wave field. Waves coming into the area from elsewhere will also contribute to the range of wave sizes, as will interaction between w a v e s - a process we explain in Section 1.4.2.

Oceanographers find it convenient to consider a wave field as a spectrum of wave energies (Figure 1.4). The energy contained in an individual wave is proportional to the square of the wave height (see Section 1.4).

:igure 1.4 Wave energy spectra for three fully Jeveloped seas, related to wind speeds of 20, 30 ]nd 40 knots (about 10, 15 and 20 rn s -1 9

espectively). The area under each curve is a

~easure of the total energy in that particular Jvave field.

17

1.1.4 WAVE HEIGHT AND WAVE STEEPNESS

As was hinted in Section 1.1.3, the height of any real wave is determined by many component waves, of different frequencies and amplitudes, which move into and out of phase with, and across each other ('in phase' means that peaks and troughs coincide). In theory, if the heights and frequencies of all the contributing waves were known, it would be possible to predict the heights and frequencies of the real waves accurately. In practice, this is rarely possible. Figure 1.5 illustrates the range of wave heights occurring over a short time at one location - there is no obvious pattern to the variation of wave height.

:igure 1.5 A typical wave record, i.e. a record 9f variation in water level (displacement from

~quilibrium) with time at one position.

For many applications of wave research, it is necessary to choose a single wave height which characterizes a particular sea state. Many oceanographers use the significant wave height,

H1/3,

which is the average height of the highest one-third of all waves occurring in a particular time period. In any wave record, there will also be a maximum wave height,/-/max. Prediction of /-/max for a given period of time has great value in the design of structures

such as flood barriers, harbour installations and drilling platforms. To build these structures with too great a margin of safety would be unnecessarily expensive, but to underestimate Hmax could have tragic consequences.

However, it is necessary to emphasize the essentially random nature of /-/max. Although the wave Hmax(25 years), will occur on a v e r a g e once every 25 years, this does not mean such a wave will automatically occur every 25 years - there may be periods much longer than that without one. On the other hand, two such waves might appear next week.

As wind speed increases, s o

H1/3

in the fully developed sea increases. The relationship between sea state,

H1/3

and wind speed is expressed by the Beaufort Scale (Table 1.1, overleaf). The Beaufort Scale can be used to estimate wind speed at sea, but is valid only for waves generated within the local weather system, and assumes that there has been sufficient time for a fully developed sea to have become established (cf. Figure 1.4).

The absolute height of a wave is less important to sailors than is its steepness (H/L). As mentioned in Section 1.1.2, most wind-generated waves have a steepness in the order of 0.03 to 0.06. Waves steeper than this can present problems for shipping, but fortunately it iS very rare for wave steepness to exceed 0.1. In general, wave steepness diminishes with increasing

wavelength. The short choppy seas rapidly generated by local squalls are particularly unpleasant to small boats because the waves are steep, even though not particularly high. On the open ocean, very high waves can usually be ridden with little discomfort because of their relatively lon~ wavelengths.

Table 1.1 A selection of information from the Beaufort Wind Scale.

Beaufort No.

Name Wind speed

(mean)

knots m s -1

State of the sea-surface Significant

wave height,

H1/3 (m)

10

11

12

Calm <1 0.0-0.2

Light air 1-3 0.3-1.5

Light breeze 4-6 1.6-3.3

Gentle breeze 7-10 3.4-5.4

Moderate breeze 11-16 5.5-7.9

Fresh breeze 17-21 8.0-10.7

Strong breeze 22-27 10.8-13.8

Near gale 28-33 13.9-17.1

Gale 34-40 17.2-20.7

Strong gale 41-47 20.8-24.4

Storm 48-55 24.5-28.4

Violent storm 56-64 28.5-32.7

Hurricane >64 >32.7

Sea like a mirror

Ripples with appearance of scales; no foam crests Small wavelets; crests have glassy appearance but do not break

Large wavelets; crests begin to break; scattered white horses

Small waves, becoming longer; fairly frequent 1.5 white horses

Moderate waves taking longer form; many 2.0

white horses and chance of some spray

Large waves forming; white foam crests 3.5

extensive everywhere and spray probable

Sea heaps up and white foam from breaking 5.0 waves begins to be blown in streaks; spindrift

begins to be seen

Moderately high waves of greater length; edges 7.5 of crests break into spindrift; foam is blown in

well-marked streaks

High waves; dense streaks of foam; sea begins 9.5 to roll; spray may affect visibility

Very high waves with overhanging crests;

sea-surface takes on white appearance as foam in great patches is blown in very dense streaks;

rolling of sea is heavy and visibility reduced Exceptionally high waves; sea covered with long white patches of foam; small and medium-sized ships might be lost to view behind waves for long times; visibility further reduced

Air filled with foam and spray; sea completely >15 white with driving spray; visibility greatly reduced

0 0.1-0.2 0.3-0.5

0.6-1.0

12.0

15.0

To simplify the theory of surface waves, we assume here that the wave-form is sinusoidal and can be represented b y t h e curves shown in Figures 1.1 and 1.6. This assumption allows us to consider wave d i s p l a c e m e n t (q) as simple harmonic motion, i.e. a sinusoidal variation in water level caused by the wave's passage. Figure 1.1 shows how the displacement varies over distance at a fixed instant in time - a 'snapshot' of the passing waves - w h e r e a s Figure 1.6 shows how wave displacement varies with time at a fixed point.

Figure 1.6 The displacement of an idealized wave at a fixed point, plotted against time.

Maximum and minimum displacements are recorded in fractions of the period, T.

19 3efore examining displacement, let us remind ourselves of the relationship 9etween period and frequency.

The displacement (q) of a wave at a fixed instant in time, or at a fixed point

"n space, varies between +a (at the peak) a n d - a (in the trough).

9isplacement is zero where L = 1L on Figure 1.1 (and at intervals of L/2 along the distance axis). DiSplacement is also zero at T = 1T on Figure 1.6

~and at intervals of T/2 along the time axis).

The curves shown in Figures 1.1 and 1.6 are both sinusoidal. However, host wind-generated waves do not have simple sinusoidal forms. The steeper the wave, the further it departs from a simple sine curve. Very steep waves resemble a troctioidai curve, which is illustrated in Figure 1.7.

:igure 1.7 Profile of trochoidal waves.

A point marked on the rim of a car tyre will appear to trace out a trochoidal

~urve as the car is driven past an observer. Invert that pattern and you have :he profile of a trochoidal water wave. We do not need to delve into the

nathematical complexities of trochoidal wave forms here, because the

~inusoidal model is sufficient for our purposes.

20

1.2.1 MOTION OF WATER PARTICLES

Water particles in a wave in deep water move in an almost closed circular path. At wave crests, the particles are moving in the same direction as wave propagation, whereas in the troughs they are moving in the opposite direction. At the surface, the orbital diameter corresponds to wave height, but the diameters decrease exponentially with increasing depth, until at a depth roughly equal to half the wavelength, the orbital diameter is negligible, and there is virtually no displacement of the water particles (Figure 1.8(a)). This has some important practical applications. For

example, a submarine only has to submerge about 150 m to avoid the effects of even the most severe storm at sea, and knowledge of the exponential decrease of wave influence with depth has implications for the design of stable floating oil rigs.

It is important to realize that the orbits are only approximately circular.

There is a small net component of forward motion, particularly in waves of large amplitude, so that the orbits are not quite closed, and the water, whilst in the crests, moves slightly further forward than it moves backward whilst in the troughs. This small net forward displacement of water in the direction o f wave travel is termed wave drift (see Figure 1.8(b)). In shallow water,

where depth is less than half the wavelength and the waves 'feel' the sea- bed, the orbits become progressively flattened with depth (Figure 1.8(c) and (d)). The significance of these changes will be seen in Section 1.5, and in the Chapters on sediment movement.

The orbital motions relevant to internal waves (Section 1.1.1) are shown in Figure 1.8(e): they are in opposite directions on either side of the interface.

The passage of intemal waves can often be detected visually by secondary effects at the surface, especially if the upper layer (above the pycnocline) is not very thick and the sea is relatively calm. As the internal waves travel along, the upper layer becomes alternately thinner (over the internal wave crests) and thicker (over the troughs). The result is that there are

convergences and divergences of water at the surface, as water is displacec back and forth between the thinner and thicker parts of the upper layer.

A combination of these to-and-fro motions and the opposing particle orbits on either side of the interface may, under certain conditions, influence the movement of vessels with a draught comparable to the depth to the interface, sometimes causing an unexpected drag on the hull, thus making the vessels sluggish and difficult to handle.

Sometimes the convergences compress short wavelength surface waves, making them more visible, but commonly they bring together organic material (especially oils released by marine organisms), which increases the surface tension and tends~ to suppress ripples, so that the water is smoother than elsewhere. Alternating bands of smooth and tippled surface water at intervals of a few hundred metres may thus indicate the passage of an internal wave, but whether ripples represent convergences or divergences depends upon local conditions.

Zl

:igure 1.8 (a) Particle motion in deep-water waves (depth greater than L/2), showing

~xponential decrease of the diameters of the _~rbital paths with depth.

~b) Particle motion in large deep-water waves, showing wave drift.

~c) Particle motion in waves where water depth

"s less than U2 but greater than U20 (see Section 1.2.3), showing both decrease in

lorizontal orbital diameter and progressive :lattening of the orbits near the sea-bed.

~d) Particle motion in shallow-water waves

~depth less than L/20, see Section 1.2.3), showing progressive flattening of the orbits

~but no decrease in horizontal diameter) near the sea-bed.

~e) Particle motion in internal waves (Section 1.1.1), and the convergences and divergences 3ssociated with their passage. The orbits will )nly be truly circular if the layers are thick

~nough (i.e. greater than half the wavelength).

The orbital diameters decrease linearly (i.e. not ]xponentially) with distance from the interface )ut may not reach zero at the free surface above :he interface (where any undulations do not lecessarily reflect those of the internal waves).

]rbital motions are in opposite directions above ]nd below the interface.

1.2.2 WAVE SPEED

As we have already hinted, there are mathematical relationships linking the characteristics of wavelength (L), wave period (T) and wave height (H) to wave speed in deep water and to wave energy. First, let us consider wave speed (c) (the c stands for

'celerity of propagation ' ).

T h e speed of a wave can be ascertained from the time taken for one wavelength to pass a fixed point. As one wavelength (L) takes one wave period (T) to pass a fixed point, then:

c= L/T

(1.1)

which is simply a form of the well-known expression: speed = distance/time.

So, if we know any two of the variables in Equation 1.1, we can calculate the third.

Two other terms you may meet in oceanographic literature are the

wave number, k,

which is

2re~L,

and the

angular frequency

or, which is

2re~T;

both of these relate to the sinusoidal nature of the idealized wave form. The units of k a r e m - 1 (i.e. number of waves per metre), and the units of cr are s -1 (i.e. number of cycles (waves) per second).

1.2.3 WAVE SPEED IN DEEP AND IN SHALLOW WATER

You may have noticed that when wave speeds have been mentioned we have been careful to state that the waves described were travelling in deep water. Thus you might have suspected that in shallow water, water depth has an effect on wave speed, because of interaction with the sea-bed. If so, you were quite right. Wave speed in any water depth can be represented by the general equation:

c = tanh (1.2)

where the acceleration due to gravity g = 9.8 m s -2, L = wavelength (m), and d = water depth (m). Tanh is a mathematical function known as the hyperbolic tangent. All you need to know about it in this context is that if x is small, say less than 0.05, then tanh x -- x, and if x is larger than n:, then tanh x -- 1.

In summary, the implications of your answers to Question 1.7 in terms of factors affecting wave speed are as follows (cf. Figure 1.8):

1 In water deeper than half the wavelength, wave speed depends upon the wavelength, and Equation 1.2 approximates to:

~ gL

c = (1.3)

9,rc

Diameters of circular particle orbits decrease exponentially downwards to near zero at depth = L/2 (Figure 1.8(a)).

2 In water very much shallower than the wavelength (in practice, when d < L / 2 0 ) , wave speed is determined by water depth, and Equation 1.2 approximates to:

c = ~f-gd i (1.4)

Horizontal diameters of particle orbits remain constant in size with depth, but ellipticity increases downwards (Figure 1.8(d)).

3 When d lies between L/20 and L/2, the full form of Equation 1.2 is required. Hence, to calculate wave speed you would need to know wavelength and depth, and have access to a set of hyperbolic tangent tables, or a calculator with hyperbolic functions on its keyboard. Particle orbits decrease in size downwards and become progressively more elliptical (Figure 1.8(c)).

The answer to Question 1.7(a) (i.e. Equation 1.3) allows us to explore further the relationships between T and L. We saw in Equation 1.1 that c = L/T, so it is possible to combine Equations 1.1 and 1.3.

The answer to Question 1.8 provides an equation expressing L in terms of T, i.e.

L = gT2 (1.5)

2n:

A similar exercise, substituting the expression obtained for L from Equation 1.5 into Equation 1.1, will give c in terms of T:

c = ~ gT (1.6)

2to

Thus, it is possible, given only one of the wave characteristics c, T or L, to za!culate either of the other two. Moreover, by substituting the numerical values of the constants involved, the equations can be simplified as follows:

Equation 1.3 becomes c = ~/i.56L Equation 1.5 becomes L = 1.56T 2 Equation 1.6 becomes c = 1.56T

(1.7) (1.8) (1.9)

The answer to Question 1.10(c) highlights an important conclusion about wave speed in shallow water. Provided that depth is less than 1/20 of their :vavelengths, all waves will travel at the same speed in water of a .~iven depth.

9_4

1.2.4 ASSUMPTIONS MADE IN SURFACE WAVE THEORY

The simple wave theory introduced above is a first-order approximation, and makes the following assumptions:

1 The wave shapes are sinusoidal.

2 The wave amplitudes are very small compared with wavelengths and depths.

3 Viscosity and surface tension can be ignored.

4 The Coriolis force (see Section 2.3) and vorticity (Section 1.1.1), which result from the Earth's rotation, can be ignored.

5 The depth is uniform, and the bottom has no bumps or hummocks.

6 The waves are not constrained or deflected by land masses, or by any other obstruction.

7 That real three-dimensional waves behave in a way that is analogous to a two-dimensional model.

None of the above assumptions is valid in the strictest sense, but predictions based on simple models of surface wave behaviour approximate closely to how wind-generated waves behave in practice.

Those deep-water waves that have the greatest wavelengths and longest periods travel fastest, and thus are first to arrive in regions distant from the storm which generated them. This separation of waves by virtue of their differing rates of travel is known as dispersion, and Equation 1.3

(c = ~/gL / 2 Jr) is sometimes known as the dispersion equation, because it shows that waves of longer wavelength (L) travel faster than shorter wavelength waves.

The simple experiment of tossing a stone into a still pond shows that a band of ripples is created, which gets wider with increasing distance from the original disturbance. Ripples of longer wavelength progressively out- distance shorter ones - an example of dispersion in action. There is a second feature of the tipple band, which is not obvious at first sight. Each individual tipple travels faster than the band of tipples as a whole. A tipple appears at the back of the band, travels through it, and disappears out of the front. The speed of the band, called the group speed, is about half the wave speed of the individual tipples which travel through that band.

To understand the relationship between wave speed and group speed, the additive effect of two sets of waves (or wave trains) needs to be examined.

If the difference between the wavelengths of two sets of waves is relatively small, the two sets will 'interfere' and produce a single set of resultant waves. Figure 1.9 shows a simplified and idealized example of interference.

Where the crests of the two wave trains coincide (i.e. they are 'in phase'), the wave amplitudes are added, and the resultant wave has about twice the amplitude of the two original waves. Where the two wave trains are 'out of phase', such that the crests of one wave train coincide with the troughs of the other, the amplitudes cancel out, and the water surface has minimal displacement.

25

:igure 1.9 (a) The merging of two wave trains ishown in red and blue) of slightly different

~vavelengths (but the same amplitudes), to form .rave groups (b).

:igure 1.10 The relationship between wave speed (phase speed) and group speed. As the wave advances from left to right, each wave

~noves through the group to die out at the front

~e.g. wave 1), as new waves form at the rear

~e.g. wave 6). In this process, the distance :ravelled by each individual wave as it moves :rom the rear to front of the group is twice that :ravelled by the group as a whole. Hence, the wave speed is twice that of the group speed.

The two component wave trains thus interact, each losing its individual identity, and combine to form a series of wave groups, separated by regions almost free from waves. The wave group advances more slowly than individual waves in the group, and thus in terms of the occurrence and propagation of waves, group speed is more significant than speeds of the individual waves. Individual waves do not persist for long in the open ocean, only as long as they take to pass through the group. Figure 1.10 shows the relationship between wave speed (sometimes called p h a s e speed) and group speed in the open ocean.

The group speed is h a l f the average speed of the two wave trains, and for your interest we present below an abbreviated form of how this relationship is derived. It is not necessary to follow all the steps that lead to Equation

1.10 (overleaf), still less to memorize them.

If two sets of waves are interfering to produce a succession of wave groups, the group speed (Cg) is the difference between the two angular frequencies (or I and or2) divided by the difference between the two wave numbers (k 1 and k2 respectively), i.e.

(71 - - 0"2

Cg = kl - k2

_)15

We have seen (Question 1.6) that c = or~k, and we know (Section 1.2.2) that angular frequency, or, can be expressed in terms of T, also that wave number k, can be expressed in terms of L. In addition, we know that Equations 1.6 and 1.3, respectively, enable us to express both T and L in terms of c (Section 1.2.3). Hence, Cg can be expressed in terms of the respective speeds, cl and c2, of the two wave trains. The equation obtained is:

Cg " -C 1 X C 2 ~ C 1 + C 2

If C l is nearly equal to c2, this equation simplifies to"

Cg~ c 2 / 2 c

o r r c/2 (1.10)

where c is the average speed of the two wave trains.

Equation 1.2 shows that as the water becomes shallower, wavelength becomes less important, and depth more important, in determining wave speed. As a result, in shoaling water, wave (phase) speed decreases, becoming closer to group speed. Eventually, at depths less than L/20, all waves travel at the same depth-determined speed, there will be no wave- wave interference, and therefore in effect each wave will represent its own

'group'. Thus, in shallow water, group speed can be regarded as equal to wave (phase) speed.

The energy possessed by a wave is in two forms:

1 kinetic energy, which is the energy inherent in the orbital motion of the water particles; and

2 potential energy possessed by the particles as a result of being displaced from their mean (equilibrium) position.

For a water particle in a given wave, energy is continually being converted from potential energy (at crest and trough) to kinetic energy (as it passes through the equilibrium position), and back again.

The total energy (E) per unit area of a wave is given by"

E = 1 -~ (pgH 2 ) (1.11)

where p is the density of the water (in kg m-3), g is 9.8 m s -z, and H is the wave height (m). The energy (E) is then in joules per square metre (J m-Z).

Equation 1.11 shows that wave energy is proportional to the square of the wave height.

27

1.4.1 PROPAGATION OF WAVE ENERGY

Figures 1.9 and 1.10 show that waves travel in groups in deep water, with areas of minimal disturbance between groups. Individual waves die out at the front of each group. It is obvious that no energy is being transmitted across regions where there are no waves, i.e. between the groups. It follows that the energy is contained within the wave group, and travels at the group speed. The rate at which energy is supplied at a particular location (e.g. a beach) is called wave power, and is the product of group speed (Cg) and wave energy per unit area (E), expressed p e r unit l e n g t h . o f w a v e crest.

1.4.2 ATTENUATION OF WAVE ENERGY

Wave attenuation involves loss or dissipation of wave energy, resulting in a reduction of wave height. Energy is dissipated in four main ways:

1 White-capping, which involves transfer of wave energy to the kinetic energy of moving water, thus reinforcing the wind-driven surface current (Section 1.1.2).

2 Viscous attenuation, which is only important for very high frequency .~apillary waves, and involves dissipation of energy into heat by friction between water molecules.

3 Air resistance, which applies to large steep waves soon after they leave the area in which they were generated and enter regions of calm or contrary winds.

4 Non-linear wave-wave interaction, which is more complicated than the simple (linear) combination of frequencies to produce wave groups as autlined in Section 1.3.

Non-linear interaction appears to be most important in the frequency range 3.2 to 0.3 s -1. Groups of three or four frequencies can interact in complex :aon-linear ways, to transfer energy to waves of both higher and lower

frequencies. A rough but useful analogy is that of the collision of two drops of water. A linear combination would simply involve the two drops coalescing

(adding together) into one big drop, whereas a non-linear combination is akin

~.o a collision between the drops so that they split into a number of drops of differing sizes. The total amount of water in the drops (analogous to the total amount of energy in the waves) is the same before and after the collision.

Thus, non-linear wave-wave interaction involves no loss of energy in itself,

~ecause energy is simply 'swapped' between different frequencies.

Jowever, the total amount of energy available for such 'swapping' will

~radually decrease, because higher frequency waves are more likely to dissipate energy in the ways described under 1 and 2 above. For example, aigher frequency waves are likely to be steep, and thus more prone to white-capping. Wave attenuation is greatest in the storm-generating area, where there are waves of many frequencies, and hence more opportunities

:or energy exchange between them.

28

1.4.3 SWELL

The sea-surface is rarely still. Even when there is no wind, and the sea 'looks like a mirror', a careful observer will notice waves of very long wavelength (say 300 to 600 m) and only a few centimetres amplitude. At other times, a sea may include locally generated waves of small wavelength, and travelling through these waves, possibly at a large angle to the wind, other waves of much greater wavelength. Such long waves are known as swell, which is simply defined as waves that have been generated elsewhere and have travelled far from their place of origin. If you look out to sea on a calm day, the waves that you see will be swell waves from a distant storm.

Figure 1.11 (a) The spreading of swell from a storm centre, showing the area in which swell might be expected. As distance from the storm increases, the length of the wave crest increases, with a corresponding decrease in wave height and energy per unit length of wave - spreading loss, discussed opposite.

(b) Wave record near a storm centre.

(c) Wave record of swell, well away from the storm centre.

29 Systematic observations show that local winds and waves have very little effect on the size and progres s of swell waves, and swell seems able to pass through locally generated seas without hindrance or interaction. Once swell waves have left the storm area, their wave height gradually diminishes, chiefly because the wave crests lengthen over a progressively wider front (Figure 1.11 (a)) but also because of energy loss caused by air resistance to steep waves (item 3 of Section 1.4.2). Once wave height has diminished to a few tens of centimetres, swell waves are not steep enough to be

significantly influenced by local winds.

In the ocean, we find waves travelling in many directions, resulting in a confused sea. To achieve a complete description of such a sea-surface, the amplitude, frequency and direction of travel of each component are needed.

The energy distribution of the sea-surface (cf. Figure 1.4) can then be calculated, but, as you might imagine, such a complex process requires expensive equipment to measure the wave characteristics, and computer facilities to perform the necessary calculations.

One or more components of a confused sea may be long waves or swell resulting from distant storms. In practice, about 90% of the sea-surface energy generated by the storm propagates within an angle of 30 ~ to 45 ~ either side of the wind direction. Consequently, waves generated by a storm in a localized region of a large ocean radiate outwards as a segment of a circle (Figure 1.11 (a)). As the circumference of the circle increases, the energy per unit length of wave crest must decrease (and so must wave height), so that the total energy of the wave front remains the same. This decrease in energy per unit length of wave crest is known as spreading loss (of wave energy), and in the case of established swell waves there is very little loss of wave energy apart from that caused by spreading over a progressively wider front.

The waves with the longest periods travel fastest, and progressively out- distance waves of higher frequencies (shorter periods). Near to the storm, dispersion is likely to be minimal (Figure 1.11 (b)), but the further one moves from the storm location, the more clearly separated waves of differing frequencies become, resulting in the regular wave motions we know as swell (Figure 1.11 (c)).

-igure 1.12 Wave-energy spectra, each :letermined from wave heights measured over a 3hort time interval, for two areas (a) and (b) in :he same ocean (not to scale). One area is a

~torm centre, and the other is far away from the

~torm. (For use with Question 1.13.)

If you recorded the waves arriving from a storm a great distance (over 1000 km) away, you would, as time progressed, see the peak in the wave- energy spectrum move progressively towards higher frequencies (i.e.

shorter periods). By recording the frequencies of each of a series of swell waves arriving at a point, it would be possible to calculate each of their speeds. From the set of speeds, a graph could be plotted to estimate the time and place of their origin. Before the days of meteorological satellites, this method was often used to pinpoint where and when storms had occurred in remote parts of the oceans.

3U

1.4.4 USES OF WAVE ENERGY

Wave energy is a potential source of pollution-free 'alternative energy', and has been used for some time on a small scale, e.g. to recharge batteries on buoys carrying navigation lights (see below). Harnessing wave power on a large scale presents a number of problems:

1 Prevailing sea conditions must ensure a supply of waves with amplitudes sufficient to make conversion worthwhile.

2 Installations must not be a hazard to navigation, or to marine ecosystems. The nature of wave energy is such that rows of converters many kilometres in length are needed to generate amounts of electricity comparable with conventional power stations. These would form offshore barrages which might interfere with shipping routes, although sea

conditions would be made calmer on the shoreward side. Calmer conditions, however, lead to reduced water circulation, less sediment transport, and increased growth of quiet-water plants and animals.

Pollutants are less easily flushed away from such an environment.

Some of these problems can be avoided by use of another type of wave- energy converter, the circular 'clam' or 'atoll', designed so that waves from any ~ direction are guided by a system of radiating vanes to a central vertical channel where the water spirals downwards to drive a turbine and generator. Such converters would be deployed in arrays rather than in long rows, but they are big, and many of them would be required to generate useful amounts of power commercially. For example, a 60 m diameter

'clam' could generate an average of about 600 kW, so many hundreds would be needed to equal the power provided by even one conventional fossil-fuel or nuclear power station.

3 The capital cost of such floating power stations and their related energy transmission and storage systems is enormous. Installations need to be robust enough to withstand storm conditions, yet sensitive enough to be able to generate power from a wide range of wave sizes. Such

conditions are expensive to meet, and make it difficult for large-scale wave-energy schemes to be as cost effective as conventional energy sources.

A large (2 MW) oscillating water column converter (OSPREY = Ocean Swell Powered Renewable EnergY) was installed off northern Scotland in

1995. Soon afterwards it was extensively damaged by storms and had to be taken ashore for repairs, which set back the UK wave-energy

programme by some years. Relatively small-scale utilization of wave power is more feasible. For example, oscillating water-column converters provide power for both buoys and lighthouses in several parts of the world.

Other approaches have been demonstrated by the Norwegians, who in 1985 brought into operation a wave-powered generator of 850 kW. This machine was sited on the west coast of Norway, where waves were funnelled into a narrow bay and increased substantially in height and thus in energy. Unfortunately, this installation also was wrecked by storms a few years later, but similar schemes have been successfully developed elsewhere.

31

-igure 1.13 Plan view illustrating changes in :he speed of waves approaching the shore. Grey ines represent wave crests at depths dl and d2.

Vave rays (dashed blue lines) are at right angles ii.e. normal) to the wave crests. For further 3xplanation, see text.

It is a matter of common observation that waves coming onto a beach increase in height and steepness and eventually break. Figure 1.13 shows a length of wave crest, s, which is directly approaching a beacK As the water is shoaling, the wave crest passes a first point where the water depth (dl) is greater than at a second point nearer the shore (where the depth is d2). We assume that the amount of energy within this length of wave crest remains constant, the wave is not yet readyto break, and that water depth is less than 1/20 of the wavelength (i.e. Equation 1.4 applies: c = ~/gd ). Because wave speed in shallow water is related to depth, the speed Cl at depth dl is greater than the speed c2 at depth d2.

If energy remains constant per unit length of wave crest, then

t

ElClS = E2c2s

or E 2 = Cl

(1.12)

E1 c2

and because energy is proportional to the square of the wave height (Equation 1.11) then we can write

E2 Cl H2 2

E1 r H12 (1.13)

Thus, both the square of the wave height and wave energy are inversely proportional to wave speed in shallow water.

This is quite a difficult question, best answered by considering the highly simplified case illustrated in FigUre 1.14. Imagine waves travelling shoreward over deep water (depth greater than half the wavelength). Wave speed is then governed solely by wavelength (Equation 1.3, c = ~/gL / 2 Jr. The energy is being propagated at the group speed (Cg) which is approximately half the wave speed (c), Section 1.3. As the waves move into shallower water, wave speed becomes governed by both depth and wavelength (Equation 1.2), but once the waves have moved into shallow water, where d < L/20, wave speed becomes governed solely by depth (Equation 1.4) and is much reduced.

Remember from Section 1.3 that in shallow water group speed is equal to wave speed. The rate at which energy arrives from offshore (Figure 1.14, overleaf) must be equal to the rate at which energy moves inshore; so if the group speed in shallow water is less than half the original wave speed (and hence less than the original group speed) in deep water, the waves will show corresponding increases in height and in energy per unit area.

However, it is essential to realize that while the energy and height of individual waves will increase as they enter shallow water, the rate of supply of wave energy (wave power, Section 1.4.1) must remain constant (ignoring frictional losses).

As mentioned earlier, when waves move into shallow water, the waves begin to 'feel the bottom', the circular orbits of the water particles become flattened (Figure 1.8(c) and (d)), and wave energy will be dissipated by friction at the sea-bed, resulting into-and-fro movement of sediments. The gentler the slope of the immediate offshore region, the sooner the incoming waves will 'feel' the bottom, the greater will be the friction with the sea-bed and the greater the energy loss before the waves finally break (see Section 15.2).

~z

Cl >c2 >c3

Cgl > Cg2 > Cg 3

individual waves travel at Cl =

~fgL/2n

b

energy travels at cgl (= 89 cl) energy travels at cg2 (where c2 > %2 > -} c2)

l

I

progressive decrease in wave speed (c) and group speed (co) individual

:/gL

t a n h ( - ~ )

waves c2 ~ all waves travel at c3 =

travel at b "-

IF ^~

energy travels at Cg 3 ( = C3)

Figure 1.14 Vertical section (not to scale) illustrating changes in the relationship between wave speed and group speed, and how this affects the speed of energy propagation as waves move from deep to shallow water. The energy is being brought in from offshore at the same rate as it is being removed as the waves break at the shore (see Section 1.5.2). As group speed (cg) in shallow water is less than in deep water, then the waves in shallow water must have more energy per unit length of wave crest, and a greater wave height than the waves in deep water.

1.5.1 WAVE REFRACTION

Figure 1.15 shows an idealized linear wave crest (length S1, between A and B) approaching a shoreline at an angle. Because the waves are travelling in shallow water, their speed is depth-determined (Equation i.4, c = ~/gd ).

The depth at A exceeds the depth at B, so the wave at A will travel faster than the wave at B, leading to the phenomenon known as refraction, which will tend to 'swing' the wave crest to an alignment parallel with the depth contours.

Refraction of waves in progressively shallowing (shoaling) water can be described by a relationship equivalent to Snell's law, which describes refraction of light rays through materials of different refractive indices.

Rays can be drawn perpendicular to the wave crests, and will indicate the direction of wave movement. The angle between these wave rays and lines drawn perpendicular to the depth contours can be related to wave speeds at various depths. In Figure 1.15, a wave ray approaching shoaling water at an angle 01, where water depth is d 1, will be at an angle 02 when it reaches depth d2. Angles 01 and 02 are related to wave speed by:

sin01 : c 1 :

g~ _ ~ _ I dl

sin02 c2 - - ~ 2 - ~ 2 - (1.14;

where cl and c~ are the respective wave speeds at depths dl and d2.

;5;5

:igure 1.15 Plan view illustrating the 9 elationship between wave approach angle (0);

Jvater depth (d), and length of wave crest (s).

The wave rays (broken blue lines), are normal to -he wave crests, and are the paths followed by )oints on the wave crests. For further

~xplanation, see text.

Well, of course, one could do that, and obtain analogous relationships between the relevant angles, depths and wave speeds. However, as you will see presently, wave rays are often more useful than wave crests in

determining regions that are likely to experience waves that are higher or lower than normal beCause of refraction.

Consider a length, Sl, of ideal wave crest, with energy per unit length E l, which is bounded by two wave rays, as in Figure 1.15. To a first approximation, we may assume that the total energy of the wave crest between these two rays will remain constant as the wave progresses. Therefore, if the two rays converge, the same amount of energy is contained within a shorter length of wave crest, so that, for the total wave energy to remain constant, the wave height will have to increase (Equation 1.11). Conversely, if the wave rays were to diverge, then the wave height would decrease.

If, as they finally approach the shore, the two wave rays are separated by a length s2, as in Figure 1.15, and if the wave energy is conserved, then the final wave energy must equal the initial wave energy, i.e. E l S 1 = E2s2, or in terms of wave heights (remember Equation 1.11):

H12s1 = H22s2 (1.15)

For simplicity, s2 in Figure 1.15 is the same length as s l, but it is common for wave rays to converge or diverge, and in general they converge (focus) on headlands and diverge in bays (Figure 1.1 6).

In more complicated situations, wave refraction diagrams can be plotted for a region by using the wave of most common period and the most common direction of approach, and in this way areas in which wave rays are focused or defocused can be identified.

igure 1.16 Waves are refracted and the wave :ays show how wave energy is focused on leadlands, where erosion is active, while Jeposition occurs in the bays, where the wave

ays diverge and wave energy is less. Waves 'feel :he bottom' and are slowed first in the shallow areas off the headland. The parts of the wave

:ronts that move through the deeper water 3ading into the bays are not slowed until they are well into the bays.

:~4

:igure 1.17 (a) Bathymetric map of the continental shelf off New York harbour at the

~nouth of the Hudson River on the Atlantic coast )f the USA. The rectangle shows the area of map ib). The position of the Hudson Canyon can be Jeduced from the submarine contours (in athoms).

ib) Pattern of wave crests (wave fronts) and rave rays off part of Long Branch beach.The t~ave fronts are drawn at intervals of 45 waves of )eriod 12 s.

35 We can estimate increase or decrease in wave size by measuring the

tistances between wave rays, and applying Equation 1.15. This method is luite useful provided wave rays neither approach each other too closely nor :ross over, as in these cases the waves become high, steep and unstable, and simple wave theory becomes inadequate.

1.5.2 WAVES BREAKING UPON THE SHORE

As a wave breaks upon the shore, the energy it received from the wind is Jissipated. Some energy is reflected back out to sea, the amount depending Jpon the slope of the b e a c h - the shallower the angle of the beach slope, the less energy is reflected. Most of the energy is dissipated as heat and sound ,'the 'roar' of the surf) in the final small-scale mixing of foaming water, sand and shingle. Some energy is used in fracturing large rock or mineral

9articles into smaller ones, and yet more may be used to move sediments and increase the height and hence the potential energy of the beach form.

This last aspect depends upon the type of waves. Small gentle waves and swell tend to build up beaches, whereas storm waves tear them down (see also Chapter 5).

A breaking wave is a highly complex system. Even some distance before :he wave breaks, its shape is substantially distorted from a simple sinusoidal wave. Hence the mathematical model of such a wave is more complicated than we have assumed in this Chapter.

Four major types of breaker can be identified, though you may often see breakers of intermediate character and/or of more than one type on the same beach at the same time.

1 Spilling breakers are characterized by foam and turbulence at the wave crest. Spilling usually starts some distance from shore and is caused when a layer of water at the crest moves forward faster than the wave as a whole.

Foam eventually covers the leading face of the wave, and such waves are characteristic of a gently sloping shoreline. A tidal bore (Section 2,4.3) is an extreme form of a spilling breaker. Breakers seen on beaches during a storm, when the waves are steep and short, are of the spilling type. They dissipate their energy gradually as the top of the wave spills down the front of the crest, which gives a violent and formidable aspect to the sea because of the more extended period of breaking.

2 Plunging breakers are the most spectacular type. The classical form, much beloved by surf-riders, is arched, with a convex back and concave front. The crest curls over and plunges downwards with considerable force, dissipating its energy over a short distance. Plunging breakers on beaches of relatively gentle slope are usually associated with the long swells generated by distant storms. Locally generated storm waves seldom develop into plunging breakers on gently sloping beaches, but may do so on steeper ones. The

iii

energy dissipated by plunging breakers is concentrated at the plunge point (i.e. where the water hits the bed) and can have great erosive effect.

3 Collapsing breakers are similar to plunging breakers, exceptthat the waves may be less steep and instead of the crest curling over, the front face collapses. Such breakers occur on beaches with moderately steep slopes, and under moderate wind conditions, and represent a transition: from plun~in~to sur~in~ breakers.

5b

37 4 Surging breakers are found on the very steepest beaches. Surging breakers are typically formed from long, low waves, and the front faces and crests thus remain relatively unbroken as the waves slide up the beach.

Figure 1.18 illustrates the relationship between wave steepness, beach steepness and breaker type.

The way breaker shape changes from top to bottom of the picture depends upon:

1 Increasing beach slope (if considered independently from wave characteristics).

2 Increasing wavelength and period and correspondingly decreasing wave steepness, if these characteristics are considered independently of beach slope.

It is not always possible to consider 1 and 2 separately, because as you will see in later Chapters, beach slope is partly influenced by prevailing wave type and partly by the particle sizes of the beach sediments, which in turn depend upon the energy of the waves which erode, transport and deposit them.

From the descriptions, Figure 1.18, and the answer to Question 1.15, it can be seen that the four types of breaker form a continuous series. The spilling breaker, characteristic of shallow beaches and steep waves (i.e. with short periods and large amplitudes), forms one end of the series. At the other end of the series is the surging breaker, characteristic of steep beaches and of waves with long periods and small amplitudes. For a given beach, the arrival of waves steeper than usual will tend to give a type of breaker nearer the 'spilling' end of the series, whereas calmer weather favours the surging type. The dynamics of collapsing (3) and surging (4) breakers are affected by bottom slope more than those of spilling (1) and plunging (2) breakers.

Spilling and plunging breakers can also occur in deep water, partly because the sea-bed is far below and does not affect wave dynamics. Collapsing and surging breakers do not occur in deep water.

-igure 1.18 (a) The four types of breaker seen in perspective view from top to bottom

~1-4): spilling, plunging, collapsing, surging. The vertical arrow shows their

elationships to beach slope, wave period, 3ngth and steepness.

~b) Cross-sections through the four breaker types.

~c) Photograph of a breaker, part spilling, )art plunging. See text for further Jiscussion.

IMPORTANs When examining Figure 1.18, tou need to be aware that the four types of creaker illustrated are just stages in a continuous spectrum; changes from one to ]nother are gradual, not instantaneous.

3t~

Waves of unusual character may result from any one of a number of conditions, such as a particular combination of wave frequencies; the constraining effect of nearby land masses; interaction between waves and ocean currents; or a submarine earthquake. The destructive effects of abnormally large waves are well known, and prediction of where and when they will occur is of extreme importance to all who live or work beside or upon the sea.

1.6.1 WAVES AND CURRENTS

Anyone who regularly sails a small boat into and out of estuaries will be well aware that at certain states of the tide the waves can become abnormally large and uncomfortable. Such large waves are usually associated with waves propagating against an ebbing fide. Because the strength of the tidal current varies with position as well as with time, waves propagating into an estuary during an ebb tide often advance into

progressively stronger counter-currents. The only ocean waves that can disturb the estuary as a whole during an ebbing tide are those that have speeds sufficiently high to overcome the effects of the counter-current.

Consider a simple system of deep-water waves, moving from a region with little or no current (A) into another region (B) where there is a current flowing parallel to the direction of wave propagation. Imagine two points, one in region A and one in region B, each of which are fixed with respect to the sea-bed. The number of waves passing each point in a given time must be the same, otherwise waves would either have to disappear, or be

generated, between the two points. In other words, the wave frequency (and period) must be the same at each point.

How will wavelength and wave height be affected if the current is flowing (a) with or (b) against the direction of wave propagation?

Clearly, a current flowing with the waves will have the effect of increasing the speed of the waves, although the wave period (T) must remain constant, i.e.

T = L o = L

c o c + u

where Lo = wavelength when current is zero;

Co = wave speed when current is zero;

L = wavelength in the current;

c = wave speed in the current;

u = speed of the current

Because c + u is greater than Co, then if T is to remain constant, L must be greater than L0, i.e. the waves get longer. Moreover, the waves get correspondingly lower, because the rate of energy transfer depends upon group speed (half wave speed) and wave height. If the rate of energy transfer is to remain constant, then as wave speed increases, wave height

(1.16~