ocean waves -concept of wave spectrum page 9

8/8/2019 Ocean Waves -Concept of Wave Spectrum Page 9

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Chapter 16

Ocean Waves

Looking out to sea from the shore, we can see waves on the sea surface. Lookingcarefully, we notice the waves are undulations of the sea surface with a height of around a meter, where height is the vertical distance between the bottom of atrough and the top of a nearby crest. The wavelength, which we might take tobe the distance between prominent crests, is around 50-100 meters. Watchingthe waves for a few minutes, we notice that wave height and wave length arenot constant. The heights vary randomly in time and space, and the statisticalproperties of the waves, such as the mean height averaged for a few hundredwaves, change from day to day. These prominent offshore waves are generatedby wind. Sometimes the local wind generates the waves, other times distantstorms generate waves which ultimately reach the coast. For example, wavesbreaking on the Southern California coast on a summer day may come fromvast storms offshore of Antarctica 10,000 km away.

If we watch closely for a long time, we notice that sea level changes fromhour to hour. Over a period of a day, sea level increases and decreases relative toa point on the shore by about a meter. The slow rise and fall of sea level is dueto the tides, another type of wave on the sea surface. Tides have wavelengths of thousands of kilometers, and they are generated by the slow, very small changesin gravity due to the motion of the sun and the moon relative to Earth.

In this chapter you will learn how to describe ocean-surface waves quantita-tively. In the next chapter we will describe tides and waves along coasts.

16.1 Linear Theory of Ocean Surface WavesSurface waves are inherently nonlinear: The solution of the equations of mo-tion depends on the surface boundary conditions, but the surface boundary

conditions are the waves we wish to calculate. How can we proceed?We begin by assuming that the amplitude of waves on the water surfaceis innitely small so the surface is almost exactly a plane. To simplify themathematics, we can also assume that the ow is 2-dimensional with wavestraveling in the x-direction. We also assume that the Coriolis force and viscositycan be neglected. If we retain rotation, we get Kelvin waves discussed in 14.2.

273


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274 CHAPTER 16. OCEAN WAVES

With these assumptions, the sea-surface elevation of a wave traveling inthe x direction is:

= a sin(k x t) (16.1)with

= 2 f =2T

; k =2L

(16.2)

where is wave frequency in radians per second, f is the wave frequency inHertz (Hz), k is wave number, T is wave period, L is wave length, and wherewe assume, as stated above, that ka = O(0).

The wave period T is the time it takes two successive wave crests or troughsto pass a xed point. The wave length L is the distance between two successivewave crests or troughs at a xed time.

Dispersion Relation Wave frequency is related to wave number k by thedispersion relation (Lamb, 1932 228):2 = g k tanh( kd) (16.3)

where d is the water depth and g is the acceleration of gravity.Two approximations are especially useful.

1. Deep-water approximation is valid if the water depth d is much greaterthan the wave length L. In this case, d L, kd 1, and tanh( kd) = 1.

2. Shallow-water approximation is valid if the water depth is much less thana wavelength. In this case, d L, kd 1, and tanh( kd) = kd.

For these two limits of water depth compared with wavelength the dispersionrelation reduces to:

2 = g k Deep-water dispersion relation (16.4)d > L/ 4

2 = g k2 d Shallow-water dispersion relation (16.5)d < L/ 11

The stated limits for d/L give a dispersion relation accurate within 10%. Be-cause many wave properties can be measured with accuracies of 510%, theapproximations are useful for calculating wave properties. Later we will learnto calculate wave properties as the waves propagate from deep to shallow water.

Phase Velocity The phase velocity c is the speed at which a particular phaseof the wave propagates, for example, the speed of propagation of the wave crest.In one wave period T the crest advances one wave length L and the phase speedis c = L/T = /k . Thus, the denition of phase speed is:

c k

(16.6)


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16.1. LINEAR THEORY OF OCEAN SURFACE WAVES 275

The direction of propagation is perpendicular to the wave crest and toward thepositive x direction.

The deep- and shallow-water approximations for the dispersion relation give:

c = gk = g Deep-water phase velocity (16.7)c = g d Shallow-water phase velocity (16.8)

The approximations are accurate to about 5% for limits stated in (16.4, 16.5).In deep water, the phase speed depends on wave length or wave frequency.

Longer waves travel faster. Thus, deep-water waves are said to be dispersive.In shallow water, the phase speed is independent of the wave; it depends onlyon the depth of the water. Shallow-water waves are non-dispersive.

Group Velocity The concept of group velocity cg is fundamental for under-standing the propagation of linear and nonlinear waves. First, it is the velocityat which a group of waves travels across the ocean. More importantly, it is alsothe propagation velocity of wave energy. Whitham (1974, 1.3 and 11.6) givesa clear derivation of the concept and the fundamental equation (16.9).

The denition of group velocity in two dimensions is:

cg k

(16.9)

Using the approximations for the dispersion relation:

cg =g

2=

c2

Deep-water group velocity (16.10)

cg = g d = c Shallow-water group velocity (16.11)For ocean-surface waves, the direction of propagation is perpendicular to thewave crests in the positive x direction. In the more general case of other typesof waves, such as Kelvin and Rossby waves that we met in 14.2, the groupvelocity is not necessarily in the direction perpendicular to wave crests.

Notice that a group of deep-water waves moves at half the phase speed of thewaves making up the group. How can this happen? If we could watch closely agroup of waves crossing the sea, we would see waves crests appear at the backof the wave train, move through the train, and disappear at the leading edge of the group. Each wave crest moves at twice the speed of the group.

Do real ocean waves move in groups governed by the dispersion relation? Yes.Walter Munk and colleagues (1963) in a remarkable series of experiments in the1960s showed that ocean waves propagating over great distances are dispersive,and that the dispersion could be used to track storms. They recorded wavesfor many days using an array of three pressure gauges just offshore of SanClemente Island, 60 miles due west of San Diego, California. Wave spectra were


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= 1 2 1

o= 2

0 5 o

9 11 13 15 17 19 21September 1959

10 10

10

10

.01.01.01

.01

0.1

1.0

F r e q u e n c y

[ m H z ]

0.01

0

0.07

0.08

= 1 1 5

o= 2

0 5 o

= 1 2 6

o = 2 0 0

o

= 1 3 5

o = 2 0 0

o

0.02

0.03

0.04

0.05

0.06

Figure 16.1 Contours of wave energy on a frequency-time plot calculated from spectra of waves measured by pressure gauges offshore of southern California. The ridges of high waveenergy show the arrival of dispersed wave trains from distant storms. The slope of the ridgeis inversely proportional to distance to the storm. is distance in degrees, is direction of arrival of waves at California (from Munk et al. 1963).

calculated for each days data. (The concept of a spectra is discussed below.)From the spectra, the amplitudes and frequencies of the low-frequency wavesand the propagation direction of the waves were calculated. Finally, they plottedcontours of wave energy on a frequency-time diagram (gure 16.1).

To understand the gure, consider a distant storm that produces waves of many frequencies. The lowest-frequency waves (smallest ) travel the fastest(16.11), and they arrive before other, higher-frequency waves. The further awaythe storm, the longer the delay between arrivals of waves of different frequencies.The ridges of high wave energy seen in the gure are produced by individualstorms. The slope of the ridge gives the distance to the storm in degrees along a great circle; and the phase information from the array gives the angle tothe storm . The two angles give the storms location relative to San Clemente.Thus waves arriving from 15 to 18 September produce a ridge indicating the

storm was 115

away at an angle of 205

which is south of new Zealand nearAntarctica.The locations of the storms producing the waves recorded from June through

October 1959 were compared with the location of storms plotted on weathermaps and in most cases the two agreed well.


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16.1. LINEAR THEORY OF OCEAN SURFACE WAVES 277

Wave Energy Wave energy E in Joules per square meter is related to thevariance of sea-surface displacement by:

E = w g 2 (16.12)

where w is water density, g is gravity, and the brackets denote a time or spaceaverage.

0 20 40 60 80 100 120 -200

-100

0

100

200

Time (s)

W a v e A m p l i t u d e ( m )

Figure 16.2 A short record of wave amplitude measured by a wave buoy in the NorthAtlantic.

Signicant Wave Height What do we mean by wave height? If we look ata wind-driven sea, we see waves of various heights. Some are much larger thanmost, others are much smaller (gure 16.2). A practical denition that is oftenused is the height of the highest 1/3 of the waves, H 1 / 3 . The height is computedas follows: measure wave height for a few minutes, pick out say 120 wave crestsand record their heights. Pick the 40 largest waves and calculate the averageheight of the 40 values. This is H 1 / 3 for the wave record.

The concept of signicant wave height was developed during the World WarII as part of a project to forecast ocean wave heights and periods. Wiegel (1964:p. 198) reports that work at the Scripps Institution of Oceanography showed

. . . wave height estimated by observers corresponds to the average of the highest 20 to 40 per cent of waves . . . Originally, the term signicantwave height was attached to the average of these observations, the highest30 per cent of the waves, but has evolved to become the average of thehighest one-third of the waves, (designated H S or H 1 / 3 )

More recently, signicant wave height is calculated from measured wave dis-placement. If the sea contains a narrow range of wave frequencies, H 1 / 3 isrelated to the standard deviation of sea-surface displacement ( nas , 1963: 22;Hoffman and Karst, 1975)

H 1 / 3 = 4 2 1 / 2 (16.13)

where 21 / 2 is the standard deviation of surface displacement. This relation-

ship is much more useful, and it is now the accepted way to calculate waveheight from wave measurements.


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16.2 Nonlinear wavesWe derived the properties of an ocean surface wave assuming waves were in-nitely small ka = O(0). If the waves are small ka 1 but not innitely small,the wave properties can be expanded in a power series of ka (Stokes, 1847). Hecalculated the properties of a wave of nite amplitude and found:

= a cos(kx t) +12

ka 2 cos2(kx t) +38

k2 a3 cos3(kx t) + (16.14)

The phases of the components for the Fourier series expansion of in (16.14) aresuch that non-linear waves have sharpened crests and attened troughs. Themaximum amplitude of the Stokes wave is amax = 0 .07L (ka = 0 .44). Suchsteep waves in deep water are called Stokes waves (See also Lamb, 1945, 250).Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a,1963b, 1966), using the tools of high-energy particle physics, worked out to 6thorder the interactions of three or more waves on the sea surface. He, Phillips(1960), and Longuet-Higgins and Phillips (1962) showed that n free waves onthe sea surface can interact to produce another free wave only if the frequenciesand wave numbers of the interacting waves sum to zero:

1 2 3 n = 0 (16.15a)k 1 k 2 k 3 k n = 0 (16.15b)

2i = g ki (16.15c)

where we allow waves to travel in any direction, and k i is the vector wavenumber giving wave length and direction. (16.15a,b) are general requirementsfor any interacting waves. The fewest number of waves that meet the conditions

of (16.15) are three waves which interact to produce a fourth. The interaction isweak; waves must interact for hundreds of wave lengths and periods to producea fourth wave with amplitude comparable to the interacting waves. The Stokeswave does not meet the criteria of (16.15) and the wave components are not freewaves; the higher harmonics are bound to the primary wave.

Wave Momentum The concept of wave momentum has caused considerableconfusion (McIntyre, 1981). In general, waves do not have momentum, a massux, but they do have a momentum ux. This is true for waves on the seasurface. Ursell (1950) showed that ocean swell on a rotating Earth has no masstransport. His proof seems to contradict the usual textbook discussions of steep,non-linear waves such as Stokes waves. Water particles in a Stokes wave movealong paths that are nearly circular, but the paths fail to close, and the particlesmove slowly in the direction of wave propagation. This is a mass transport, andthe phenomena is called Stokes drift. But the forward transport near the surfaceis balanced by an equal transport in the opposite direction at depth, and thereis no net mass ux.

Solitary Waves Solitary waves are another class of non-linear waves, and theyhave very interesting properties. They propagate without change of shape, and


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16.3. WAVES AND THE CONCEPT OF A WAVE SPECTRUM 279

two solitons can cross without interaction. The rst soliton was discovered byJohn Scott Russell (18081882), who followed a solitary wave generated by aboat in Edinburghs Union Canal in 1834.

Scott witnessed such a wave while watching a boat being drawn along theUnion Canal by a pair of horses. When the boat stopped, he noticed thatwater around the vessel surged ahead in the form of a single wave, whoseheight and speed remained virtually unchanged. Russell pursued the waveon horseback for more than a mile before returning home to reconstructthe event in an experimental tank in his garden.Nature 376, 3 August1995: 373.

The properties of a solitary waves result from an exact balance between disper-sion which tends to spread the solitary wave into a train of waves, and non-lineareffects which tend to shorten and steepen the wave. The type of solitary wavein shallow water seen by Russell, has the form:

= a sech23a4d3

1 / 2

(x ct) (16.16)

which propagates at a speed:

c = c0 1 +a2d

(16.17)

You might think that all shallow-water waves are solitons because they arenon-dispersive, and hence they ought to propagate without change in shape.Unfortunately, this is not true if the waves have nite amplitude. The velocityof the wave depends on depth. If the wave consists of a single hump, thenthe water at the crest travels faster than water in the trough, and the wavesteepens as it moves forward. Eventually, the wave becomes very steep andbreaks. At this point it is called a bore. In some river mouths, the incomingtide is so high and the estuary so long and shallow that the tidal wave enteringthe estuary eventually steepens and breaks producing a bore that runs up theriver. This happens in the Amazon in South America, the Severn in Europe,and the Tsientang in China (Pugh, 1987: 249).

16.3 Waves and the Concept of a Wave SpectrumIf we look out to sea, we notice that waves on the sea surface are not simplesinusoids. The surface appears to be composed of random waves of variouslengths and periods. How can we describe this surface? The simple answeris, Not very easily. We can however, with some simplications, come close to

describing the surface. The simplications lead to the concept of the spectrumof ocean waves. The spectrum gives the distribution of wave energy amongdifferent wave frequencies of wave lengths on the sea surface.

The concept of a spectrum is based on work by Joseph Fourier (17681830),who showed that almost any function (t) (or (x) if you like), can be repre-sented over the interval T/ 2 t T/ 2 as the sum of an innite series of sine


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and cosine functions with harmonic wave frequencies:

(t) = a0

2+

n =1(an cos2nft + bn sin2nft ) (16.18)

where

an =2T

T/ 2

T/ 2 (t)cos2nftdt, (n = 0 , 1, 2, . . . ) (16.19a)

bn =2T

T/ 2

T/ 2 (t)sin2nftdt, (n = 0 , 1, 2, . . . ) (16.19b)

f = 2 /T is the fundamental frequency, and nf are harmonics of the funda-mental frequency. This form of (t) is called a Fourier series (Bracewell, 1986:204; Whittaker and Watson, 1963:

9.1). Notice that a 0 is the mean value of

(t) over the interval.Equations (16.18 and 16.19) can be simplied using

exp(2 inft ) = cos(2 nft ) + i sin(2nft ) (16.20)

where i = 1. Equations (16.18 and 16.19) then become:

(t) =

n = Z n exp i

2 nft (16.21)

where

Z n =1T

T/ 2

T/ 2 (t)exp i 2 nft

dt, (n = 0 , 1, 2, . . . ) (16.22)

Z n is called the Fourier transform of (t).The spectrum S (f ) of (t) is:

S (nf ) = Z n Z

n (16.23)

where Z is the complex conjugate of Z . We will use these forms for the Fourierseries and spectra when we describing the computation of ocean wave spectra.

We can expand the idea of a Fourier series to include series that representsurfaces (x, y ) using similar techniques. Thus, any surface can be representedas an innite series of sine and cosine functions oriented in all possible directions.

Now, lets apply these ideas to the sea surface. Suppose for a moment that

the sea surface were frozen in time. Using the Fourier expansion, the frozensurface can be represented as an innite series of sine and cosine functionsof different wave numbers oriented in all possible directions. If we unfreezethe surface and let it evolve in time, we can represent the sea surface as aninnite series of sine and cosine functions of different wave lengths moving inall directions. Because wave lengths and wave frequencies are related through


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the dispersion relation, we can also represent the sea surface as an innite sumof sine and cosine functions of different frequencies moving in all directions.

Note in our discussion of Fourier series that we assume the coefficients(an , bn , Z n ) are constant. For times of perhaps an hour, and distances of per-haps tens of kilometers, the waves on the sea surface are sufficiently xed thatthe assumption is true. Furthermore, non-linear interactions among waves arevery weak. Therefore, we can represent a local sea surface by a linear super-position of real, sine waves having many different wave lengths or frequenciesand different phases traveling in many different directions. The Fourier seriesin not just a convenient mathematical expression, it states that the sea surfaceis really, truly composed of sine waves, each one propagating according to theequations we wrote down in 16.1.

The concept of the sea surface being composed of independent waves can becarried further. Suppose I throw a rock into a calm ocean, and it makes a bigsplash. According to Fourier, the splash can be represented as a superpositionof cosine waves all of nearly zero phase so the waves add up to a big splashat the origin. Furthermore, each individual Fourier wave then begins to travelaway from the splash. The longest waves travel fastest, and eventually, far fromthe splash, the sea consists of a dispersed train of waves with the longest wavesfurther from the splash and the shortest waves closest. This is exactly what wesee in gure 16.1. The storm makes the splash, and the waves disperse as seenin the gure.

Sampling the Sea Surface Calculating the Fourier series that represents thesea surface is perhaps impossible. It requires that we measure the height of thesea surface (x,y,t ) everywhere in an area perhaps ten kilometers on a side forperhaps an hour. So, lets simplify. Suppose we install a wave staff somewhere inthe ocean and record the height of the sea surface as a function of time (t). Wewould obtain a record like that in gure 16.2. All waves on the sea surface willbe measured, but we will know nothing about the direction of the waves. Thisis a much more practical measurement, and it will give the frequency spectrumof the waves on the sea surface.

Working with a trace of wave height on say a piece of paper is difficult, solets digitize the output of the wave staff to obtain

j (t j ), t j j (16.24) j = 0 , 1, 2, , N 1

where is the time interval between the samples, and N is the total numberof samples. The length T of the record is T = N . Figure 16.3 shows the rst

20 seconds of wave height from gure 16.2 digitized at intervals of = 0 .32 s.Notice that j is not the same as (t). We have absolutely no informationabout the height of the sea surface between samples. Thus we have convertedfrom an innite set of numbers which describes (t) to a nite set of numberswhich describe j . By converting from a continuous function to a digitizedfunction, we have given up an innite amount of information about the surface.


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Time, j [s] 0 5 10 15 20

-200

-100

0

100

200

W a v e

H e i g h t

, j

[ m ]

Figure 16.3 The rst 20 seconds of digitized data from gure 16.2. = 0 .32 s.

The sampling interval denes a Nyquist critical frequency (Press et al,1992: 494)

Ny

1/ (2) (16.25)

The Nyquist critical frequency is important for two related, but dis-tinct, reasons. One is good news, the other is bad news. First the goodnews. It is the remarkable fact known as the sampling theorem : If a con-tinuous function (t ), sampled at an interval , happens to be bandwidth limited to frequencies smaller in magnitude than Ny , i.e., if S (nf ) = 0for all |nf | N y , then the function (t ) is completely determined by itssamples j . . . This is a remarkable theorem for many reasons, amongthem that it shows that the information content of a bandwidth lim-ited function is, in some sense, innitely smaller than that of a generalcontinuous function . . .

Now the bad news. The bad news concerns the effect of sampling acontinuous function that is not bandwidth limited to less than the Nyquistcritical frequency. In that case, it turns out that all of the power spectral

density that lies outside the frequency range N y nf Ny is spuri-ously moved into that range. This phenomenon is called aliasing . Anyfrequency component outside of the range ( Ny,Ny ) is aliased (falselytranslated) into that range by the very act of discrete sampling . . . Thereis little that you can do to remove aliased power once you have discretelysampled a signal. The way to overcome aliasing is to (i) know the naturalbandwidth limit of the signal or else enforce a known limit by analogltering of the continuous signal, and then (ii) sample at a rate sufficientlyrapid to give at least two points per cycle of the highest frequency present.Press et al 1992, but with notation changed to our notation.

Figure 16.4 illustrates the aliasing problem. Notice how a high frequencysignal is aliased into a lower frequency if the higher frequency is above thecritical frequency. Fortunately, we can can easily avoid the problem: (i) use

instruments that do not respond to very short, high frequency waves if we areinterested in the bigger waves; and (ii) chose t small enough that we lose littleuseful information. In the example shown in gure 16.3, there are no waves inthe signal to be digitized with frequencies higher than Ny = 1 .5625 Hz.

Lets summarize. Digitized signals from a wave staff cannot be used to studywaves with frequencies above the Nyquist critical frequency. Nor can the signal


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t = 0.2 s

f = 4 Hz f = 1 Hz

one second

Figure 16.4 Sampling a 4 Hz sine wave (heavy line) every 0.2 s aliases the frequency to 1 Hz(light line). The critical frequency is 1/(2 0.2 s) = 2.5 Hz, which is less than 4 Hz.

be used to study waves with frequencies less than the fundamental frequencydetermined by the duration T of the wave record. The digitized wave record

contains information about waves in the frequency range:1T

< f p (16.37d)where F is the distance from a lee shore, called the fetch , or the distance overwhich the wind blows with constant velocity.


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16.4. OCEAN-WAVE SPECTRA 289

0.8

0.6

0.4

0.2

0

4x10 - 6

3

2

1 T ( n ) ( m 2 )

-1

-2

-3

-4

( n ) ( m

2 s e c )

(n)

T(n)

n(rad sec - 1 ) 1 2 3 4 5

x x x

x x

x x

x x x

x x

x x

x

x x

x

x

Figure 16.10 The function transferring wave energy from one part of the wave spectrum toanother part via nonlinear wave-wave interactions T (n ) where n is frequency. Dashed curveis for the theoretical spectrum ( n ), is for a measured spectrum. The interactionsextract energy from high frequency waves and transfer it to low frequency waves, causingthe spectrum to change with time and generating waves going faster than the wind, (FromPhillips, 1977: 139).

The energy of the waves increases with fetch:

2 = 1 .67 10 7 (U 10 )

2

gx (16.38)

where x is fetch.The jonswap spectrum is similar to the Pierson-Moskowitz spectrum except

that waves continues to grow with distance (or time) as specied by the term,and the peak in the spectrum is more pronounced, as specied by the term.The latter turns out to be particularly important because it leads to enhancednon-linear interactions and a spectrum that changes in time according to thetheory of Hasselmann (1966).

Generation of Waves by Wind We have seen in the last few paragraphsthat waves are related to the wind. We have, however, put off until now justhow they are generated by the wind. Suppose we begin with a mirror-smoothsea (Beaufort Number 0). What happens if the wind suddenly begins to blowsteadily at say 8 m/s? Three different physical processes begin:

1. The turbulence in the wind produces random pressure uctuations at thesea surface, which produces small waves with wavelengths of a few cen-timeters (Phillips, 1957).

2. Next, the wind acts on the small waves, causing them to become larger.Wind blowing over the wave produces pressure differences along the waveprole causing the wave to grow. The process is unstable because, as the


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wave gets bigger, the pressure differences get bigger, and the wave growsfaster. The instability causes the wave to grow exponentially (Miles, 1957).

3. Finally, the waves begin to interact among themselves to produce longerwaves (Hasselmann et al. 1973). The interaction transfers wave energyfrom short waves generated by Miles mechanism to waves with frequenciesslightly lower than the frequency of waves at the peak of the spectrum(gure 16.10). Eventually, this leads to waves going faster than the wind,as noted by Pierson and Moskowitz.

16.5 Wave ForecastingOur understanding of ocean waves, their spectra, their generation by the wind,and their interactions are now sufficiently well understood that the wave spec-trum can be forecast using winds calculated from numerical weather models. If we observe some small ocean area, or some area just offshore, we can see waves

generated by the local wind, the wind sea , plus waves that were generated inother areas at other times and that have propagated into the area we are ob-serving, the swell . Forecasts of local wave conditions must include both sea andswell, hence wave forecasting is not a local problem. We saw, for example, thatwaves off California can be generated by storms more than 10,000 km away.

Various techniques have been used to forecast waves. The earliest attemptswere based on empirical relationships between wave height and wave lengthand wind speed, duration, and fetch. The development of the wave spectrumallowed evolution of individual wave components with frequency f travelling indirection of the directional wave spectrum (f, ) using

0t

+ c g 0 = S i + S nl + S d (16.39)

where 0

= 0

(f, ; x , t ) varies in space ( x ) and time t, S i is input from the windgiven by the Phillips (1957) and Miles (1957) mechanisms, S nl is the transferamong wave components due to nonlinear interactions (gure 16.10), and S d isdissipation.

The third-generation wave-forecasting models now used by meteorologicalagencies throughout the world are based on integrations of (16.39) using manyindividual wave components (The swamp Group 1985; The wamdi Group, 1988;Komen et al, 1994). The forecasts follow individual components of the wavespectrum in space and time, allowing each component to grow or decay de-pending on local winds, and allowing wave components to interact according toHasselmanns theory. Typically the sea is represented by 300 components: 25wavelengths going in 12 directions (30 ). Each component is allowed to propa-gate from grid point to grid point, growing with the wind or decaying in time,

all the while interacting with other waves in the spectrum. To reduce comput-ing time, the models use a nested grid of points: the grid has a high density of points in storms and near coasts and a low density in other regions. Typically,grid points in the open ocean are 3 apart.

Some recent experimental models take the wave-forecasting process one stepfurther by assimilating altimeter and scatterometer observations of wind speed


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16.6. MEASUREMENT OF WAVES 291

Global 1 x 1,25 grid 20 August 1998 00:00 UTCWind Speed in Knots

Figure 16.11 Output of a third-generation wave forecast model produced by noaa s OceanModelling Branch for 20 August 1998. Contours are signicant wave height in meters, arrowsgive direction of waves at the peak of the wave spectrum, and barbs give wind speed in m/sand direction. From noaa Ocean Modelling Branch.

and wave height into the model. Forecasts of waves using assimilated satellitedata are available from the European Centre for Medium-Range Weather Fore-casts. Details of the third-generation models produced by the Wave AnalysisGroup ( wam ) are described in the book by Komen et al. (1994).

noaa s Ocean Modelling Branch at the National Centers for EnvironmentalPredictions also produces regional and global forecasts of waves. The Branch usea third-generation model based on the Cycle-4 wam model. It accommodatesever-changing ice edge, and it assimilates buoy and satellite altimeter wavedata. The model calculates directional frequency spectra in 12 directions and 25frequencies at 3-hour intervals up to 72 hours in advance. The lowest frequencyis 0.04177 Hz and higher frequencies are spaced logarithmically at incrementsof 0.1 times the lowest frequency. Wave spectral data are available on a 2.5 2.5 grid for deep-water points between 67.5 S and 77.5 N. The model is drivenusing 10-meter winds calculated from the lowest winds from the Centers weather

model adjusted to a height of 10 meter by using a logarithmic prole (8.19). TheBranch is testing an improved forecast with 1 1.25 resolution (gure 16.11).16.6 Measurement of WavesBecause waves inuence so many processes and operations at sea, many tech-niques have been invented for measuring waves. Here are a few of the more


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commonly used. Stewart (1980) gives a more complete description of wavemeasurement techniques, including methods for measuring the directional dis-tribution of waves.Sea State Estimated by Observers at Sea This is perhaps the most com-mon observation included in early tabulations of wave heights. These are thesignicant wave heights summarized in the U.S. Navys Marine Climatological Atlas and other such reports printed before the age of satellites.

Accelerometer Mounted on Meteorological or Other Buoy This is aless common measurement, although it is often used for measuring waves dur-ing short experiments at sea. For example, accelerometers on weather shipsmeasured wave height used by Pierson & Moskowitz and the waves shown ingure 16.2. The most accurate measurements are made using an accelerometerstabilized by a gyro so the axis of the accelerometer is always vertical.

Double integration of vertical acceleration gives displacement. The doubleintegration, however, amplies low-frequency noise, leading to the low frequencysignals seen in gures 16.4 and 16.5. In addition, the buoys heave is not sensitiveto wavelengths less than the buoys diameter, and buoys measure only waveshaving wavelengths greater than the diameter of the buoy. Overall, carefulmeasurements are accurate to 10% or better.Wave Gages Gauges may be mounted on platforms or on the sea oor inshallow water. Many different types of sensors are used to measure the heightof the wave or subsurface pressure which is related to wave height. Sound,infrared beams, and radio waves can be used to determine the distance fromthe sensor to the sea surface provided the sensor can be mounted on a stableplatform that does not interfere with the waves. Pressure gauges described in

6.8 can be used to measure the depth from the sea surface to the gauge. Arrays

of bottom-mounted pressure gauges are useful for determining wave directions.Thus arrays are widely used just offshore of the surf zone to determine offshorewave directions.

Pressure gauge must be located within a quarter of a wavelength of thesurface because wave-induced pressure uctuations decrease exponentially withdepth. Thus, both gauges and pressure sensors are restricted to shallow water orto large platforms on the continental shelf. Again, accuracy is 10% or better.Satellite Altimeters The satellite altimeters used to measure surface geo-strophic currents also measure wave height. Altimeters were own on Seasat in1978, Geosat from 1985 to 1988, ers1 &2 from 1991, Topex/Poseidon from1992, and Jason from 2001. Altimeter data have been used to produce monthlymean maps of wave heights and the variability of wave energy density in time

and space. The next step, just begun, is to use altimeter observation with waveforecasting programs, to increase the accuracy of wave forecasts.The altimeter technique works as follows. Radio pulse from a satellite al-

timeter reect rst from the wave crests, later from the wave troughs. Thereection stretches the altimeter pulse in time, and the stretching is measuredand used to calculate wave height (gure 16.12). Accuracy is 10%.


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16.7. IMPORTANT CONCEPTS 293

-15 -10 -5 0 5 10 15 20 Time (ns)

R e c e i v e d p o w e r

0.56m SHW

1.94m SWH

Figure

16.12 Shape of radio pulse received by the Seasat altimeter, showing the inuence of oceanwaves. The shape of the pulse is used to calculate signicant wave height. (From Stewart,1985).

Synthetic Aperture Radars on Satellites These radars map the radar re-ectivity of the sea surface with spatial resolution of 625 m. Maps of reectivityoften show wave-like features related to the real waves on the sea surface. I saywave-like because there is not an exact one-to-one relationship between waveheight and image density. Some waves are clearly mapped, others less so. Themaps, however, can be used to obtain additional information about waves, espe-cially the spatial distribution of wave directions in shallow water (Vesecky andStewart, 1982). Because the directional information can be calculated directlyfrom the radar data without the need to calculate an image (Hasselmann, 1991),data from radars and altimeters on ers 1 & 2 are being used to determine if theradar and altimeter observations can be used directly in wave forecast programs.

16.7 Important Concepts1. Wavelength and frequency of waves are related through the dispersion

relation.

2. The velocity of a wave phase can differ from the velocity at which waveenergy propagates.

3. Waves in deep water are dispersive, longer wavelengths travel faster thanshorter wavelengths. Waves in shallow water are not dispersive.

4. The dispersion of ocean waves has been accurately measured, and obser-vations of dispersed waves can be used to track distant storms.

5. The shape of the sea surface results from a linear superposition of waves of all possible wavelengths or frequencies travelling in all possible directions.

6. The spectrum gives the contributions by wavelength or frequency to thevariance of surface displacement.

7. Wave energy is proportional to variance of surface displacement.


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8. Digital spectra are band limited, and they contain no information aboutwaves with frequencies higher than the Nyquist frequency.

9. Waves are generated by wind. Strong winds of long duration generate thelargest waves.

10. Various idealized forms of the wave spectrum generated by steady, homo-geneous winds have been proposed. Two important ones are the Pierson-Moskowitz and jonswap spectra.

11. Observations by mariners on ships and by satellite altimeters have beenused to make global maps of wave height. Wave gauges are used on plat-forms in shallow water and on the continental shelf to measure waves.Bottom-mounted pressure gauges are used to measure waves just offshoreof beaches. And synthetic-aperture radars are used to obtain informationabout wave directions.

ocean waves -concept of wave spectrum page 9

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