Ocean Waves

Texas A&M OCNG 410, Rob Hetland

Guest Lecturer: Kristen M. Thyng

April 17 and 19, 2013

Contents

1 Introduction 1

2 Linear Theory 22.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Phase and Group Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Wave Representation 53.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Wave Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Sampling Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Idealized Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Application of Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Modeling 11

5 Wave Power 11

1 Introduction

• Waves are generated by a restoring force to a disturbance

• Examples of waves: sound waves (due to pressure disturbances), tides (caused bymoon/sun gravitation), ocean waves

• Ocean waves are generally a disturbance due to wind and a restoring force fromgravity

• Think of them on the surface of the ocean, but can be within the water column aswell (along density surfaces)

1

If x-axis is space: wave lengthIf x-axis is time: wave period

amplitude

Figure 1: Wave length and wave period are shown, dependent on what is representated bythe x-axis. Amplitude is also shown. Snapshot from wave illustration tool made by RobHetland.

• Waves coming onto the beach, waves in the open ocean, wave ripples moving awayfrom a rock dropped into water

2 Linear Theory

Waves are complicated, so let’s start with some assumptions to narrow our scope to thebasics:

• Amplitude of waves on water surface is small

• Flow is 2D

• Waves travel in one direction only

• No rotation or viscosity

Form for sea surface elevation, ζ, of a wave in the x-direction is

ζ = a sin (kx− ωt), (1)

• ω wave frequency in radians/sω = 2πf = 2π

T , T wave periodT is the time between successive waves to pass

• k wave number in 1/mk = 2π

L , L wavelengthL is the distance between two waves for a fixed time

Various wave properties are illustrated in Figure 1. The form of a wave (Equation 1)tells us that a wave can be approximately described as a constant amplitude multiplied bya sinusoidal function, which varies in time between -1 and 1.

2

ζ

d

-d

Figure 2: Notation for a wave on the sea surface. From Kundu and Cohen (2004).

2.1 Dispersion Relation

The dispersion relation tells us how ω and k depend on each other. Given our assumptions,the dispersion relation is

ω2 = gk tanh kd, (2)

where d is the depth of the water and g is the gravitational constant (9.81 m/s2). SeeFigure 2 for an illustration.

Taking extreme values of parameters allows us to simplify and examine the importantfeatures of specific cases. Here we look at two physical cases: when the depth d is muchgreater or much less than the wavelength L of a wave.

Deep water (d� L or d/L� 1):

2πd

L→∞ as d/L→∞ (3)

⇒ tanh (kd) = tanh

(2πd

L

)→ 1 as d/L→∞ (4)

⇒ ω2 ≈ gk. (5)

You can check the behavior of tanh() by plotting or trying different numbers.Shallow water (d� L or d/L� 1):

tanh

(2πd

L

)→ 2πd

Las d/L→ 0 (6)

⇒ ω2 ≈ gk2d (7)

3

2.2 Phase and Group Speed

Phase speed c is that speed that a particular wave travels at; it is the wave movement thatwe can most readily see. It can be derived as:

velocity = distance/time (8)

c = L/T (9)

=2π

k

ω

2π(10)

=ω

k. (11)

We can use again learn about the phase speed by looking at special cases.Deep water:

c =ω

k(12)

=

√gk

k(13)

=

√g

k(14)

=g

ω. (15)

Shallow water:

c =ω

k(16)

=

√gk2d

k(17)

=√gd. (18)

The important thing to note here is that in the deep water case, the phase speed is afunction of the wave number/frequency. This means that waves of different wave lengthwill travel at different speeds. These waves are then called dispersive. Oppositely, thephase speed for shallow water waves does not depend on the wave properties, only gravityand the depth of the water, and waves will travel at the same speed. These waves arenondispersive.

Group speed is the rate of travel of a packet of waves; it is the speed at which energyand information propagate. It is defined as

cg =∂ω

∂k. (19)

4

Deep water:

cg =∂

∂k

((gk)1/2

)(20)

=1

2(gk)−1/2g (21)

=1

2

√g/k (22)

=1

2c (23)

Shallow water:

cg =∂

∂k

((gk2d)1/2

)(24)

=1

2

(gk2d

)−1/22gkd (25)

=√gd (26)

= c (27)

For shallow water waves, the group and phase speeds are equal.For deep water waves, the group speed is half the phase speed. This means that individualwaves travel faster than groups of waves. This effect is possible because deep water wavesare dispersive, and the phase speed depends on the wave properties.

In general, examples of the different scenarios possible between two waves interactingcan be found using Rob’s twoWave simulator. Different scenarios include:

• c > cg, cg > 0 : ω1 = 0.5, ω2 = 0.44, k1 = 1.72, k2 = 1.44

• c < cg, cg > 0 : ω1 = 0.5, ω2 = 0.39, k1 = 1.72, k2 = 1.44

• c > cg, cg < 0 : ω1 = 0.5, ω2 = 0.72, k1 = 1.72, k2 = 1.44

Thus, the wave frequency/number can be adjusted for the simulated waves to cause thephase speed to be greater then or less than the group speed, and the group speed can bepositive or negative.

Examples can also be found online. In this case, the waves are approximately movingin just one direction (outward from the boat track), as we assumed. It is hard to see thepacket of waves moving, but we can see individual waves appear from the tail end andmove forward, as in the model case.

3 Wave Representation

In order to study waves and the surface of the sea, we need to be able to represent thesurface. For waves, this is typically done using an energy spectrum, which is based on theconcept of Fourier series.

5

4 3 2 1 0 1 2 3 44

3

2

1

0

1

2

3

4Fourier series of f(x) with various numbers of terms included

f(x)=x

1

5

10

30

50

500

5000

Figure 3: Different numbers of terms in the Fourier series representation of the functionf(x) have been included and overlaid. Example from O’Neil (2003)

3.1 Fourier Series

Fourier series are summations of an infinite number of terms made up of sine and cosinesfunctions, unlike Taylor series, which are made up of polynomial terms. Fourier seriescan be used to represent a surface on a particular interval of space or time. They are acommonly-used tool for information representation and manipulation.

For example, let us consider the function f(x) = x for −π ≤ x ≤ π. This can berepresented using a Fourier series for the interval −π < x < π as

f(x) = x =

∞∑

n=1

2

n(−1)n+1 sin(nx).

The function actually equals the summation when it includes an infinite number ofterms. However, in practice we select a finite number of terms to approximate our function.How many terms we include will affect how good of a representation it is.

Figure 3 shows a plot of f(x) = x overlaid with multiple representations of the functionas Fourier series with differing numbers of included terms from the series. Clearly, as moreterms are included, the approximation to the function is improved, and it will be exactwhen an infinite number of terms are included. Note that the endpoints are never correct,but −π and π are not included in the interval of representation for the Fourier series. Thisexample was made using Python script fourier.py and is available if anyone is interested.

6

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 = "wg!!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)

Wave

Am

plit

ude (

m)

Figure 16.2 A short record of wave amplitude measuredby a wave buoy in the north Atlantic.

Significant 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 (figure 16.2). A practical definition that is oftenused is the height of the highest 1/3 of the waves, H1/3. 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 H1/3 for the wave record.

The concept of significant 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 thehighest 20 to 40 per cent of waves. . . Originally, the term significant waveheight was attached to the average of these observations, the highest 30per cent of the waves, but has evolved to become the average of the highestone-third of the waves, (designated HS or H1/3)

More recently, significant wave height is calculated from measured wave dis-placement. If the sea contains a narrow range of wave frequencies, H1/3 isrelated to the standard deviation of sea-surface displacement (nas , 1963: 22;Ho!man and Karst, 1975)

H1/3 = 4!!2"1/2

(16.13)

where!!2"1/2

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.

Figure 4: From Stewart (2008).

3.2 Wave Spectrum

The wave energy spectrum gives the distribution of the variance of sea-surface height asa function of frequency. There is a mathematical way, based on Fourier series, to movebetween information in time space to frequency space, to get this representation. Spectraare used as a convenient way to represent information in some applications. For example,tracking any one wave in the ocean is very difficult and not necessarily very helpful, butknowing a measure of the strength of the waves (wave height) and the frequency of thewaves (which also gives period and, if you know the dispersion relation, the wave number)gives the relevant information.

Wave energy is calculated as:

E = ρg〈ζ2〉, [J/m2]

where ρ is the water density and 〈ζ2〉 gives the variance of the sea surface displacement.Wave energy then can be represented using the wave energy spectrum.

We should specify what we mean by wave height. Two ways of calculating specific waveheight are:

• As the height of the highest 1/3 of the waves, averaged together

• H1/3 = 4〈ζ2〉1/2

3.3 Sampling Data

Let’s start with some example wave data from Stewart (2008), which is shown in Figure 4.

7

282 CHAPTER 16. OCEAN WAVES

t=0.2 s

f=4Hz f=1Hz

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.

interested in the bigger waves; and (ii) chose !t small enough that we lose littleuseful information. In the example shown in figure 16.3, there are no waves inthe signal to be digitized with frequencies higher than Ny = 1.5625 Hz.

Let’s summarize. Digitized signals from a wave sta" cannot be used to studywaves with frequencies above the Nyquist critical frequency. Nor can the signalbe used to study waves with frequencies less than the fundamental frequencydetermined by the duration T of the wave record. The digitized wave recordcontains information about waves in the frequency range:

1

T< f <

1

2!(16.24)

where T = N! is the length of the time series, and f is the frequency in Hertz.

Calculating The Wave Spectrum The digital Fourier transform Zn of awave record !j equivalent to (16.21 and 16.22) is:

Zn =1

N

N!1!

j=0

!j exp[!i2"jn/N ] (16.25a)

!n =N!1!

n=0

Zj exp[i2"jn/N ] (16.25b)

for j = 0, 1, · · · , N ! 1; n = 0, 1, · · · , N ! 1. These equations can be summedvery quickly using the Fast Fourier Transform, especially if N is a power of 2(Cooley, Lewis, and Welch, 1970; Press et al. 1992: 542).

This spectrum Sn of !, which is called the periodogram, is:

Sn =1

N2

"|Zn|2 + |ZN!n|2

#; n = 1, 2, · · · , (N/2 ! 1) (16.26)

S0 =1

N2|Z0|2

SN/2 =1

N2|ZN/2|2

Figure 5: Beware of subsampling a wave and aliasing the signal (Stewart (2008))

Aliasing is an important concept to keep in mind when collecting data. Figure 5represents the problem: when you are trying to sample an unknown function that lookslike the 4 Hz line, but you unknowingly sample too infrequently and capture what looks likethe 1 Hz line. Waves with frequencies above the Nyquist critical frequency (Ny = 1/(2∆),where ∆ is the sampling frequency) cannot be captured for a given instrument setting,so instrument settings must be established such that the frequencies of interest for theapplication can be captured. The signal in question must also be sampled for a longenough time to capture the period of the desired wave, in order to have a full sample ofthe wave. These requirements can be summarized as (Stewart, 2008):

1

N∆< f <

1

2∆,

where N is the number of samples taken and ∆ is the length of the time series acquired.The example wave data, digitized, is shown in Figure 6. This basically just points out

that we only know information about the waves at a discrete number of points. This datais sampled often enough to capture the necessary wavelengths.

Two example spectra are shown in Figure 7. The spectrum, Sn, of ζ, is shown in Figure7(a) and is called a periodogram. Because the periodogram is noisy, we average severaltogether, called the spectrum of the wave height, shown in Figure 7(b). This gives thedistribution of the variance of sea surface height as a function of frequency. We wouldnormally use three hours of data to compute this spectrum. Note that data in the spectrado not exceed the stated Nyquist frequency of 1.5625 Hz.

8

16.3. WAVES AND THE CONCEPT OF A WAVE SPECTRUM 281

Time, j! (s)

0 5 10 15 20-200

-100

0

100

200

Wa

ve H

eig

ht,

"j (

m)

Figure 16.3 The first 20 seconds of digitized data from figure 16.2. ! = 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 infinite set of numbers which describes !(t) to a finite set of numberswhich describe !j . By converting from a continuous function to a digitizedfunction, we have given up an infinite amount of information about the surface.

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

Ny ! 1/(2!) (16.23)

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 bandwidthlimited to frequencies smaller in magnitude than Ny, i.e., if S(nf) = 0for all |nf | ! Ny, then the function !(t) is completely determined by itssamples !j . . . This is a remarkable theorem for many reasons, among themthat it shows that the “information content” of a bandwidth limited func-tion is, in some sense, infinitely smaller than that of a general continuousfunction. . .

Now the bad news. The bad news concerns the e"ect 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 spectraldensity that lies outside the frequency range "Ny # 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 analogfiltering of the continuous signal, and then (ii) sample at a rate su#cientlyrapid 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) useinstruments that do not respond to very short, high frequency waves if we are

Figure 6: From Stewart (2008).

16.3. WAVES AND THE CONCEPT OF A WAVE SPECTRUM 283

Pow

er

Spect

ral D

ensi

ty (

m2/H

z)

104

106

102

100

10-2

10-4

10-3 10-1 101

Frequency (Hz)

10-2 100

108

Figure 16.5 The periodogram calculated from the first 164 s of datafrom figure 16.2. The Nyquist frequency is 1.5625 Hz.

where SN is normalized such that:

N!1!

j=0

|!j |2 =

N/2!

n=0

Sn (16.27)

The variance of !j is the sum of the (N/2+ 1) terms in the periodogram. Note,the terms of Sn above the frequency (N/2) are symmetric about that frequency.Figure 16.5 shows the periodogram of the time series shown in figure 16.2.

The periodogram is a very noisy function. The variance of each point is equalto the expected value at the point. By averaging together 10–30 periodogramswe can reduce the uncertainty in the value at each frequency. The averagedperiodogram is called the spectrum of the wave height (figure 16.6). It gives thedistribution of the variance of sea-surface height at the wave sta! as a functionof frequency. Because wave energy is proportional to the variance (16.12) thespectrum is called the energy spectrum or the wave-height spectrum. Typicallythree hours of wave sta! data are used to compute a spectrum of wave height.

Summary We can summarize the calculation of a spectrum into the followingsteps:

1. Digitize a segment of wave-height data to obtain useful limits according to(16.26). For example, use 1024 samples from 8.53 minutes of data sampledat the rate of 2 samples/second.

284 CHAPTER 16. OCEAN WAVES

Pow

er

Spect

ral D

ensi

ty (

m2/H

z)

104

105

103

102

101

100

10-1

10-2

10-2 10-1 100

Frequency (Hz)

Figure 16.6 The spectrum of waves calculated from 11 minutes of data shown in figure 7.2by averaging four periodograms to reduce uncertainty in the spectral values. Spectral valuesbelow 0.04 Hz are in error due to noise.

2. Calculate the digital, fast Fourier transform Zn of the time series.

3. Calculate the periodogram Sn from the sum of the squares of the real andimaginary parts of the Fourier transform.

4. Repeat to produce M = 20 periodograms.

5. Average the 20 periodograms to produce an averaged spectrum SM .

6. SM has values that are !2 distributed with 2M degrees of freedom.

This outline of the calculation of a spectrum ignores many details. For morecomplete information see, for example, Percival and Walden (1993), Press etal. (1992: §12), Oppenheim and Schafer (1975), or other texts on digital signalprocessing.

16.4 Ocean-Wave Spectra

Ocean waves are produced by the wind. The faster the wind, the longer thewind blows, and the bigger the area over which the wind blows, the bigger thewaves. In designing ships or o!shore structures we wish to know the biggestwaves produced by a given wind speed. Suppose the wind blows at 20 m/s formany days over a large area of the North Atlantic. What will be the spectrumof ocean waves at the downwind side of the area?

Figure 7: From Stewart (2008).

9

16.4. OCEAN-WAVE SPECTRA 285

0

10

30

50

60

80

90

110

100

70

40

20

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35Frequency (Hz)

Wa

ve S

pe

ctra

l De

nsi

ty (

m2

/ H

z)

20.6 m/s

18 m/s

15.4 m/s

12.9 m/s

10.3 m/s

Figure 16.7 Wave spectra of a fully developed sea for di!erentwind speeds according to Moskowitz (1964).

Pierson-Moskowitz Spectrum Various idealized spectra are used to answerthe question in oceanography and ocean engineering. Perhaps the simplest isthat proposed by Pierson and Moskowitz (1964). They assumed that if thewind blew steadily for a long time over a large area, the waves would come intoequilibrium with the wind. This is the concept of a fully developed sea. Here, a“long time” is roughly ten-thousand wave periods, and a “large area” is roughlyfive-thousand wave lengths on a side.

To obtain a spectrum of a fully developed sea, they used measurements ofwaves made by accelerometers on British weather ships in the north Atlantic.First, they selected wave data for times when the wind had blown steadily forlong times over large areas of the north Atlantic. Then they calculated the wavespectra for various wind speeds (figure 16.7), and they found that the function

S(!) ="g2

!5exp

!!#"!0

!

#4$

(16.28)

was a good fit to the observed spectra, where ! = 2$f , f is the wave frequencyin Hertz, " = 8.1 " 10!3, # = 0.74 , !0 = g/U19.5 and U19.5 is the wind speedat a height of 19.5 m above the sea surface, the height of the anemometers onthe weather ships used by Pierson and Moskowitz (1964).

For most airflow over the sea the atmospheric boundary layer has nearly

Figure 8: Idealized wave spectrum dependent on wind speed. From Stewart (2008).

3.4 Idealized Spectra

Wave energy can also be represented with idealized functional forms of spectra. Spectra forcertain situations, such as wind blowing on the ocean for a long time over a large area, canbe approximated using known functions, which enables quick calculations, with a possiblesacrifice in accuracy depending on how realistic the idealized functions are. See Figure 8.

3.5 Application of Spectra

Stewart shows an example in his book of people using dispersion and wave spectra to trackstorms in the 1960s. They recorded waves for many days using three pressure gauges 60miles west of San Diego. They calculated wave spectra for each day’s data and found theamplitudes, frequencies, and propagation direction of the low-frequency waves.

Shown in Figure 9 is wave energy as a function of both frequency and time. In orderto interpret the plot, consider a distant storm generating waves. These waves will travelat different speeds due to dispersion. The wave energy will travel at the group speed,and for deep water waves, cg = 1/2 ∗ g/ω, so that lower frequency waves will travel fasterthan higher frequency waves. This means that lower frequency wave energy will reach thewave gauges before higher frequency wave energy, and the lag between the arrival of thewaves will align for a tell-tale signal in the resulting data. For example, the storm arrivingbetween September 15th and 18th can be seen in the plot due to the ridgeline of wave

10

276 CHAPTER 16. OCEAN WAVES

9 11 13 15 17 19 21

September 1959

10 10

10

10

.01.01.01

.01

0.1

1.0

Fre

quency

(m

Hz)

0.01

0

0.07

0.08

!=115o "=205

o

!=126o "=200

o

!=135o "=200

o

0.02

0.03

0.04

0.05

0.06

!=121o "=205

o

Figure 16.1 Contours of wave energy on a frequency-time plot calculated from spectra ofwaves measured by pressure gauges o!shore 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 ofarrival of waves at California. After Munk et al. (1963).

calculated for each day’s 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 (figure 16.1).

To understand the figure, consider a distant storm that produces waves ofmany 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 di!erent frequencies.The ridges of high wave energy seen in the figure 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 storm’s location relative to San Clemente.Thus waves arriving from 15 to 18 September produce a ridge indicating thestorm 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 throughOctober 1959 were compared with the location of storms plotted on weathermaps and in most cases the two agreed well.

Figure 9: From Stewart (2008).

energy, spread out in time and frequency according to the above reasoning. The angle ofthe ridge indicates the distance away of the storm, in degrees, and the angle of directionof the storm. This information helps to pinpoint the origin of the storm, which from thisinformation was south of New Zealand near Antarctica.

4 Modeling

One way of finding wave information is by running a numerical model. An example of asystem with wave modeling included in COAWST. Waves tend to correlate with the windvectors in the Gulf of Mexico simulation.

5 Wave Power

Energy production from sources like this tend to be based around making something moverelative to something else. The PowerBuoy is a point absorber that moves up and downin the water relative to the outer part of the structure to create energy (to be built off

11

(a) Ocean Power Technology PowerBuoy(Wikipedia, 2013b)

(b) Pelamis (Wikipedia, 2013a)

Figure 10: Several wave energy technologies

the Oregon coast, Reedsport, this year!). The different segments of Pelamis move up anddown at different rates to generate energy. The WaveDrageon uses overtopping to capturewater, store it, and then drain it through turbines downward to generate electricity.

References

Kundu, P. K. and Cohen, I. M. (2004). Fluid Mechanics. Elsevier Academic Press, thirdedition.

O’Neil, P. V. (2003). Advanced Engineering Mathematics. Brooks/Cole Thomson Learning,fifth edition.

Stewart, R. H. (2008). Introduction to Physical Oceanography.

Wikipedia (2013a). Pelamis Wave Energy Converter. http://en.wikipedia.org/wiki/

Pelamis_Wave_Energy_Converter.

Wikipedia (2013b). Powerbuoy. http://en.wikipedia.org/wiki/File:Optbuoy.jpg.

Wikipedia (2013c). Wave Dragon. http://en.wikipedia.org/wiki/Wave_Dragon.

12

(a) WaveDragon (Wikipedia, 2013c) (b) WaveDragon diagram

Figure 11: WaveDragon

13