Energy from Waves and Tidal Currents - Marine Renewables Canada

ENERGY FROM WAVES AND TIDAL CURRENTSTowards 20yy ?Emile BaddourInstitute for Ocean TechnologyNational Research CouncilAugust 2004Acknowledgements:The author wishes to thank IC for their support of the project, Geoff Lewis of IC for numerousdiscussions through phone, emails and meetings, CISTI for their collection and their help ingathering related information and NRC for allowing the time spent on the project.TABLE OF CONTENTSSummary1. Introduction2. Ocean waves and marine currents2.1 Origins, generation and propagation of wind waves2.2 Description and modification of waves2.3 Real sea characteristics2.4 Wave energy resource distribution2.5 Data about waves2.6 Marine currents origins2.7 Marine currents resource2.8 Marine currents characteristics and resource distribution3. Ocean energy harnessing systems3.1 Wave energy harnessing technology3.2 Wave energy developments and activities3.3 Ocean currents energy technology3.4 Ocean currents developments and activities1

4. Transfer of power systems4.1 Interfaces4.2 Storage of energy4.3 Electric power generation and conversion4.4 Power transmission to the grid5. Ocean power economics5.1 Factors affecting the evaluation of costs5.2 Capital and Operating costs of Ocean Energy systems5.3 Generation costs of wave and current energy5.4 Comparison with costs of other renewables and electricpower prices6. R&D programmes6.1 In Canada7. Timeline proposal7.1 Ocean energy influential events7.2 Timeline7.3 Scientific, technical, societal and economic challenges7.4 Constraints and opportunities8. References9. Recent scientific and technical research bibliographyAppendices:Appendix 1. Formulation and computation of wave power in longcrestedregular and irregular seasAppendix 2: Useful wave & tidal power/energy linksAppendix 3: Recent publications from IOT2

Summary:Renewable energy sources have been known for centuries. It is only recently in the past30 years that modern technology and investment got together to produce viablealternatives to fossil fuel that can produce energy on a useful, economical scale. Thestimulus to develop renewables is being driven by several factors. It is found that fossilfuels are causing global warming and there is pressure to reduce the amount of carbondioxide produced in energy production. There is prospect of rising pricing of oil andgas as well as depletion of the reserves and relying on importing fuel. Now formally onaccepting the Kyoto protocol Canada has committed towards reducing gas emitions thatcause house effects.In its Millennium Statement, "Energy for Tomorrow’s World – Acting Now!", thereport of the World Energy Council, presents three principles for energy development:Accessibilityis the provision of reliable, affordable modern energy services.Availabilityaddresses the quality and reliability of the service.Acceptabilityaddresses environmental goals and public attitudes, specifically local pollution andglobal climate change.These principles are underpinned by ten policy actions which include: keeping allenergy options open and ensuring adequate and appropriate research funding. (WorldEnergy Council, 2000)Keeping with the recommendation of keeping all energy options open, the presentdocument reports on Energy from the Ocean. To put this energy source in perspectivewe have to note that presently this new renewable is part of the Other sources makingthe 0.5% of the World Total Primary Energy Supply. See Figure 1.1 below.This report presents an overview of the main developments in ocean energy fromwaves and currents and could be considered as an introduction to the topic of OceanEnergy Systems, specifically from ocean waves and currents. Limited time and notbeing in the scope of this review precluded verification of some of the claims herein,using different or independent sources from the ones reported. However, it can indicatethe trend. Further development and assessment of the resource are needed. The reportconcludes with a list of tasks classified under 6 main possible threads for an action planfor research and development in ocean energy systems and a proposal to identify theInstitute of Ocean Technology of the National Research Council, http://iot-ito.nrccnrc.gc.ca/about.htmlas the Center for Testing, Evaluation, and Research in OceanEnergy Systems. The center is proposed as a node within an Ocean Energy TechnologyNetwork that would allow Industry, Academia and Government a much-neededcollaboration within their identified respective roles.3

ever closer to commercial utilization. A number of commercial plants are being built inEurope, Australia and elsewhere. A number of devices have proven their applicabilityon a large scale in harsh operational environments and other are in different stages oftheir Research and Development cycle with different levels of their implementation.These will be reported herein. From studying the experts reports it is concluded thatextensive R&D work is needed at both fundamental and application levels with theobjective of improving the cost estimation, performance, feasibility of ocean energysystems to establish their position in the renewable energy market.The societal, political, industry and academic sectors are converging towards a needand demand for more "green", renewable energy. This trend is clearly seen in the Kyotoagreement and expressed in action plans in Europe, Canada, USA and other parts of theworld. The following excerpt from of the Speech from the Throne in BC speaks foritself:Speech from the Throne, The Honourable Iona Campagnolo, Lieutenant Governor at the Opening of the FifthSession, Thirty-Seventh Parliament of the Province of British Columbia, February 10, 2004

non-commercial energy and covers all major renewable energy resources. Combustiblerenewables and waste account for nearly 80% of the renewables share, hydro for 16.5%and "new" renewables: geothermal, solar, tidal, wave, wind and other, together accountfor 0.5%. See Figure 1.1 below.Figure 1.1 Fuels Shares of World Total Primary Energy Supply (source: IEA, 2002)The domination of the energy supply by fossil fuels is set to continue for aforeseeable future, since their resource base remains adequate and the adverseenvironmental impacts are attracting a considerable effort to identify and deploycleaner fossil fuel technologies, an area in which World Energy Council (WEC) hasbeen active for several years.In its Millennium Statement, "Energy for Tomorrow’s World – Acting Now!", thereport of the World Energy Council, presents three principles for energy development:Accessibilityis the provision of reliable, affordable modern energy services for which payment ismade under policy specifically targeted on meeting the needs of the poor.Availabilityaddresses the quality and reliability of the serviceAcceptabilityaddresses environmental goals and public attitudes, specifically local pollution andglobal climate change.These principles are underpinned by ten policy actions which include keeping allenergy options open and ensuring adequate and appropriate research funding. (WorldEnergy Council, 2000). See also International Energy Agency, Ocean Energy SystemsAnnual Report (2003).Keeping with the recommendation of keeping all energy options open, the presentdocument reports on Energy from the Ocean. To put this energy source in perspective6

we note that this new renewable is part of the Other sources making the 0.5% of all therenewables, See Figure 1.1.2 Ocean waves and marine currents2.1 Origins and generation of wind waves2.2 Wave energy resource distribution2.3 Description and modification of waves2.4 Real sea characteristics2.5 Data about waves2.6 Marine currents origins2.7 Marine currents resource2.8 Marine currents characteristics and resource distribution2.1 Origins and generation of wind wavesThere are many kinds of waves in the ocean. They differ in form, velocity andorigin. Some waves are too long and low to see, also there exist waves that travel ondensity interfaces below the sea surface. Waves may be generated by ships orlandslides or the passage of the moon or by earthquakes or changes in the atmosphericpressure. The waves which are of interest in this report are those mainly raised by thewinds.The energy in the waves comes from the sun through the winds as they blow over theoceans due to the differential heating of the earth. The winds transfer their energy to thesurface of the sea creating waves. See Kinsman (1984). Wave energy is henceconsidered as a concentrated form of solar energy. The mechanism of the transfer ofenergy from the atmosphere to the surface of the sea is complex. The main phenomenacan be reduced to:Air flowing over the free surface of the water activates a tangential stress on thewater surface and result in the formation and growth of the waves.Variable shear stresses and pressure fluctuations are created by turbulent air flowing onthe sea surface. Further wave increase and development happens when theseoscillations and fluctuations are in phase with the waves. Some kind of resonanceeffect.Wave reaching a certain height will be directly affected by the wind forcing on theupwind face of the wave causing further growth. At each of the above steps energy intransferred to the water. The amount of energy transferred and hence the size of theresulting waves is a function of the wind speed, the length of time it blows and thedistance over which it blows called the fetch.As the wave continues to grow the surface facing the wind becomes higher andsteeper and the process of wave building becomes more efficient. However up to apoint because there is a limit on how steep a wave can be. Steepness is the ratio of theheight of the wave (distance between a crest and the following trough) to its length7

(distance between a crest and the following one) is approximately 1:7 in deep water. Inthe generating area, often a storm, wind waves form what is called a "sea". At theupwind end of the fetch the waves are small but with distance they develop i.e. theirperiod and height increase and eventually they reach maximum dimensions possible forthe wind that is raising them. The sea is then said to be fully developed. The waveshave absorbed then as much energy as they can from wind of that velocity. Anextension of the fetch or a lengthening of the time would not produce larger waves.(Kinsman 1987, Bascom 1976) See also Sverdrup and Munk: Wind, Sea and Swell.See Figure 2.1 for a schematic of waves development and propagation stages.Figure 2.1. Concept of wave generation and propagation. The fetch is within the dashed line.Source: R. Silvester.2.2 The wave energy resource distributionAt each of the above steps energy in transferred to the water. The amount of energytransferred and hence the size of the resulting waves is a function of the wind speed, thelength of time it blows and the distance over which it blows called the fetch. It is foundthat at each step power is concentrated and that solar power levels of about 100W/m 2can eventually be transformed into waves with power levels of over 1000 kW per meterof wave crest .The distribution of the wave energy resource over the globe and its daily andseasonal variability during the year are dependent on the major wind distributions andsystems that are the main cause for the generation of ocean waves. The main windsystems are due to extra-tropical storms and trade winds. Ocean currents like the GulfStream in the North Atlantic and the Kuroshio current in the pacific feed energy intoextra-tropical cyclones creating low pressure systems with wind speed that can reach upto 25 m/s and blow over a thousand kilometer fetch for two to four consecutive daysbefore subsiding by hitting the coast. The storms are most frequent during the winter.Extra-tropical cyclones follow a north-easterly track and continually build the waves in8

the southern sector of the storm. These waves will travel in the same direction as thestorm that generates them. However, waves generated in the northern part of anorthern-hemisphere cyclone travel opposite to the direction of the storm and get lessexposure to the storm’s wind energy. The result is that one gets comparatively lesswave energy in the western sides of an ocean basin than in the eastern side. Thisfeature is found in the northern hemisphere sectors of the Atlantic as well as the PacificOcean. The annual average of the wave-power levels is estimated along the edge ofNorth America from 10 to 20 kW/m reaching a level of 50 kW/m along Newfoundlandcost while along the edge of the eastern continental shelf of the north Atlantic isestimated to vary from 40kW/m off Portugal up to 75kW/m off the Irish and Scottishcoast dropping to about 30kW/m off the Northern part of the Norwegian coast. Figure2.2 shows the geographical distribution of coastal wave power levels. Along the coastof California the level is estimated as 30 kW/m and increasing to 60 kW/m along thecoast of Northern British Colombia. Local variation in the estimate of wave poweravailability is certain and more accurate estimates are needed. Accurate knowledge ofthe directionality of the wave regimes is also needed. Similar analysis has been done forthe southern hemisphere and is not presented here. (See for example Wavegen 2000)As for the trade winds systems off tropical coasts they carry an annual offshore wavepower levels of the order of 10 to 20 kW/m. In comparison to the extra-tropical stormwinds the trade-winds are more persistent and the variations in these winds intensitybetween the seasons are smaller. Some of these tropical areas might also be affected byswells generated in storm regions further north or south. Therefore some believe thatthis sustained source of wave energy around the equator is qualitatively a potential areafor wave energy systems deployment since these areas avoid the extreme conditionsfound in higher density locations. (See for example Wavegen 2000)Figure 2.2 Approximate global distribution of time-average deep water wave power.Wavepower is given in kW/m of wave front. Source: Thorpe (1998) and World Energy CouncilOrganisation (2004).9

2.3 Description and modification of wavesWaves in the ocean are generally irregular. They are short-crested as opposed tolong-crested, directional and more or less random in nature. No two waves have exactlythe same height and they travel across the surface at different speeds and in differentdirections. Techniques for the coping with the chaotic nature of these waves on the realsea surface are discussed in the next section. It is first necessary to give an overview ofthe characteristics of ideal regular waves. Such waves rarely occur in the real oceanenvironment although they can be produce in laboratory wave tanks. They areimportant also because of the fact that the theory of irregular waves is based on theassumption that they can be represented by superposing or adding together a suitablenumber of regular waves.In order to discuss waves we use a standard set of definitions and terms for parts ofthe wave. The principal ones are defined as follows (Figure 2.3):Crest : The high point of a waveTrough: The low point of a wave.Wave height: Vertical distance from trough to crest.Wave length:Wave period:Horizontal distance between adjacent crests.The time in seconds for a wave crest to travel a distance equal toone wave length.Wave frequency: The inverse of the wave period.Wave celerity: The ratio of wave length and wave period.Note that there is a direct relationship between wave period and wavelength but waveheight is independent of either.Figure 2.3 The important features of water surface waves.After © Craton 1992, Brock Univ.In the offshore deep water regions waves vary slowly over space. However, as theyapproach towards the coast interaction with the seabed and currents could lead tosignificant changes in the characteristics of these waves. There also exist wave-coastline interactions resulting in what is called focusing, defocusing and sheltering of thewaves. These changes in the wave climate will affect the energy densities andcharacteristic. In general near shore wave systems carry less energy than their offshorecounterparts.10

Shallow water phenomena are generally classified according to their features inmaintaining energy or not. Hence we classify wave changes phenomena as conservativeor non-dissipative and dissipative.The main conservative processes include:i- shoaling and refractionii- diffraction and reflectionShoaling is defined as the variation of wave height due to changes in water depth.As the water depth decreases the wave height first decreases then would increase veryrapidly. The sudden increase in wave height causes the wave to break.Refraction like shoaling is caused by a varying depth seabed resulting in the focusingand scattering of waves and the turning of the wave crests becoming parallel to thebottom contours lines. Refraction can also be caused by the interaction of the waveswith currents. Refraction in the form of focusing of the waves can be a positive factorin wave energy extraction.Diffraction is a negative factor. In wave energy extraction applications diffractionaffect the smoothing of the distribution of wave energy in space.The main dissipative processes of interest include a reduction of the total amount ofwave energy by converting it into current, water turbulence, sediment transport or heat.Also included in such dissipative phenomena are wave breaking, bottom friction, wavereflection from sloping or rough surfaced structures or beds and percolation.Waves loose energy by friction at the sea bottom. This is more important once thewaves travel in shallower waters. The losses increase with travel distance and bottomroughness.2.4 Real sea characteristicsWaves range in size from the ripples in a pond to the great storm waves of the oceanand the tides whose wavelength is half the distance around the earth. Waves areclassified according to their period (or frequency) that ranges from less than one secondto more than hundred thousand seconds (tides). The energy spectrum diagram of Prof.Munk shows that the energy in the ocean is distributed among several major groups ofwaves each with a characteristic range of periods.At the lower end of the spectrum with the very short period waves we have:Ripples:of periods of fractional secondsWind chop: of periods between 1 to 4 secondsFully developed seas: of periods ranging between 5 to 12 secondsSwells:of periods ranging between 6 to 22 secondsSurf beats:of periods ranging between 1 to 3 minutesTsunamis:of periods ranging between 10 to 20 minutesTides:with periods of 12 or 24 hours.11

Each are generated and developed in a special way. See section 2.1 for themechanisms generating wind waves. The above waves are also called gravity waves,since once they are created gravity is the force that drives them by attempting to restorethe original flat surface. Also the small tiny ripples of one or two millimeters in height,generated by a small breeze are sometimes also called capillary waves because they arecontrolled by surface tension. Tsunamis and tides are generated by differentmechanisms than the wind. See Figure 2.4.Figure 2.4 Types of waves that occur on the oceans classified by their wave period. The red barsindicate the mean wave period for each type and the yellow bars show the range of wave periods.After © Craton 1992, Brock University.Figure 2.5 after W. Munk shows the relative amount of energy in each of the wavesystems described above. See also Bascom (1976).Figure 2.5 The Ocean Wave Spectrum (Source Munk. See Bascom 1976)12

Real seas include waves that are random in height, period and direction. It is usuallyassumed that within a short length of time the characteristics of real seas remain thesame hence defining what is called a sea state. Statistical parameters derived from thewave spectrum are used to describe such sea states and characteristics relevant to theirenergy content. See Appendix 1.The following wave height and period parameters are most often used:The significant wave height H s : is the average height of the highest one-third waves.The energy period T e : is the mean wave period with respect to the spectral distributionof transport of energy. T p the peak period is defined as the period corresponding to thepeak in the variance density spectrum of sea surface elevation. It is the harmonicfrequency component having the greatest amount of energy at a place passed by arandom wave system. Some ocean energy systems can be tuned to this frequency. Itresembles the tuning of a radio circuit to an electromagnetic field. (Thorpe, 1998).In deep water the power in each sea state P is given by as:P = 0.5 Hs 2 T e kW/mWhere H s is expressed in meters and T e in seconds and density of water taken as 1000kg/m 3 . See Thorpe, 1998.In Thorpe, the annual variation in sea states is represented in a scatter diagramindicating how often a sea state with a combination of characteristic parameters occurannually. The annual average wave power level can be determined from the scatterdiagram in the form:P ave =∑ P i W i /∑ W iWhere sea states of power level P i occur W i times per year.Several models for the formulation of the wave energy density distribution in termsof the frequency have been proposed. They are based on the velocity of the wind thatgenerated the waves. The most notable are the Pierson-Moskowitz, the JONSWAPspectral models (Joint North Sea Wave Project) and others. Reference is here made to,for example, Kinsman, 1984 for further details.Some of the properties of wind waves are shown in Figure 2.6. In the figure the waveperiod is plotted against the amount of energy contained for three wind velocities. Eachcurve (spectrum) represents the distribution of energy between various periods in afully developed sea. The area under each curve gives an estimation of the total energy.As an example we can consider a 20 knot wind (10 m/s). A knot is about 0.5 m/s. Thisrelatively modest wind raises waves whose average is 5 feet or 1.52m and whoseenergy is spread over a range of periods ranging between 7 to 10 seconds (orfrequencies between 0.15 Hz to 0.1 Hz) . If the wind increases to 30 knots (15 m/s) thewaves increases substantially and the period gets longer. There is more energy availableand these longer waves store it better, (Kinsman 1984). The average height of thewaves are now 13.6 feet or 4.14 m and the maximum energy is centered around aperiod of 12 seconds or frequency of 0.08 Hz. See Figure 2.6. For a 40 knots ( or 20m/s) wind the spectrum shows a sharp peak at 16.2 seconds ( or 0.062 Hz) and theaverage height of the waves are now in this case 28 feet (or 8.53 m).13

Figure 2.6 Wave spectra for fully developed seas for winds of 20, 30 and 40 knots.After Pierson, Neumann and James see Bascom 1984.The following Table 2.1 shows the important characteristics of seas that are fullydeveloped for winds of various velocities. An important point to note here is that aparticular wind at a certain speed must blow for at least some time (shown in the table)along a minimum fetch length to raise fully the waves it is capable to generate. Fromthe table for a 50-knot or 25 m/s wind blowing for 3 days over a 1500 miles fetch thehighest tenth of the waves would average about 100 feet (about 30 m) high. Stormsrarely reach such dimensions or durations.Table 2.1 Conditions in fully developed seasAfter Munk , Kinsman 1984.Wind Distance Time WavesVelocity(knots)Length of fetch innautical miles (km)(hours)Averageheight in ft(m)H sSignificantheight (ft)H 10Average ofthe highest10% (ft)Period wheremost ofenergy isconcentrated(sec)10 10 (16.7km) 2.4 0.9 (0.27m) 1.4 (0.43m) 1.8 (0.55m) 415 34 (57.8km) 6 2.5 (0.76m) 3.5 (1.07m) 5 (1.53m) 620 75 (127.5km) 10 5 (1.52m) 8 (2.44m) 10 (3.05m) 825 160 (272.1km) 16 9 (2.74m) 14 (4.27m) 18 (5.50m) 1030 280 (476.2km) 23 14 (4.26m) 22 (6.71m) 28 (8.54m) 1240 710 (1207.5km) 42 28 (8.52m) 44 (13.41m) 57 (17.4m) 1650 1420 (2415.1km) 69 48 (14.63m) 78 (23.77m) 99 (30.17m) 2014

2.5 Data about ocean wavesFor the evaluation and estimation of long-term series of wave data twomethodologies have been suggested. The first is based on measurement andobservations and the second on building time series with numerical wind-wave models.A wide variety of on site and remote sensing measuring methods are available thatproduce accurate wave data. Visual observations made from sea-going ships are theearliest type of wave data for the oceans. Wind-wave models are mathematicalalgorithms encapsulated in computer programs that numerically generate and propagatewave energy based on input wind data or other relevant data. The accuracy is good foropen ocean resource assessment in large basins such as the North-Atlantic and PacificOceans. These models are implemented at most meteorology centers. The availablewave data is not an easy task to collect since the data are archived at several institutionsthat have different procedures to access it.1) Measurements and Observations of ocean (wave/current) climate:MeasurementsA wide choice of measuring systems exists. The choice of a system depends on anumber of parameters, namely: on depth, access and wave conditions of themeasurement site and of the required details, for example directionality. Figure 2.7shows a diagram of several wave measurement systems. Figure 2.8 shows a sketch ofan in situ measuring system based on a buoy with local data storage as well astransmission to an on-shore processing data station.Figure 2.7: Diagrammatic Sketch of several measurement systemsSource: Pontes T. Department of Renewable Energies, Lisbon, Portugal. Original in Earle andBishop (1984). See also Brooke (2003).15

Figure 2.8: Diagrammatic sketch of an in situ wave measuring system.Source: Pontes T. Department of Renewable Energies, Lisbon, Portugal and Laboratorio nacionalde Engenharia e Tecnologia, Portugal . See also Brooke (2003).Some devices can provide wave directional information by their own or whencoupled with other devices. In situ devices can store information or transmit by cable ortelemetry to an on-shore data processing stations. Remote sensing could use laser orradar devices mounted on a satellite, aircraft, ship, or could be based on land. Studiescomparing device accuracy have been performed and could be found in the literature.Generally, the types of measurements are classified as:i) In situ measurementsii) Remote sensing measurementsWave-recording buoys are used extensively in the open-sea. Their data are generallyin the form of time series of sea surface elevation from which wave height, period anddirection parameters could be calculated. Spectral and direct analyses of the time seriesare used for this purpose. Beside buoys other devise are used. For example in coastalareas submerged or suspended pressure and acoustic probes, wave staffs, current meterscould be used to obtain non-directional information about wave activities characteristic.When such probes are used in arrays directional information could be calculated.Energy devices utilizing Oscillating Water Column principle would generally includesome measuring system of the water surface inside the chamber. Difficulties could arisein the presence of water spray.For resource assessment long period measurements are needed. However these are noteasy to find. Short-term wave data acquisition is usually performed for coastalengineering projects as well as for offshore oil platforms.Remote sensing systems are used to provide spatial information about the sea surfacein contrast to the information gathered at one point (in situ) as described above. Thesimplest remote-system measuring devices rely on aerial photography. These could be16

used for studies of wave refraction and diffraction near and along coastlines. Theywould be important for optimizing near-shore and shoreline plants sites.Satellites have also been advanced as a means to remote sensing wave on the surfaceof the ocean. The main limitation with this technology is that the data would only beintermittent. This strongly limits the data available in a specific zone. The positive sideof it is that the method is statistically unbiased since the satellite sampling method isnot under the effects of the sea conditions they try to measure. Altimeter data valueshave been used.The cost of performing and collecting wave data is not discussed in this report.This is by no means an exhaustive review and evaluation of the methods used for wavedata collection.ObservationsEarliest types of wave data resulted from visual observations made from travelingships. These have been archived from 1850s onward. Presently the visual observationsare performed using well-defined procedures and techniques. Reference is here made tothe world Meteorological Organization publications and standards. Several authorsstudied the accuracy of visual data. The report on these studies is outside the scope ofthis report. However, a number of reports conclude that in general, visually wavedirections are most reliable, wave heights are considered satisfactory but wave periodsare much less accurate. Visual data are considered to supplement the data obtainedfrom measurements. Several global and regional atlases of visual wave climates couldbe found in the literature. They would also be relied upon wherever measurements arenot available.2) Theoretical and/or Computational ocean wave modelsTheoretical and computational models have been used to assess shoreline, near-shore oroffshore wave power resource. Deepwater models are available to simulate andcompute the propagation of deepwater waves and their energy transport. Calculationsof the transformation of deepwater wave systems when they approach shallower depthare also required. The size of the coastal area in these studies can vary depending on theseabed topography and the length the coastal line. Scaled physical models aresupplemented with computational computations that are based on mathematical modelsof wave propagation and transformation. This is an ongoing area of research in offshoreand coastal engineering and the results could well suit the application discussed in thisreport. Such models except perhaps wave breaking and some other complexinteractions can satisfactorily describe a wide range of wave propagation phenomena.Simplified representations are hence used in these cases. The purpose being to evaluateand determine the amount of ocean (wave/current) energy transported and dissipated.Without being exhaustive Table 2.2 and Table 2.3 present an example of an attemptto classify computational shallow water wave models that are found to be appropriatefor the assessment of ocean energy resource. The information in the tables follow thecriteria presented by Southgate (1987, 1993). These tables help to illustrate only thevast amount of studies on waves and currents, their generation, propagation, forecastingand modeling. A separate study of the resource evaluation methodology is needed toassess the progress in this field of research and development.17

Table 2.2 Wave processes incorporated in shallow water computational models(source after Southgate 1987, 1993)Computational modelRay modelsGrid modelsForwardtrackingBacktrackingRefractionHyperbolicrefractiondiffractionParabolicrefractiondiffractionEllipticrefractiondiffractionInternaldiffractionNot modeled,but numericalsmoothing byray averagingSame asaboveSame asaboveWave processes modeledExternal Reflections BottomdiffractionfrictionOnly forspecialsituationsWavebreakingYes Yes (1) Yes (1,2)No Yes NoNo, apartfrom check atinshore pointsNo No (3) Yes Yes (2)Yes Yes No Yes Yes (2)YesYes, butdifficult ingeneralNo (3) Yes Yes (2)Yes Yes Yes Yes YesNonlinear Yes Yes Yes Yes YesNote: All models require reasonable gentle depth variations(1) Except for intersecting wave trains(2) Approximate energy loss in shallow water only(3) Backscattered waves cannot be modeledTable 2.3 Shallow water computational models: model suitability parametersComputationalmodelRay modelsGrid modelsForwardtrackingBathymetry(specialrequirements)Regular depthvariation;poor for shoalsystems(source after Southgate 1987)Extent of seaareaUnlimitedBacktracking No UnlimitedRefractionHyperbolicrefractiondiffractionParabolicrefractiondiffractionEllipticrefractiondiffractionRegular depthvariationSame asaboveNoNoUnlimitedType ofcoastlineAnyAny, exceptwhere depthvariation isimportantReasonablystraight coastfacing open seaNumber ofinshore pointsCovering entiremodeled areaOneCovering entiremodeled areaUnlimited Same as above Same as aboveLimited to afew kilometersat mostSame as aboveOffshorewaveconditionsRegularwave (1)FullspectrumRegularwave (1)Regularwave (1)Same as above Same as above Regularwave (1)Same as above Same as above Regularwave (1)Nonlinear No Same as above Same as above Same as above Surfaceelevation18

Note: All models require reasonable gentle depth variations(1) Full spectrum S(fθ ) may be covered by multiple runs.Wave-current interaction models have also been studied. The review of this topic isbeyond the scope of this report. However, the work of Baddour et al (1990, 1991) onthis particular subject of wave-current interaction and more recently in 2003- 2004 onnonlinear wave generation and propagation is worth noting, see Appendix 3. Thesemodels also allow the development of a computational tool for energy flux estimationsin deep and shallow water for specified sea states. Other CFD tools and computerprograms could theoretically be used for such purposes. An evaluation campaign ofthese tools are perhaps in order.2.6 Marine currents originsCurrents within the oceans are determined, for the most part, by the large scalestructure of atmospheric circulation; currents at the surface of the oceans are dictated bythe prevailing winds that blow over the water surface. Surface waters move due tofriction between the moving air and the water surface. Figure 2.9 shows the majorsurface currents of the world’s oceans. This displacement of water at the surface, inturn, contributes to the generation of currents that extend to great depth within theoceans. See: Craton 1993, Brock university.We are here interested in locally generated currents due to tides that are simply therhythmic rising and falling of the surface of the ocean over the course of a day. Tidesare not currents, themselves, but as the water surface rises and falls along a coast thewater must flow to accommodate the geometry and topography of the coastline. Therising and falling of the ocean surface is due to the rotation of the approximatelyspherical Earth that is covered by a slightly elliptical ocean. The form of the surface ofthe oceans is dictated by gravitational interaction between the Earth, the Sun and theMoon.The centrifugal force about the earth (due to its rotation) acts outward from thecentre of the Earth and is equal in all directions. The presence of the Moon, in a circularorbit around the Earth, results in a gravitational attraction between the two bodies.Gravitational attraction is strongest at the closest point between the two bodies anddiminishes away from that point. Thus, the moon’s gravitational attraction is strongeston the side of the Earth that faces the moon and weakest on the opposite side of theEarth. The average gravitational force exerted on the Earth by the moon is balancedexactly by the centrifugal force of the spinning Earth. On the side of the Earth facingthe moon the gravitational force is strongest and exceeds the centrifugal force so thatthe ocean surface is pulled slightly towards the moon (creating a bulge on the oceansurface). On the opposite side of the Earth the centrifugal force exceeds thegravitational force, pushing the ocean surface slightly away from the Earth (creating asecond bulge on the ocean surface). The result is an ocean surface with two bulges at180 degrees to each other (and shallower water at 90 degrees to each of the bulges).19

The position of these two bulges are fixed with respect to the position of the Moonwhile the Earth rotates about its polar axes and the ocean’s ellipsoidal axis is the linethat joins the moon and the Earth. The tides are produced as the Earth rotates throughthese two bulges on the ocean surface. See Craton 1992. Every 24 hours a point on theocean will rotate through the bulge twice, creating two high tides, and through therelatively shallow areas at 90 degrees to the axis of the bulge, creating two low tides.The water level at any point varies in a constant manner from high to low tide and backagain twice over every day.Fig 2.9: Major surface currents (Source © Craton 1992, Brock University)2.7 Marine current resourceThe global marine current energy resource that we are interested in, is mostly drivenby the tides. Other technologies would be interested in currents driven by thermal anddensity effects and are not presented here. As discussed above the tides cause water toflow inwards twice each day (flood tide) and seawards twice each day (ebb tide) with aperiod of approximately 12 hours and 24 minutes (a semi-diurnal tide), or once bothinwards and seawards in approximately 24 hours and 48 minutes (a diurnal tide). Inmost locations the tides are a combination of the semi-diurnal and diurnal effects, withthe tide being named after the most dominant type.The strength of the currents varies, depending on the proximity of the moon and sunrelative to Earth. The magnitude of the tide-generating force is about 68% moon and32% sun due to their respective masses and distance from Earth (Open University,1989). Where the semi-diurnal tide is dominant, the largest marine currents occur atnew moon and full moon (spring tides) and the lowest at the first and third quarters ofthe moon (neap tides). With diurnal tides, the current strength varies with thedeclination of the moon (position of the moon relative to the equator). The largestcurrents occur at the extreme declination of the moon and lowest currents at zerodeclination. Further differences occur due to changes between the distances of the20

moon and sun from Earth, their relative positions with reference to Earth and varyingangles of declination. These occur with a periodicity of two weeks, one month, one yearor longer, and are entirely predictable (Bernstein et al, 1997, and World Energy Council2000).2.8 Marine currents characteristics and resource distributionGenerally the marine current resource follows a sinusoidal curve with the largestcurrents generated during the mid-tide. The ebb tide often has slightly larger currentsthan the flood tide. At the turn of the tide (slack tide), the marine currents stop andchange direction by approximately 180 0 .The strength of the marine currents generated by the tide varies, depending on theposition of a site on the earth, the shape of the coastline and the bathymetry (shape ofthe sea bed). Along straight coastlines and in the middle of deep oceans, the tidal rangeand marine currents are typically low. Generally, but not always, the strength of thecurrents is directly related to the tidal height of the location. However, in land-lockedseas such as the Mediterranean, where the tidal range is small, some sizeable marinecurrents exist.There are some locations where the water flows continuously in one direction only,and the strength is largely independent of the moon’s phase. These currents aredependent on large thermal movements and run generally from the equator to coolerareas. The most obvious example is the Gulf Stream, which moves approximately 80million cubic metres of water per second (Gorlov, 1997). Another example is the Straitof Gibraltar where in the upper layer a constant flow of water passes into theMediterranean basin from the Atlantic (and a constant outflow in the lower layer).Areas that typically experience high marine current flows are in narrow straits, betweenislands and around headlands. Entrances to lochs, bays and large harbours often alsohave high marine current flows (EECA, 1996). Generally the resource is largest wherethe water depth is relatively shallow and a good tidal range exists, Bay of Fundy is anexcellent example. In particular, large marine current flows exist where there is asignificant phase difference between the tides that flow on either side of large islands.There are many sites world-wide with velocities of 5 knots (2.5 m/s) and greater.Countries with an exceptionally high resource include the UK (E&PDC, 1993), Ireland,Italy, the Philippines, Japan and parts of the United States. Few studies have beencarried out to determine the total global marine current resource, although it isestimated to exceed 450 GW (Blue Energy, 2000). See World Energy Council reports.A recommendation is here hence made to initiate a study to estimate the energy inocean currents in Canada. We all know of the currents available in the Bay of Fundy forexample. We need an accurate estimate of the energy due to currents and itsdistribution along the coast. Similar to a wave energy-prospecting project we need amarine current energy detailed survey. See Figure 2.10.21

Figure 2.10 Variation in Tidal range along the coast of North America. Tidal heights areproportional to the currents induced. Ranges given are the differences in water levels from low tideto high tide, in meters. The bare map is courtesy of Theodora.com.3 Ocean energy harnessing systems3.1 Wave energy harnessing technology3.2 Wave energy developments and activities3.3 Ocean currents energy technology3.4 Ocean currents developments and activities3.1 Wave energy harnessing technologyWave power developments face a number of difficulties. Basically, they include thefollowing: Large loading in extreme, harsh weather conditions, corrosive environment,randomness in power input or low transmission frequencies. The design of a waveenergy harnessing system to be efficient and competitive has to deal with thesedifficulties in an efficient way. This means that the system must be beneficial andeconomically reasonable.Starting with a conceptual idea a wave energy device goes through a long evolution:usually starting with theoretical analyses and design the project goes through extensive22

experimental R&D work in the wave tanks at small and intermediate scales. This R&Dwork is required before the first prototype can be deployed in the sea. Freak loads inthe sea may exceed the estimated values and are difficult to predict. High degree ofknowledge and sophistication are needed so that the design of a wave energy systemmay operate safely in extreme conditions and be economically viable. This knowledgeand sophistication could be found nowadays in the offshore engineering industry.In ocean energy resource utilization, in contrast to other renewables, there are a largenumber of ideas and concepts for wave energy harnessing or conversion. In Japan,North America and Europe there are over a 1000 wave energy harnessing techniquespatents.Harnessing wave energy could involve three levels:The first level called primary conversion of wave energy is gained by an oscillatingsystem. These systems include for example: a floating body, an oscillating solidelement or oscillating water within a structure. The system will then be able to storesome kinetic and/or potential energy extracted from the wave.A second level called secondary conversion may be required to convert the storedenergy into some useful form. In this level devices for control and power take offinvolve controllable valves, hydraulic rams and various hydraulic and pneumaticcomponents as well as electronic hardware and software. This secondary conversion isusually obtained by means of a turbine through rotation of a shaft.Tertiary conversion could be needed if electric generators are used for the conversionof the harnessed power into electricity. Brooke 2003.Classification:In the literature a number of ways are found to classify wave energy converters(WEC).According to their horizontal size and orientation:If the size of the system is small compared to the typical wavelength then the WEC iscalled a point absorber. See Budal and Falnes (1975).On the other hand if the extension is large and comparable to the typical wavelengththen the WEC is called a line absorber. Terminator and attenuator have also been usedto denote these WECs. A WEC is a terminator if it is aligned along the prevailingdirection of wave crests and is an attenuator if aligned normal to the prevailing wavecrests.According to their different location with respect to the coastline:WECs may be located onshore, nearshore or offshore. Onshore WEC’s are on the coastline, however nearshore is the designation given to WEC’s if located in shallow watersand within 10-15 km distance from the coastline, while offshore systems are the oneswhich would be developed beyond that.According to their locations with respect to the mean water level:WECs are found partly above and partly below the mean water level. They may becompletely submerged and placed on the seabed below the mean water level. Devicesmay be moored in a floating on the free surface or partly submerged either nearshore oroffshore. Some systems could be called hybrid in the sense that nearshore units couldbe pumping fluid in a closed loop to an elevated reservoir on the shore from whichenergy would be extracted.23

Figure 3.1 from Hagerman (1995) identifies twelve distinct process variations. Themain features that distinguish one concept from another are the mode of oscillation forenergy absorption, type of absorber, and type of reaction point. Hence energy can beabsorbed from heave motion, surge, pitch and yaw or combinations of these as shownin Figure 3.1. Table 3.1 is a modification of Hagerman (1995) classification andpresented in Brooke (2003).The wave energy conversion is described as follows. See Brooke (2003). The waveforce acts on a movable absorbing member which reacts against a fixed point on land orsea-bed based structure, or against another movable, but force–resisting structure.Heave forces may be reacted against a submerged horizontal plate. Wave forces mayalso be reacted against a long spine. The wave force results in oscillatory motion of theabsorbing member. The product of wave force and corresponding motion representsabsorbed wave energy.Figure 3.1 Classification of Wave Energy Converters Systems. Hagerman(1995)24

PrimaryLocationSystemofTable 3.1 Classification of wave energy devices processesSource Brooke (2003) with reference to Hagerman (1995) of Fig. 3.1Wave Energy Conversion Process# DescriptionOnshore 1.1 Fixed oscillating water column 21.2 Reservoir filled by wave surge 11.3 Pivoting flaps 4Nearshore 2.1 Freely floating oscillating water column 32.2 Moored floating oscillating water column 32.3 Bottom mounted oscillating water column 22.4 Reservoir filled by direct wave action 12.5 Flexible pressure device 112.6 Submerged buoyant absorber with sea floor reaction 122.7 Heaving float in bottom mounted or moored floating caisson 52.8 Floating articulated cylinder with mutual force reactionOffshore 3.1 Freely heaving float with sea floor reaction point 63.2 Freely heaving float with mutual force reaction 73.3 Contouring float with mutual force reaction 83.4 Contouring float with sea floor reaction point 93.5 Pitching float with mutual force reaction 103.6 Flexible bag with spine reaction point 113.7 Submerged pulsating-volume body with sea floor reaction point3.8 Reservoir filled by direct wave action 1ReferencetoHagermanFig 3.1The main generic types of wave energy harnessing schemes could hence be listed as:See Brooke (2003) and Hagerman (1995).1. The Oscillating Water Column systems:which in general include a partially submerged, hollow structure open to the sea belowthe water line.2. Overtopping systems:that collect the water of incident waves to create a head to drive one or more low headturbines.3. Point absorbers systems:these are either floating or mounted on the sea bed and provide a heaving motion that istransformed by mechanical and/or hydraulic subsystems into linear or rotational motionto drive electric generators.4. Surging devices :these devices use the particle velocity in a wave to drive a deflector or to generatepumping effect of a flexible bag facing the wave.5. Other devices:25

that do not fall under the above 1-4 classes. These include important developments ofwhich we find: the Salter duck, the Cockerell raft and the McCabe Wave Pump.3.2 Wave energy developments and activitiesRecently there has been a renewed interest in ocean wave and tidal current energy.Recent WATTS conferences attest to that. See WATTS 2004 conference proceedingsfor more details and status of recent developments.New ocean energy companies have been involved in the development of newtechnologies. Examples of these are the Pelamis, the Archimedes Wave Swing and theLimpet. The plan is to increase the worldwide ocean energy capacity to 6 MW in thenear future. See also Thorpe (2000). As of 2002 the installed capacity (around theworld) was about 1 MW, mainly from demonstration projects. Source Thorpe (2000)and Brooke (2003). See the following tables from the same sources.Shoreline wave energy systems:EnergyConversionProcessCountry and location Site StatusFixed OWC Australia, Port Kembla Breakwater Adv. Stage Develop." China, Dawanshan 1 Shoreline Operational" China, Shanwei Adv. Stage Develop." India, Vizhinjam, Harbour Operational" Japan, Sanze Shoreline gully Operational" Japan, Sakata port Breakwater Operational" Japan, Kujukuri-Cho Breakwater Operational" Japan, Haramachi Operational" Mexico Operational, sea water pump" Norway, Toftoy Cliff wall Operational" Portugal, Pico, Azores Rocky gully Operational" UK, Isle of Islay Shoreline gully Operational" UK, Isle of Islay Cliff face OperationalReservoir filled Norway, Toftoy Gully and Operationalby wave surgeinterior bayPivoting flaps Japan, Muroran Port Seawall OperationalPivoting flaps Japan, Wakasa Bay Seawall OperationalNearshore wave energy systemsEnergyProcessConversionDevice name Country and location StatusFreely floating OWC - China, various Op., navigation buoy" - Japan, various Operational" Kaimei floating Japan, Yura OperationalFixed floating OWC Mighty Whale Japan, Gokasho Bay Opearational" Sperbuoy UK, Plymouth Adv. Stage Dev.26

" Shim wind-wave syst. South Korea Adv. Stage Dev.Bottom mounted OWC Osprey UK, Thurso -Reservoir filled by wave Floating wave power Sweden Adv Stage Dev.Flexible pres. device SEA Clam UK Adv Stage Dev.Submerged buoyant- - -absorber sea-floor RPHeaving float in bottommountedConWEC Norway Adv. Stage Dev.or mooredcaissonFloating articulated Pelamis UK, Shetland/Isle of Islay Adv. Stage Dev.cylinder with inertial RPOffshore wave energy systemsDevice name Country and location StatusEnergy ConversionProcessFreely heaving float with OPT Wave Power USA, Australia, Portalnd, Adv. Stage Dev.sea floor RPSystemAustralia" Danish Heaving buoy Denmark, Hanstholm Adv. Stage Dev." Phase-controlled Norway, Trondheim Fjord Adv. Stage Dev.Power Buoy" DELBUOY US OperationalFreely heaving float with Hosepump Sweden Adv. Stage Dev.inertial RP" IPS Buoy Sweden Adv. Stage Dev.Contouring float with McCabe Wave Pump Ireland, Shannon River Adv. Stage Dev.inertial RPEstuary" Wave Energy Module US Adv. Stage Dev.Contouring float with seafloorKaiyo Jack-up Rig Japan, Iriomote Island Adv. Stage Dev.RPOkinawa" Contouring Raft UK Adv. Stage dev." Contouring Raft US Adv. Stagre Dev.Pitching float with inertial - - -reaction pointFlexible bag with spine - - -reaction pointSubmerged pulsatingvolumeArchimedes Wave Netherlands, Viano do Adv. Stage Dev.body with sea-SwingCastello, Portugalfloor RP3.3 Ocean currents energy technologyThere are basically two ways to convert tidal currents energy into useful power. Theyrely on the types of turbines to be used.27

1) Turbines harnessing the potential energy in a low hydrostatic headThe first type needs a hydrostatic head for it to transformation the stored potentialenergy into power. The technology is very similar to the technology used in traditionalhydroelectric power plants. For this purpose arises the requirement of a dam or barrageacross a tidal bay or estuary. Example of this technology is found in the AnnapolisRoyal Tidal Power plant and the one at La Rance in France.Fig. 3.2 The phases of operation at La Rance, France. The darkened areas in the graphindicate the amount of head or quantity of energy available. The upper diagram shows the cycle withoutpumping. The lower one shows more energy available in sipite the expenditure on pumping. © AfterClancy (1968) The tides : Pulse of the EarthBuilding dams is an expensive process and its footprint and environmental impactcould be large. In any case, at certain points along the dam gates and turbines areinstalled. When there is an adequate difference in the elevation of the water on thedifferent sides of the barrage the gates are opened. The hydrostatic head causes thewater to flow through the turbines to produce electricity. Technically, power can begenerated by water flowing both into and out of the bay. Basically the barrage holds thewater in the estuary as the tide falls. Then, gates are opened and the water rushes28

seaward through the turbine(s). Later, the rising tide will be held back by the barragethen released to flow through another turbine (or the same if it is designed to do so) intothe river estuary. See Figure 3.2.The major factors in determining the cost effectiveness of a tidal power site are thesize (length, height) of the required dam, and the difference in height between high andlow tides. These factors are expressed in what is called a site Gibrat ratio. The Gibratratio is the ration of the length of the dam in meters to the annual energy production inKilowatt hours. (1 Kilowatt hour = 1 KWH = 1000 watts used for 1 hour). The smallerthe Gibrat site ratio the more desirable is the site.Very little is understood about how altering the tides can affect complex aquatic andshoreline ecosystems. Unfortunately one of the only methods of increasing ourknowledge about how tidal barrages affect the environment may be the study of theeffects before and after such plants have been built. This strategy perhaps might then betoo late!!2) Turbines harnessing the kinetic energy of the flowUseful energy can be generated from marine currents using completely submergedturbines comprising of rotor blades and a generator. They are sometimes called waterturbines. Water turbines work on the same principle as wind turbines by using thekinetic energy of moving fluid and transferring it into useful rotational and electricalenergy. The velocities of the currents are lower than those of the wind, however owingto the higher density of water (835 times that of air) water turbines are smaller thantheir wind counterparts for the same installed capacity. The power that can be extractedfrom the currents is dependent on the velocity of the water flow and the area andefficiency of the water turbine, and can be calculated asfollows:P = 1 2 ρ A v3 C PWhere ρ is the density of sea water 1025 kg/m 3 A is the area of the rotor blades in m 2v is the marine current velocity in m/s and C p is the power coefficient, a measure of theefficiency of the turbine.Two types of turbines have been proposed.i) Horizontal axis turbines (axial flow turbine).ii) Vertical axis turbines (cross flow turbine).See Figure 3.3 for the conceptual designs and possible configurations.29

Figure 3.3 Conceptual diagrams of types of current turbines and their potentialconfigurations. Source © World Energy CouncilIn order for marine current energy to be utilised, a number of potential problems willneed to be addressed, including:Avoidance of cavitation by reducing tip speeds to approximately 8 m/s. This suggestsa turbine with a higher solidity than a wind turbine;Prevention of marine growth building up on the blades or ingress of debris;Proven reliability, as operation and maintenance costs are potentially high;Corrosion resistance, bearing systems and sealing;Turbines may be suspended from a floating structure or fixed to the seabed. In largeareas with high currents, it will be possible to install water turbines in groups or clustersto make up a marine current farm, with a predicted density of up to 37 turbines persquare km. This is to avoid wake-interaction effects between the turbines and to allowfor access by maintenance vessels (DTI, 1999).30

As there are currently no commercial turbines in operation, it is difficult to assess thecost of energy and competitiveness with other energy sources. Initial studies suggestthat for economic exploitation, velocities of at least 2 m/s (4 knots) will be required,although it is possible to generate energy from velocities as low as 1 m/s. As thetechnology matures and with economies of scale, it is likely that the costs will reducesubstantially, Rudkin, 2001.Future of Marine Current EnergyCompared with other renewable technologies, there has been little research intoutilising marine current energy for power generation. However, in principle marinecurrent energy is technically straightforward and may be exploited using systems basedon proven engineering components (FMP, 1999). In particular, knowledge gained fromthe oil and gas industry, the existing hydro industry and the emerging wind energyindustry can be used to overcome many of the hurdles facing marine current energy.The global marine current energy resource is very large, and it has a number ofadvantages over other renewables. The table below shows a comparison of the oceanwaves and marine current energy resource with other renewables and conventionalenergy sources. It is clear that there are many benefits to utilising current energy,including : (Vortec Energy)The resource has four times the energy density of a good wind site, so thediameter of water turbines can be less than half that of a wind turbine for the sameenergy output.The water velocities and therefore power outputs are completely predictable,once accurate site measurements have been taken.Water turbines will not need to be designed for extreme atmosphericfluctuations as required with wind turbines, meaning that the design can be better costoptimised.With increased conflicts over land use, water turbines offer a solution that willnot occupy land and has minimal or zero visual impact.The greatest resource is in close proximity to coastlines and many areas withhigh population densities.The technology is potentially modular and avoids the need for large civilengineering works.The environmental impact resulting from marine current energy use is likely tobe minimal. Project planning will need to be cognisant of species protection includingfish and marine mammals, although since the blade velocities and pressure gradientsare low this is unlikely to cause any serious problems (Fraenkel, 1999). In the processof locating turbines, consideration of shipping routes and present recreational uses suchas fishing and diving will be required. Fishery exclusion zones might be necessary to beestablished.31

Table 3.2: Comparison of wave and marine current energy with other energy resourcesSource: Emily Rudkin, Vortec Energy (2001).The table (Rudkin et al, 2001) shows that marine current energy is one of the mostpromising new renewable energy sources, and is deserving of further investment.Furthermore, the know-how is now available to combine existing technologies to utilisemarine current energy for power generation.It is likely that water turbines will initially be deployed in island or coastalcommunities with strong marine currents and which are isolated from national gridsystems, where they are most likely to offer a cost-effective alternative. However,marine currents have the potential to supply significant quantities of energy into thegrid systems of many countries. As interest grows, marine current energy is likely toplay an increasing role in complementing other energy technologies and contributing tothe future global energy supply mix. See World Energy Council report.3.4 Ocean currents developments and activitiesThe applications and activities in this sector are also divided according to thetechnology. We have two types: the applications that use the tidal energy in generatinga hydrostatic head hence needing a dam and the applications that use water turbines toharness the marine current kinetic energy.1) Tidal energy plantsi) La Rance in Bretagne, France: This is a major tidal generating station inoperation. It is a 240 MW (1megawatt= 1MW=1 million watts) at the mouth of LaRance River estuary on the northern coast of France. Note that a large coal or nuclearpower plant generates about 1000 MW of electricity. It has been in operation since1966 and has been a very reliable source of electricity. The energy resource is exactlypredictable relying of course on the tides. This plant was supposed to be one of manytidal power plants in France, until the nuclear program in that country was greatlyexpanded in the late 1960’s.ii) Annapolis Royal Tidal Power plant, on the Annapolis river, Bay of Fundy, NS.This was designed as an experimental facility and is of the order of 20 MW. It is in32

operation since 1984. It would have been interesting to know the economics of the plantif it had more than one generating unit.iii) Murmansk tidal power plant: of the order of 0.4 MW. Not much is known aboutthis one.iv) Studies have been undertaken to examine the potential of several other tidalpower sites worldwide. It has been estimated that a barrage across the Severn River inthe west of England could supply as much as 10% of that country’s electricity needs.(12 GW). The proposed facility on the Severn would have a construction cost of $15Billion . Similarly, several sites in Cook Inlet in Alaska and the White Sea in Russiahave been found to have potential to generate large amounts of electricity.2) Water turbines plantsMarine current energy is at an early stage of development, with only a small numberof prototypes and demonstration units having been tested to date. There are nocommercial grid-connected turbines currently operating. A number of configurationshave been tested on a small scale that are essentially wind turbines adapted for themarine environment. Generally speaking, turbines are either of horizontal axis orvertical axis type. Variants of these two types have been investigated, includingturbines using concentrators or shrouds, and tidal fences.© World Energy Council.i) Horizontal axis turbines (axial flow turbine).This is similar in concept to the widespread horizontal axis wind turbine. Prototypeturbines of up to 10 kW have been built and tested using this concept. There arecurrently plans to install a demonstration machine of 300 kW off the south coast of theUnited Kingdom (MCT, 2000).Concentrators (or shrouds) may be used around the blades to increase the flow andpower output from the turbine. This concept has been tested on a small scale in anumber of countries, including New Zealand (Rudkin, 2001).More information on horizontal axis turbines and their use could be found in:http://www.europeanenergyfair.com/download/marine_current_turbines.pdfSee also Marine Current Turbines Ltd. information and their web site at:http://www.marineturbines.com/ . The company has been formed to developtechnology for exploiting flowing water in general and tidal streams in particular. Thegoal is both to arrive at cost-effective and reliable power systems and to develop thesecommercially on a large scale. This is being achieved through what they call a phasedR&D programme and in partnership with an industrial consortium together with variousstrategic partners who are shareholders in this company. According to their plan MarineCurrent Turbines Ltd has no intention of manufacturing the technology; this function isachieved by its partners and "third parties", but the company "does research theresource with a view to developing future projects". The company will develop, ownand deliver the technology and the resource to go with it. The company's shareholdersconsist of the management, private investors and several corporate investors - namelySeacore Ltd., London Power Company plc, Carrs Milling plc and IT Power Ltd. Furtherdetails could be found from the company information site.From early December 2003, a tidal turbine or underwater windmill started running apower-generating propeller mounted on the seabed of the Kvalsund channel. The33

turbine is equipped with blades of 15-16m and has been connected to the nearby townof Hammerfest’s power grid in Northern Norway via a shore connecting cable. SeeFigure 3.4 and http://www.e-tidevannsenergi.com/index.htm. It will be imperative tofollow up on the study of the economics of the project.Fig. 3.4 Diagrammatic sketch of the underwater tidal current mill operating inKvalsund channel (~70 0 N) in Northern Norway since the fall of 2003.© World Energy Councilii) Vertical axis turbines (cross flow turbine).Both drag and lift turbines have been investigated, although the lift devices offermore potential. The best-known example is the Darrieus turbine with three or four thinblades of aerofoil cross-section. Some stand-alone prototypes have been tested,including a 5 kW Darrieus turbine in the Kurushima Straits, Japan. The concept ofinstalling a number of vertical axis turbines in a tidal fence is being pursued in Canada,with plans to install a 30 MW demonstration system in the Philippines (Blue Energy,2000).For more information see: www.bluenergy.com34

4. Transfer of power systems4.1 Interfaces4.2 Storage of energy4.3 Electric power generation and conversion4.4 Power transmission to the gridWaves or currents action has to be converted from one form to another form that isamenable for transmission and/or storage. Usually the harnessed energy is convertedinto electricity that is then either fed into a grid system or transformed into some otherform for storage purposes. Hence to achieve the first transformation, power transfersystems have to turn the slowly varying oscillating forces of incoming waves into thefast, unidirectional forces required to drive generators that will produce electricity.With very few exceptions most of the systems consists of a two-stage operation: amechanical rotary device coupled to an electrical generator. A wide range of options isavailable to convert and transfer energy. After the initial energy-harnessing device mostsystems will consist of a mechanical interface an electrical generator and a way fortransmission of power and deliver it to the grid.4.1 InterfacesThe following interfaces have been proposed, adapted or used.Direct mechanical interfaceOnly few designs for a pure mechanical power transfer have been proposed. Theyhave not been tested in practice. Haggerman (1995) and the European CommissionReview (1993) do not report these as options. Devising mechanical components toconvert the oscillating variable forces into RPM unidirectional output is difficult.Thorpe (1992) indicates that to deal with such large force will require large sizecomponents making the option uneconomic. Recently the following has been reported:The Wave Rotor: see Retzler (1996). This concept uses the Magnus effect on twocontra-rotating cylinders to absorb wave power. It has been tested on a small-scalemodel. The extrapolation to a full scale capability is unknown.The OLASS system: see Rebello et al (1995). This is an Oscillating Water Columndevice in which the energy is extracted through a float system on the oscillating watercolumn. This is then mechanically coupled to a generator. This is done by means ofwhat is called a mechanical rectifier and speed multiplier. A small prototype developedabout 15 kW. Few details are available.Air turbines interfaceA simple way of transforming the low velocities and high forces of air compressedby sea waves into high speeds and low forces required by conventional generators is35

provided by air turbines. They seem to fulfil the requirements of a power transfersystem. The combination of air column and turbine seem to provide a cost efficientapproach to gearing. The most popular air turbine is found to be the Wells turbine. Tohave the property to rotate in the same direction even when the direction of the air flowchanges the turbine uses symmetrical air foils with their chords being in the plane ofrotation. No pitch angles are proposed. Blades are hence cheap to manufacture incomparison and the losses are small during idling.Some of the configurations that are found in the literature include the following:Monoplane turbines: In steady air flow these turbines were found to have amaximum efficiency of about 60% in small scale testing in steady air flow. Monoplaneturbines with guide vanes: their efficiency can reach 70% in small scale testing andsteady airflows. They have poorer behaviour against stall.Contra-rotating turbines: this configuration is made up of two Wells turbines placedclose together with their blades rotating in opposite directions. The efficiency about70% was found for small scale testing in steady airflows. A wider operating range isfound in this case in comparison to the ones with vanes.See Gato et al (1996, 1997).Efficiency measurements on Wells turbines (and any other turbines in that matter seebelow) under realistic oscillating flows is a priority area for further research work.Other air turbines configurations that have been considered:Variable pitch turbines: See European Commission pilot OWC in the Azores.Russel et al (1996).Impulse turbines: with adjustable guide vanesUse of valves: to allow some of the airflow to bypass the turbines hence avoidingstall. Valves can be used to delay the movement of the water column hence allowing itsvelocity to come into phase with the wave force. Better suited for hydraulic conversion.See Korde (2002).Water Turbines:In this sector of the industry these are considered to be a well-developed technology.The surrounding fluid provides a good supply of working fluid. The various designsproposed claim to offer control over the volume of water flow allowing the devise tomanage variations in wave power levels enabling relatively conventional electricalgenerators to be used.According to the European Commission report (1994) several types of turbine couldbe found specifically designed for different working heads or pressures.A summary of some of these water turbines is as follows:Pelton wheels:This configuration is suitable for high pressure operations. These include forexample:The original Bristol Cylinder with high pressure Pelton wheel36

The Hose-pump Swedish design with a high pressure Pelton wheel.Francis turbines:This configuration is suitable for medium pressure operationsKaplan turbines:This configuration is suitable for low pressure operations. These include for example:The Danish water pump with low pressure propeller,The Tapchan with low pressure Kaplan turbine.Two of these schemes The Tapchan and the Hose-pump have been successfullydemonstrated in pilot plants. In a sense showing that water turbines can be used in wavepower devices with little development providing well known problems in the use ofthese turbines are taken care of. Cavitation is the first that would come to the front.Hydraulic systemsFor several wave energy devices high pressure oil systems have been proposed andused. As part of power take-off technology these systems have the followingadvantages:They are capable of handling high power levels in a small volume.They can be made to adapt to various types of input motions (circular or linear) andso could be utilised on a range of types of devices.They can be computer controlled hence allowing optimization of the device on awave by wave basis.They can accommodate a wide range of input power levels.The reliability of hydraulic systems has been proven in their wide use in manyapplications. In wave energy harnessing systems they will be subject to much differentenvironments, namely:Hydraulic systems will be in contact with seawater. The design will have to preventand make sure that seawater does not come in contact with the oil in the system.Corrosion problems and ways to mitigate with underwater systems specific problems ingeneral have been very well researched and studied within the Offshore Engineeringcommunity. Use of new materials like ceramics and ceramic coatings have provensuccessful in such applications.Hydraulic rams might be affected with high loads due to large displacements inextreme conditions. High dynamic loadings in general will have to be taken intoaccount.Some wave energy applications are predicting their operations under high torque andvariable displacement. In general, current rotary hydraulic machinery applicationsoperate under relatively low torque and fixed displacement.Nebel (1992) describes the problems with control systems to optimize the capture andconversion efficiencies of wave energy schemes. Current methods of controllinghydraulic mechanisms could involve much power losses.The European Energy Programme (Russel and Diamantaras 1996, Eshan et al 1996)predicted the need to design tailor-made hydraulic systems and is an intrinsic part of theprogramme. However current hydraulic systems can be used in wave energy devices.Research in their efficiency and reliability will be needed.37

4.2 Storage of energyPower is proportional to velocity squared and the surface velocity of waves varieswith wave height and period of waves. Wave power subsequently is not steady andconstant but varies significantly. This variation has an impact on the capital cost of awave power installation and its power losses. In general it is found that to facilitate theintegration of the system to the grid short-term energy storage is needed.For ocean waves systems the three main short-term energy storage are: the flywheel,pressure accumulators and water reservoirs.Flywheels provide energy storage in the form of rotational kinetic energy. They incursome losses. Found to be more efficient in large systems.Gas accumulators have been suggested and would use an inert gas contained in steelaccumulators to store energy in oil hydraulic systems. Gas is stored at high pressure andlow volume and the energy is released as the gas expands to a greater volume and lowerpressure.Water reservoirs as in particular the Norwegian Tapchan systems could be used as ashort –term energy storage.Further details could be found in Edinburgh-Scopa-Laing 1979, Hotta 1996, Brook2003.There may also be the need to longer-term energy storage or energy transformationfor future use. Use of batteries and Hydrogen technology might then be included in theequation. This topic is beyond the scope of our discussion.4.3 Electric power generation and conversionPower generationElectrical generators are used to convert mechanical power at the shaft to electricalpower. For high conversion efficiency from wave energy to electrical energy withoutstorage large rotation differences are used in the turbine-generator system. Specialgenerators can be found to satisfy these requirements. Examples of these include multipolegenerators, linear generators and generators with reluctance changes. Lineargenerators are suitable for direct connection to the mechanical reciprocating motion ofwave energy systems. Research and Development in the field of linear generators andsuitable electrical generation systems for the feasible applications here discussed areneeded.Power conversionTo get grid-quality power from ocean energy systems it might be required to convertfrom alternating current to direct current and vice-versa. A number of electric powerconversion methodology and designs have been put forward in projects, proposals andpilot schemes and cannot be extensively reviewed in the present report. The techniquesare based on previous developments in the electrical and electronics fields. This is anissue that must be addressed and reviewed by any future proposal and scheme.Agreements and discussions with hydro-power authorities and grid operators are also amust during the feasibility and development stages.38

See for example SEASUN (1988) and Beattie et al (2000) for detailed outlines anddiscussions of alternatives.Briefly we can classify the types of power conversion as follows: See Brooke (2003).i) Constant or nearly constant RPM power conversionand they are of two types: Synchronous generators and Asynchronous generators. SeeBrooke (2003).ii) Electronic power conversionHigh power electronics has been used to convert electrical power between AC and DC.AC systems of different frequencies can in general be connected. Configurations couldinclude a direct converter (AC/AC) or pass via a DC interconnection (AC/DC/AC).iii) Variable or free generator rpm power conversionLarge variations in RPM are possible through power electronics and using a DCinterconnection between the generator and the power grid. For low speeds or RPM agenerator is heavy and consequently expensive. The generator produces voltage thatwill be frequency and amplitude modulated by the random wave field and cannot bedirectly connected to the grid. The voltage variations are reduced in the DCinterconnection ig the DC voltage from several generators running at random RPM areadded in series.4.4 Power transmission to the gridThrough power collecting systems the converters are connected to the electrical gridon land. Transmissions losses will have to be accounted for and are proportional to thedistance between the converters and the grid. The transmission to the grid is throughsea-floor cables. Hydraulic transfer may be possible depending on the distance notbeing large.Electrical transmissionElectrical transmission to the grid is a large topic and has been the object of study inelectrical engineering. Reviews could also be found for the application here discussedin SEASUN (1988). See also Brooke (2003) for more details. Also the reviews in Scott(2001) and Thorpe (1992) of the grid connection of large-scale ocean power projectsand the transmission systems pertaining to UK specific projects are worth noting.Extensive experience exists in electrical cables laying in the sea using speciallydesigned vessels for this purpose. Power can be transmitted as AC or DC. The losses inthe cable transmission must be estimated and accounted for. Maintenance of the cablesare also issues to be studied. The need of electrical transformers to transfer powerbetween grids of different voltage levels might also arise. The design of the electricaltransmission system is a techno-economical problem of optimization, Brooke (2003).Ocean power generation and transmission would gain from the experience of windpower systems. In this case major short cuts in research and development could besaved and gained from cross-fertilization between the two applications. Worth checkingfor example the international company ABB and others for their experience anddevelopment of systems of power transmission for wind farms. See ABB (2002).39

Hydraulic transmissionIn some ocean energy schemes the secondary energy transformation is realized byeither water or hydraulic oil, Brooke (2003), CONA (2003). Hydraulic oil is normallyused internally in a wave power device while water can be pumped long distances inopen or closed systems. The Swedish hose-pump system is an example using sea waterin an open system The losses are proportional to velocity squared. Hence preference tolow flow velocities combined with high pressure.The point of the discussion is that those problems and issues of generation andconversion of the electrical power have to be discussed in the design and developmentstages of any project.5. Ocean (wave/current) Power Economics5.1 Factors affecting the evaluation of costs5.2 Capital and Operating costs of Ocean Energy systems5.3 Generation costs of wave and current energy5.4 Comparison with costs of other renewables and electricpower pricesThe commercial deployment of systems for harnessing wave energy is only justrecently tentatively beginning to happen. This is in spite of extensive work in researchand development since the late seventies. In general it is considered that experience toenable the accurate assessment of the costing of such systems is lacking, Thorpe(1992), Brooke (2003). Hence an actual assessment of the degree of theircompetitiveness is difficult. No large-scale offshore ocean energy systems have beendeployed. Only prototypes have been installed giving also a much needed experience inmaking the economic evaluation. This chapter reviews some of the factors recognizedby industry to evaluate costs.5.1 Factors affecting the evaluation of costs of ocean energy systems.Usually developers of ocean energy systems define the cost of energy in cents perkilowatt-hour. The method for calculating this cost is, most of the time, not identified.Important financial assumptions are not reported. For example: rates of return, debt toequity ratio, discount rates if any, are not mentioned. Potential venture capital investorsfind it difficult to evaluate a project and viability in particular if no evaluation ofcompeting technologies or other appropriate benchmarks are available, Brooke (2003).A detailed analysis is beyond the scope of the present review. However the followingpoints are provided.The US Electric Power Research Institute in its Technical assessment Guide (EPRITAG), see Electric Power Institute 1987, 1993 and Hagerman 1995, provides what isbelieved, an accurate methodology for evaluating the costs of energy. They introducewhat is called a levelized cost of energy index that will allow also the comparisonbetween alternative designs or technologies.40

5.2 Capital and Operating costs of Ocean Energy systemsCapital costs usually include costs of construction, assembly and installation of theplant. Unfortunately no large-scale devices have been built. Currently available areprototypes and include all the additional costs involved in such a stage. Sources of costdata are hence difficult to find. Other similar or close to areas of activities in offshoreengineering could provide data that would help to calculate the capital costs of oceanenergy systems. Three approaches are found in that respect and are summarized below,see for example Brooke (2003) and Thorpe (1992).Costing by analogy: approximate costs estimates could be made from similarprojects. Adjustments must obviously be made to take care of relative sizes or othercharacteristics. Finding remotely related systems is more likely because of the scarcityof ocean energy systems.Conventional costing estimate: this is sometimes defined as a bottom-up costingmethod. In this case detailed information about the project are needed, from drawingsand construction plans. Hence a complete work breakdown structure could bedeveloped with units rates for every component. In general some aspects of the systemusing standard current technology in civil and offshore engineering would be amenablefor such costing. Compatibility between costs for different systems and by differentevaluators would be difficult.Parametric costing estimate: this method is intermediate between the two describedabove. It relies on that some functional relationship between characteristics of an itemof the system and its cost. This direct dependency is generally derived from pastexperience and/or engineering practice. These relationships are similar to those usedalso in methodology ii) above. This means that it requires also outline drawings andspecifications together with rates for materials, labor and transport. Information isavailable in general except perhaps for some aspects of the wave energy devicecomponents. Thorpe 1992 suggests that this is the method suitable for and employed tomake estimates of costing of ocean energy related projects.Table 5.1 Cost and Performance characteristics of generic wave energy systems(Source: Thorpe 2000)Cost and Performance Shoreline Near-shore OffshoreUnit Costs ($/kW) 1800-2100 1500-1800 1500-3000O&M & Insurance Costs ($/kW/year) 30-45 42-48 30-90Availability 94-96 93-96 90-95Annual Output (kWh/kW) 2000-2500 2200-2500 3000-400041

5.3 Generation costs of wave energyFor any application the determining factors are the cost per kW of the deliveredsystem or the cost per kW-hour of the delivered energy whether in the form of electricalenergy production or in kind, McCormick and Kraemer 2002. Thorpe (1998, 2000) hascompared the cost efficiencies of various wave energy conversion technologies. Asmentioned above he has divided the systems into shoreline, nearshore and offshoresystems. He found that the more recent designs both shoreline and offshore devices canproduce electricity at less than 0.1 US$ per kW. Figure 5.1 shows an unraveled designspiral as suggested by Thorpe. The figure demonstrates the cost cycling of the designprocess in time.Costestimates123456213456234561 Initial concept 2 Technical and Economic Feasibility Study3 Preliminary Design and Cost Analysis 4 Design Alterations5 Cost Analysis 6 Design AdoptionsFigure 5.1Unraveled Design-Cost Spiral for an Ocean Energy SystemSource: Thorpe 2000Also it shows the improvement in the cost estimates as time and more analysis isperformed. Table 5.1 shows the predicted costs and performance for generic types ofwave energy devices, Thorpe 2000. Thorpe 1998, also shows independently predictedelectricity generating costs of nearshore (including onshore) and offshore systemsagainst the year in which the device was designed. These costs are shown to be sitespecific and show reduction to approximately 7 cents/kWh in 2000 for both categories at8% discount rate over the lifetime of the scheme.time42

Evolution of electricity cost for onshore and nearshoresystems. Source: Thorpe (2000)NEL OWC Wavegen’s OSPREY Wavegen’s LIMPETCost of electricity (p/kWh)201510501980 1985 1990 1995 2000 2005Design yearFigure 5.2 Examples of Evolution of electricity cost for onshore and nearshore systems. SourceThorpe 2000.Evolution of electricity costs for offshore systems.Source: Thorpe (2000)Cost of electricity (p/kWh)1001011980198219851986199019921995199820002005Edinburgh DuckBristol CylinderSEA ClamPS FrogMcabe Wave PumpSloped IPS BuoyFigure 5.3 Examples of Evolution of electricity cost for offshore systems.Source Thorpe 2000.Apart from the schemes analyzed in Figure 5.2 and 5.3 there are also other systemsthat claim to be able to produce electricity at similar costs levels. This indicates thatwith a suitable climate generating costs of 3.5-8 c/kWh should be achievable. These43

systems have not been evaluated independently or else they are still in the early stagesof R&D. Therefore their costs and performance are subject to considerable uncertainty.Brooke (2003), Sjostrom et al 1996, Thorpe (1997).In Figure 5.4 below we present a plot of the resulting costs against the year in whichthe design of a device was completed. The graph is provided by the European ThematicNetwork on Wave Energy. See their report Wave Energy Utilization (2002). Itcomplements the information in Figures 5.2 and 5.3 above in that it shows forcomparison the average electricity price of wind generated electricity in the EU. At bestthe improvement in wave energy cost is similar to improvement of generating costs ofwind turbines.Figure 5.4 Predicted Electricity cost for Wave Energy Technologies (Source: European ThematicNetwork on Wave Energy and Wave Energy Utilization in Europe report 2002)In general, the economics of wave energy have shown a gradual improvement withtime. See Thorpe (1998), Brooke (2003) and the figures above. Developing moreunderstanding due to continual R&D results in reduced capital costs. This trend is alsofound in other emerging technologies. Falnes (1996) shows that a funding for R&D isrequired to bring wave energy converters to a commercial level. When this level hasbeen achieved then the potential for selling ocean energy systems in a huge marketlargely increases.Recently there are systems that propose the utilization of wave energy in conjunctionwith wind energy. These systems will then be capable of harnessing two sources ofenergy giving more flexibility in the power production. Multi device approach to oceanenergy exploitation might be the way of the future. It must be noted that these are stillvery much site dependent. Generating costs estimates will have in this case to be higherthan individual devices.Figure 5.5 from Hagerman (1995), shows a sketch of a capital cost learning curveand how R&D and experience gained in successive developments affect in time capitalcosts reducing it to a mature plant cost level.44

Figure 5.5 Hagerman (1995) sketch of a capital cost learning curve.In general the capital costs of the first individual scheme will be much higherbecause: See Hagerman (1995), Brooke (2003).• of technical innovation and immaturity. However, following a learning curveand gaining the experience the benefits follow.• of the perception of initial risk. Again this perception will initially inflate thecosts.• of the large mobilization and demobilization costs. These are accounted for onthe costs of a single device. These costs would be defrayed over a number offollowing up schemes.• initially the economies of scale are lacking. Hagerman (1995)The generating costs could, for the above reasons, be as much as twice or more than thecosts mentioned in Figures 5.2, 5.3 and 5.4.5.4 Comparison with costs of other renewables and electricity pricesThe costs of electricity from renewable resources are function of many factorsincluding:• The type of energy source and its availability.• The type of device that harnesses the energy• The efficiency of the plant• The site location.45

Unless a particular plant or project and device are specified it is very difficult to havedefinitive costs for the electricity generation. Table 5.2 below presented at the 1993World Energy Congress (Brooke, 2003) is a representing of some estimates of costs perkWh for a number of alternative renewable energy resources. The installations thatcombine both wind and ocean energy whether from waves or currents should beeconomically competitive with systems harnessing only onshore wind.Table 5.2 Typical costs of electricity from some renewable sourcesSource: World Energy Congress 1993, Thorpe 1995, Brooke 2003.System Location Date Cost (c/kWh)Solar thermal;parabolic trough New Mexico, USA 2020 7.5-11Solar Thermal; parabolic dish New Mexico, USA 2020 6.0-10Solar thermal; Central receiver New Mexico, USA 2020 5.0-9.0Photovoltaic New Mexico, USA 2020 5.0-14Photovoltaic, thin film New Mexico, USA 2020 6.0-10Photovoltaic, multiple thin film New Mexico, USA 2020 4.0-7.0Wind turbine (6-9 m/s wind spd.) - 1995 3.6-6.5Wind turbine (6-9 m/s wind spd.) - 2000 3.0-5.5Wind turbine (6-9 m/s wind spd.) - 2010-2020 2.0-4.5Combined wind-wave system South Korea 1995 11.0-18.0Combined wind-wave system UK 1995-1999 6.0-9.0Comparison with electricity pricesAnother way of evaluating the economics and competitiveness of ocean energysystems is to compare the costs of electrical generation from ocean systems against theprices that customers pay for electricity. It might become increasingly relevantespecially that local communities might be interested to invest in small local systems.The location of such communities and the lack there off of grid hook up whileproximity of ocean energy resource might be a decisive factor. The ocean energy couldthen compete with diesel generation or might in the presence of the latter complementit.Table 5.3 shows some examples of electricity prices in the USA, Japan and the UK.They are representative of prices in the industrialized countries. Note the prices ofelectricity to households. These prices would be higher in remote communities. Energyfrom the ocean might be competitive with electricity purchased by households.Table 5.3 Electricity prices in some industrialized countries. (Source: Brooke 2003)Country & Date Source reference Categories Cost (cent/kWh)USA – 1994 IEA 1994 Price to industry 4.7USA 1994 IEA 1994 Price to households 8.4Japan 1993 IEA 1994 Price to households 14-24UK 1995 DTI 1996 Price to industry 6.4UK 1995 DTI 1996 Price to households 8-13.446

6. R&D programmesR&D in ocean energy harnessing is ongoing in several countries around the world.See for example the International Energy Agency Annual Report (2003) onImplementing an Agreement on Ocean Energy Systems for further details. An overviewis here presented of some Canadian activities that came to the attention of the presentreviewer. This chapter should be upgraded for up to date information and the inclusionwhat other countries are specifically doing. Section 3.2 above lists related activitiesfound around the world.6.1 CanadaOcean Wave EnergyAs reported by the International Energy Agency (2003) Canada’s coastlines havefavourable wave energy resources. In 1995 Powertech Labs Inc. carried out a waveenergy resource assessment for the coast of BC based on the wave records obtained byMarine Environment data Services. See Bhuyan et al 1995. The result of the assessmentrecognizes that the most promising resources would be found in the Queen CharlotteSound and on the West Coast of Moresby and Graham Island. The "Green ElectricityResource Map" recently issued by BC Hydro shows an average wave power level of 33kW/m along the west coast of Vancouver Island. (International Energy Agency 2003)The total incident wave power for the west coast of Vancouver Island is estimated to be8.25 GW. A wave energy resource map for BC can be viewed at the CanadianCartographics Ltd. web site: http://www.canmap.com/index.htm.BC Hydro has initiated a pre-feasibility assessment of the potential for thedevelopment of wave energy resources in 2000. This has been done with theencouragement obtained by the rising electricity demand and inline with a voluntarycommitment to meeting 10% of increased demand through a variety of new greenenergy sources. Two specific sites in Ucluelet and Winter Harbour have been identifiedeach with over 200MW of potential wave power capacity. In 2001BC Hydro selectedUcluelet site as the initial as the initial site for the wave demonstration projects inVancouver Island. The International Energy Agency reports that presently waveparameters near the Ucluelet site are being monitored using a moored tri-axis buoy.Also in the report is that BC Hydro has signed memorandum of understanding withEnergetech of Australia and Ocean Power Delivery (UK) to build two demonstrationplants in 2004, both through joint ventures. Each of the plants will have installedcapacity of 2MW. It is intended that both of these demonstration plants be connected tothe existing grid of BC Hydro.The report goes on to say that on the east coast in the Maritimes, NS, the WavemillEnergy Corp is marketing a wave energy converter called "wavemill" with a patentedsuction chamber capable of being factory produced as an off the shelf unit. Seehttp://www.wavemill.com/products.htm .A series of performance trials of a wave mill device were conducted in 1998 at theHydraulic Laboratory of NRC.47

Presently at the NRC Institute for Ocean Technology computational tools are beingdeveloped for modelling the generation and propagation of non-linear, regular andirregular steep waves in infinite and finite depths. The computations also allow theaccurate calculation of wave energy flux in various sea states and the modeling of theinteractions between the component waves. See Appendix 3 for examples of recentpublications on that subject.Tidal current energyAs part of BC Hydro’s initiatives on green energy technologies, the feasibility ofexploiting tidal current as one of the energy resources has been examined by TritonConsultants in 2002. See Triton Consultants 2002. The report is available on BC hydroweb site. See http://www.bchydro.com/. A tidal resource map again could be found atCanadian Cartograhics Ltd. The resource assessment identified 55 sites with currentspeeds over 2m/s, which would yield a gross annual energy potential in the order of20,000 GWh. Twelve specific sites identified considering the deployment feasibilitywith a total energy production of 2,700 GWh per year.Blue Energy Canada a BC company is looking for financing for a tidal currentdemonstration project using heir technology particularly for a tidal fence concept. TheInternational Energy Agency report that they have through their previous associationwith another company undertaken some laboratory trials on Darrieus-type underwatervertical–axis turbines, often called Davis turbines.http://www.bluenergy.com/oceanenergy.htmlAnother BC company called Clean Current has undertaken numerical andexperimental hydrodynamic design on the ducted horizontal axis turbines and hasdeveloped two innovative ideas related to electricity generation and turbine design. Amodel of the turbine has been tested.NRC's Institute for Ocean Technology (IOT) plans to develop a capability fortesting, performance evaluation and for R&D work on Ocean Energy Systems. TheDesign and Fabrication, testing facilities and numerical modeling capabilities makes ita one-stop shop for testing concepts and designs of such systems. http://www.nrccnrc.gc.ca/48

7. Timeline Proposal7.1 Ocean energy influential events7.2 A Timeline7.3 Scientific, technical, societal and economic challenges7.4 Constraints and opportunities7.1 Ocean energy influential events:• Kyoto 1997: to target to reduce GHG emissions.Agreement that set out those targets and the options available to countries to achievethem is known as the Kyoto Protocol: -6 percent goal below 1990 levels by the periodbetween 2008 and 2012.• Investment : Canada is investing $500M in Action Plan 2000. Plus $625M overfive years announced in Budget 2000 results a total commitment of $1.1Billion toaddress climate change over the next five years and builds on the $850M that Canadaspent during the previous five years. Focus on climate change issues increases• Price of oil: Large increase in the price of oil and 9/11 syndrome• Environmental concerns: Societal changes and environmental concerns for thefuture are increasing. Awareness of related issues is increasing.49

7.2 A TimelineA T imelineNational Coordinator,E valuator & T es t Centre(NR C IOT )S horeline modelS horeline prototypedemons tratorNears hore modelNears hore prototypedemons trator20042005200620062009T echnology R es earch Programme& T echnology DevelopmentPhase IPhase IIHydrodynamicsHydrographic map: waves ,currents and iceHydrogen technologyDeployment, recoveryP ower technology: take offconvers ioncontroltrans mis s iontrans formationE conomicsF eas ibility and market analys isR eliability, MaintenancetechniquesNavigationA T imelineOffshore modelOffshore prototypedemonstrator20082010Phase IIINearshore and Offshore sitesprospectingWeather PredictionPower take off predictionPower storageMarket analysisSurvivabilityMooringCivil engineering, FoundationsSynergy with offshoreindustry…20yyT echnology Development50

NRC IOTIndustry, Government ,AcademiaNL WAVE POWER?Working Group onPolicy Statement onOcean Energy GenerationWorking Group onenvironmental issuesRefined Strategy onImpact AssessmentPublic Education andAcceptabilityMarket ExpansionOcean Energy T echnologyNetwork20042009Working Group on R&D & T echnologydevelopmentNational Coordinator, Evaluator & T estCentre (NRC IOT )Resource assessmentPrediction of wind, wave and currentresource; presence of iceL onger term weather forecast?Ocean energy prospecting.outputT echnologies neededData Acquisition T echniquesCable Network, Grid connectionSurface treatments, coatings, CorrosionNon-electrical outputs: hydrogenDesalination, others?Significant contribution to x% ofT otal Energy Generation20yyEnvironmental Impact ResearchThe above Timeline is a proposal for the development of an Ocean EnergyTechnology Network that would include Industry, Academia and Governmentdepartments in Canada. Of the latter it is proposed that the National Research Counciland Industry Canada play a significant role. The Institute for Ocean Technology of theNational Research Council http://iot-ito.nrc-cnrc.gc.ca/about.html is well positioned tobe considered as the National Coordinator for the Evaluation and Testing of OceanSystems.Working groups should be developed to focus on the following tasks. The list is notintended to be exhaustive in any way and the list is not suggesting any priority. In factsome of the tasks will have to be worked upon in parallel. The following is adapted inpart from the Wave Energy Utilization report (2002) and the European Wave EnergyNetwork on Wave Energy. See International Energy Agency Annual Report (2003).The purpose of the present exercise is to initiate discussions among the proponents ofsuch developments in Canada including industry, academia and government andregulatory departments.7.3 Scientific, Technical, Societal and Economic challenges of anOcean Energy Industry:The following list of issues to be addressed are necessary for the development ofOcean Energy harnessing activities. The following is adapted from the European51

Thematic Network on Wave Energy report: Wave Energy Utilization in Europe:Current Status and Perspective (2002). Included also is an expanded list of R&D issuesand further comments on computational tools to complement any experimental physicalmodelling activities.1) R&D in Ocean Energy devices:- Waves and currents energy systems and devices status and R&Drequirements.- Tidal energy systems status and devices and R&D requirements.- Strategy and action plan for above.- Lessons learned from Offshore Industry (technology transfer) wouldbenefit ocean energy technology and its applications. Status andrequirements. See also 2) below.- Scaled model and/or demonstration project developmentsrequirements.- Development of performance evaluation procedures.- Implement performance evaluation on specific concepts.2) Generic Technologies development:- Plant control and power output prediction- Plant monitoring and performance assessment- Loads and survivability studies- Maintenance and reliability studies- Modelling and standardised design methods and calculations- Experimental and mathematical modelling of systems and resourceestimation- Hydrographic analysis and mapping- Harsh environment in the ocean studies: waves, currents, tides andice- Data base development and power prospecting- Experimental and/or numerical work of testing concepts modeling ofarrays of multiple wave and other energy devices, wave and energygeneration and propagation simulation.– Wave focusing and rogue wave simulation.– Real-time wave behavior forecasting– Mooring and long term fatigue of lines and connections– Cable construction, production, testing and laying offshore– Couplings for quick release and reattachment of mooring andcables– Flexible electrical connectors– Environmentally acceptable fluid for hydraulic systems– Power smoothing systems- Follow the advancement in hydrogen technology- International collaboration: attract prototype models to our shores,responding to Canadian-specific environment (ice).- Electrical power storage technology.52

- Control and transmission systems- Flexible electrical connectors- Environmentally acceptable fluid for hydraulic systems- Development of more efficient power generation units- Power conversion- Offshore control systems- Maintenance-free systems. Is it attainable?3) Cooperation with the Power Generation and Distribution Industry:- Development of safety standards.- Assessment of procedures, costs and facilities for power generationand transmission.- Development of power quality standards.- New regulatory and energy transfer regimes.4) Financing and Economic Issues:- Market status of ocean energy: feasibility and market analysis.- Economics of Ocean Energy whether from waves, wind or currents.- Financing of ocean energy projects issues.- Economic impact on environment and local communities.5) Social, institutional and environmental impact:- Planning considerations.- Environmental impact studies.- Institutional barriers studies and regulatory regimes.- Industrial benefit and job creation studies.6) Promotion of Ocean Energy:- Support for Ocean Energy events and meetings.- Publications in International Journals.- Dissemination of information and relevant material- Development of relevant web site.- Involvement with education curricula in schools, promotion ofrenewables in general and Ocean Energy in particular.Below are some comments on the need of accurate and robust computational modelsfor the optimization of the structural configurations and designs of Ocean energySystems. The need for resource estimation techniques was discussed in Sections 2 and3 of this report.The optimal structural configuration of an ocean energy system could be studied andis achievable by taking into consideration the complex interaction of the structure withthe environmental effects. These should include the effects in extreme and harshconditions of waves, currents, winds and ice (if present) as well as the seabed. Thisobjective can be achieved with good information on those environmental effects androbust analytical/computational techniques. Synergy with offshore engineering and thevast experience gained by this industry are critical. Fatigue of structural componentsand of the foundations/anchoring as well as the ultimate capacities are some of theissues. The dynamic excitation forces will have to consider the environmental loadingeffects due the wind, waves, currents, and possibly ice on the structure, the interaction53

etween the devices, their dynamic loading and the structure supporting it and the soilor seabed. Computational techniques of offshore or onshore structures in generalrequire the modeling of the structural responses, including dynamics of the structure,the evaluation and modeling of the environment and the accurate modeling of theinteraction of the structure with the soil through the foundation, whether it is gravitybased or flexibly moored. These are presently issues in offshore engineering R&D.7.4 Constraints and opportunities:Technological:In general the technology for wave energy and stream systems has yet to bedeveloped and proven at full size, particularly for the offshore wave systemsand the variable-pitch vertical axis turbine tidal stream systems. There are noinsurmountable technological barriers to the deployment of such systems.Opportunities abound for innovation, R&D and synergy with the offshoreengineering industry.Financial, economical:There is a need for proven economics and resource assessment methodologies.Opportunities for investors are infinite and the market for renewables in generaland the ocean energy resource in particular, will keep increasing.Institutional:A number of statutory bodies are involved in our coastline and surroundingwaters. The development of an Ocean Energy Industry will require an extensiveconsultation process.54

8. References[1] ABB (2002) http://www.abb.se (in Swedish)[2] Baddour, R.E. and Song, S.W. (1998). The rotational flow of finite amplitudeperiodic water waves on shear currents. Applied Ocean Research, 20, 163-171.[3] Baddour, R. E. and Song, S. (1990). On the interaction between waves andcurrents. Ocean Engineering, 17, 1-21.[4] Baddour, R. E. and Song, S. (1990). Interaction of higher order water waves withuniform currents. Ocean Engineering, 17, 551-568.[5] Bascom , W. (1980) Waves and beaches. Anchor Books, New York.[6] Beattie, W.C. et al (2000) Producing acceptable electrical supply quality from awave-power station. Proc. Third European Wave Power Conf., Patras, Greece,252-257.[7] Bernshtein, L.B., Wilson, E.M. and Song, W.O. (1997); Tidal Power Plants,Revised Edition; Korea Ocean Research and Development Institute, Korea.[8] Blue Energy Canada Inc. (2000); http://www.bluenergy.com ; Canada.[9] British Oceanographic Data Centre (1992) Wave Data Catalogue for EuropeanWater.[10] Brooke, J. (2003) Wave Energy Conversion. Elsevier Ocean Engineering, NewYork.[11] Budal, K and Falnes, J. (1975). A resonant point absorber of ocean wave power.Nature, 256, 478-479.[12] Bhuyan et al (1994). Powertech Inc. Assessment of Wave Energy Resources forthe West Coast of Canada, June 1994. Also http://www.powertech.bc.ca[13] Clancy, E.P. (1968) The tides: Pulse of the Earth, Doubleday and Co. New York.[14] Clement, A.H. et al (2002) Wave Energy in Europe- Current status andperspectives, Renewable and Sustainable Energy reviews, 6, 405-431.[15] Craton, W. (1992). Oceanography html site, Brock University[16] Department of Trade and Industry (1999); New and Renewable Energy: Prospectsin the UK for the 21 st Century: Supporting Analysis; ETSU R-122, UnitedKingdom; www.dti.gov.uk/renewable/index.html.55

[17] EECA and CAE (1996); New and Emerging Renewable Energy Opportunities inNew Zealand; Christchurch, New Zealand; www.eeca.govt.nz/default.asp;www.cae.canterbury.ac.nz/cae.htm.[18] E&PDC (Engineering & Power Development Consultants Ltd) (1993); TidalStream Energy Review; ETSU T/05/00155/REP, United Kingdom.[19] European Energy Conferences Proceedings. (1998) and subsequent.[20] European Thematic Network on Wave Energy (2002) Wave energy utilization inEurope report: current status and perspective, Centre for Renewable EnergySources (CRES).[21] Eshan, M. et al. (1996). Simulation and dynamic response of computer controlleddigital hydraulic pump/motor use in wave power conversions. Proc. Sec.European Wave Power Conf., Lisbon, Portugal, 305-311.[22] FMP (Foresight Marine Panel) (1999); Energies from the Sea – Towards 2020; AMarine Foresight Report, United Kingdom.[23] Falnes. J. (1996). A crude estimate of benefit-cost ratio for utilization of the waveenrgy of the world’s oceans. Proc. Sec. European Wave Power Confernece,Lisbon, Portugal, 102-104.[24] Falnes, J. (2000) Optimum control of oscillation of wave-energy converters. Int. J.Offshore & Polar Engineering, 12, 147-155.[25] Falnes, J. and Budal, K. (1982) Wave power absorption by parallel rows ofinteracting oscillating bodies. Applied Ocean Research, 4, 194-207.[26] Fraenkel, P.L. (1999); Tidal Currents: A Major New Source of Energy for theMillennium; Sustainable Developments International, United Kingdom.[27] Gato, L.M.C. et al (1996). Performance of a high-solidity Wells turbinefor OWCpower plant. Trans. ASME, J. Energy ResourcesTechnology, 118, 263-268.[28] Gato, L.M.C. and Curren, R. (1997). The energy conservation performance ofseveral types of Wells turbine designs. Proc. I. Mech. E. Part A, J. PowerEngineering, 211, 133-145.[29] Gorlov, A. and Rogers, K. (1997); Helical Turbine as Undersea Power Source;Sea Technology, United States.[30] Hagerman, G. (1992) Wave resource assessment review and evaluation. Incomprehensive review and analysis of Hawaii’s renewable Energy assessments.State of Hawaii Department of Business, Economic Development and Tourism,Honolulu, Hawaii.56

[31] Hagerman, G. (1995) A standard economic assessment methodology forrenewable ocean energy projects. Proceedings of the Interantional Symposium onCoastal Ocean Space Utilization, COSU 1995,129-138.[32] Hagerman,G. (1996) Wave power: an overview of recent internationaldevelopment and potential US projects. Proceedings of the 1996 AnnualConference, American Solar Energy Society, North Carolina, 195-200.[33] International Energy Agency (2002) Renewables in Global Energy Supply.[34] International Energy Agency (2003) Implementing Agreement on Ocean EnergySystems, Annual Report .[35] Institute for Ocean Technology, National research Council Canada, 2004,http://iot-ito.nrc-cnrc.gc.ca/about.html[36] Kinsman, B. (1984) Wind Waves, Their generation and propagation on the OceanSurface, Dover Publications, New York.[37] Korde, U.A. (2002). Active control in wave energy conversion. Sea Technology,43, 47-52.[38] Leishman, J.M. and Scobie, G. (1976) The development of wave power – a technoeconomical study. Department of Industry, NEL report EAU M25.[39] Lewis, T. (1985) Wave energy Evaluation for C.E.C., EUR982EN.[40] Mellor,G.L.(1996) Introduction to Physical Oceanography. Springer Verlag,NewYork.[41] MCT (Marine Current Turbines Ltd) (2000); www.marineturbines.com; UnitedKingdom.[42] Nebel, P.(1992) Maximizing the efficiency of wave energy plants using complexconjugatecontrol. Proc. I. Mech E. Journal of Systems and Control, 206, 225-236.[43] Open University (1989); Waves, Tides and shallow water processes; PergamonPress, United Kingdom.[44] Preliminary Actions in Wave Energy (1993). Wave studies and deevlopemnt ofresource evaluation methodology, The European Economic Communities, DGXII, Final report.[45] Rebello, L. et al (1996). 1996 Project OLASS 1000-experiences, modelling andresults. Proc. Second European Wave power Conference, Lisbon, Portugal, 312-319.[46] Retzler, C.H. (1996). The wave rotor.57

Proc. Second European Wave power Conference, Lisbon, Portugal, 312-319.[47] Rudkin, E.J. and Loughman, G.L. (2001). Vortec – the marine enrgy solution.Marine Renewable Energy Conference 2001; Newcastle, United Kingdom.[48] Russel, A. and Diamantaras, K. (1996) The European Commission Wave EnergyR&D Programme. Proc. Second European Wave Power Conference, Lisbon,Portugal, 8-11. G. Elliot and Diamantaras, K. editors.[49] SEASUN Power Systems (1988). Power conditioning and transmission, waveenergy resource and technology assessment for coastal North Carolina. FinalReport. North Carolina Alternative Energy Corporation & Virginia Power.[50] Scott, N.C. (2001) Grid connection of large-scale wave energy projects. Proc.Fourth European Wave Power Conf., Aalborg, Danemark. 88-92.[51] Southgate, H.N. (1987) Wave prediction in deepwater and at the coastline, HRreport #SR114.[52] Southgate, H.N. (1993) The use of wave transformation models to evaluateinshore wave energy resource. Proc. First European Symposium on Wave Energy,Edinburgh, UK.[53] Thorpe, T.W. (1999) An overview of wave enrgy technologies: status,performance and costs. Proceedings, International One day Seminar, Institution ofMechanical Engineers, London UK.[54] Thorpe, T.W. (1999) A brief overview of wave energy. ETSU report R-120.Department of Trade and Industry, UK.[55] Thorpe, T.W. (1992) A review of wave energy. ETSU Report R-72, 1, Departmentof Trade and Industry, UK.[56] Thorpe, T.W. (1999) A brief overview of wave energy. ETSU Report R-120.Department of Trade and Industry, UK.[57] Thorpe, T.W. (1999) An evaluation of wave energy. ETSU Report AEAT/R0400. Dpartment of Trade and Industry, UK[58] University of Edinburgh, Wave Power Group. 2000.www.mech.ed.ac.uk/research/wavepower/[59] Various authors. 1993 Preliminary design and model test of a wave powerconverter: Budal 1978 Design type E. Technical reports compiled by Falnes, J.,Department of Physics, NTSU, Trondheim, Norway.58

[60] WATTS (2003, 2004). Wave and Tidal Technology Symposium Proceedings.[61] Wavegen. (2000) [62] Wave Mill Energy Corp (2004) http://www.wavemill.com/products.htm[63] World Energy Council (2004) http://www.worldenergy.org/wec-geis[64] World Energy Council (2000) http://www.worldenergy.org/wecgeis/publications/reports/etwan/execsummary/exec-summary.asp9. Recent scientific and technical research bibliographyTheme: Ocean Energy related topicsOcean Engineering - From 1995-2004D. C. Hong, S. Y. Hong and S. W. Hong, Numerical study of the motions and drift force of a floatingOWC device, Ocean Engineering, Volume 31, Issue 2, February 2004, Pages 139-164.E. Vijayakrishna Rapaka, R. Natarajan and S. Neelamani, Experimental investigation on the dynamicresponse of a moored wave energy device under regular sea waves, Ocean Engineering, In Press,Corrected Proof, Available online 2 December 2003.Y. M. C. Delauré and A. Lewis, 3D hydrodynamic modelling of fixed oscillating water column wavepower plant by a boundary element methods, Ocean Engineering, Volume 30, Issue 3, February 2003,Pages 309-330.Paolo Boccotti, On a new wave energy absorber, Ocean Engineering, Volume 30, Issue 9, June 2003,Pages 1191-1200.Umesh A. Korde, Latching control of deep water wave energy devices using an active reference, OceanEngineering, Volume 29, Issue 11, September 2002, Pages 1343-1355.W. E. Rogers, J. M. Kaihatu, H. A. H. Petit, N. Booij and L. H. Holthuijsen, Diffusion reduction in anarbitrary scale third generation wind wave model, Ocean Engineering, Volume 29, Issue 11, September2002, Pages 1357-1390.A. Brito-Melo, L. M. C. Gato and A. J. N. A. Sarmento, Analysis of Wells turbine design parameters bynumerical simulation of the OWC performance, Ocean Engineering, Volume 29, Issue 12, September2002, Pages 1463-1477.A. F. de O. Falcão, Wave-power absorption by a periodic linear array of oscillating water columns,Ocean Engineering, Volume 29, Issue 10, August 2002, Pages 1163-1186.D. J. Wang, M. Katory and Y. S. Li, Analytical and experimental investigation on the hydrodynamicperformance of onshore wave-power devices, Ocean Engineering, Volume 29, Issue 8, July 2002, Pages871-885.59

S. Neelamani and M. Vedagiri, Wave interaction with partially immersed twin vertical barriers, OceanEngineering, Volume 29, Issue 2, February 2002, Pages 215-238.Ruo-Shan Tseng, Rui-Hsiang Wu and Chai-Cheng Huang, Model study of a shoreline wave-powersystem, Ocean Engineering, Volume 27, Issue 8, August 2000, Pages 801-821.A. F. de O. Falcão and P. A. P. Justino, OWC wave energy devices with air flow control, OceanEngineering, Volume 26, Issue 12, December 1999, Pages 1275-1295.U. A. Korde, Efficient primary energy conversion in irregular waves, Ocean Engineering, Volume 26,Issue 7, July 1999, Pages 625-651.Applied Ocean Research - 1995-2004José Perdigão and António Sarmento, Overall-efficiency optimisation in OWC devices, Applied OceanResearch, Volume 25, Issue 3, June 2003, Pages 157-166.Umesh A. Korde, Systems of reactively loaded coupled oscillating bodies in wave energy conversion,Applied Ocean Research, Volume 25, Issue 2, April 2003, Pages 79-91.A. F. de O. Falcão and R. J. A. Rodrigues, Stochastic modelling of OWC wave power plant performance,Applied Ocean Research, Volume 24, Issue 2, April 2002, Pages 59-71.A. F. de O. Falcão, Control of an oscillating-water-column wave power plant for maximum energyproduction, Applied Ocean Research, Volume 24, Issue 2, April 2002, Pages 73-82.U. A. Korde, On providing a reaction for efficient wave energy absorption by floating devices, AppliedOcean Research, Volume 21, Issue 5, October 1999, Pages 235-248.H. Eidsmoen, Tight-moored amplitude-limited heaving-buoy wave-energy converter with phase control,Applied Ocean Research, Volume 20, Issue 3, June 1998, Pages 157-161.M. Greenhow and S. P. White, Optimal heave motion of some axisymmetric wave energy devices insinusoidal waves, Applied Ocean Research, Volume 19, Issues 3-4, June-August 1997, Pages 141-159.Umesh A. Korde, Performance of a wave energy device in shallow-water nonlinear waves: part I,Applied Ocean Research, Volume 19, Issue 1, February 1997, Pages 1-11.Umesh A. Korde, Performance of a wave energy device in shallow-water nonlinear waves: part II,Applied Ocean Research, Volume 19, Issue 1, February 1997, Pages 13-20.K. Thiruvenkatasamy and S. Neelamani, On the efficiency of wave energy caissons in array, AppliedOcean Research, Volume 19, Issue 1, February 1997, Pages 61-72.S. A. Mavrakos and P. McIver, Comparison of methods for computing hydrodynamic characteristics ofarrays of wave power devices, Applied Ocean Research, Volume 19, Issues 5-6, October-December1997, Pages 283-291.J. R. Chaplin and C. H. Retzler, Predictions of the hydrodynamic performance of the wave rotor waveenergy device, Applied Ocean Research, Volume 17, Issue 6, December 1995, Pages 343-347.P. McIver and M. McIver, Wave-power absorption by a line of submerged horizontal cylinders, AppliedOcean Research, Volume 17, Issue 2, 1995, Pages 117-126.60

Coastal Engineering - 1995-2004Jaak Monbaliu, Roberto Padilla-Hernández,Julia C. Hargreaves, Juan Carlos Carretero Albiach, WeiminLuo, Mauro Sclavo and Heinz Günther, The spectral wave model, WAM, adapted for applications withhigh spatial resolution, Coastal Engineering, Volume 41, Issues 1-3, September 2000, Pages 41-62.T. C. Lippmann, A. H. Brookins and E. B. Thornton, Wave energy transformation on natural profiles,Coastal Engineering, Volume 27, Issues 1-2, May 1996, Pages 1-20.Renewable Energy - 1995-2004A. Thakker and T. S. Dhanasekaran, Computed effects of tip clearance on performance of impulseturbine for wave energy conversion, Renewable Energy, Volume 29, Issue 4, April 2004, Pages 529-547.Ajit Thakker and Fergal Hourigan, Modeling and scaling of the impulse turbine for wave powerapplications, Renewable Energy, Volume 29, Issue 3, March 2004, Pages 305-317.T. Setoguchi, Y. Kinoue, T. H. Kim, K. Kaneko and M. Inoue, Hysteretic characteristics of Wells turbinefor wave power conversion, Renewable Energy, Volume 28, Issue 13, October 2003, Pages 2113-2127.A. S. Bahaj and L. E. Myers, Fundamentals applicable to the utilisation of marine current turbines forenergy production, Renewable Energy, Volume 28, Issue 14, November 2003, Pages 2205-2211.Mats Leijon, Hans Bernhoff, Marcus Berg and Olov Ågren, Economical considerations of renewableelectric energy production--especially development of wave energy, Renewable Energy, Volume 28,Issue 8, July 2003, Pages 1201-1209.T. Setoguchi, S. Santhakumar, M. Takao, T. H. Kim and K. Kaneko, A modified Wells turbine for waveenergy conversion, Renewable Energy, Volume 28, Issue 1, January 2003, Pages 79-91.Wibisono Hartono, A floating tied platform for generating energy from ocean current, RenewableEnergy, Volume 25, Issue 1, January 2002, Pages 15-20.T. Setoguchi, S. Santhakumar, H. Maeda, M. Takao and K. Kaneko, A review of impulse turbines forwave energy conversion, Renewable Energy, Volume 23, Issue 2, June 2001, Pages 261-292.Fernando Ponta and Gautam Shankar Dutt, An improved vertical-axis water-current turbineincorporating a channelling device, Renewable Energy, Volume 20, Issue 2, June 2000, Pages 223-241Apparatus for dissipating wave energy : Dorrell Donald E Rarotonga 01, Cook Island, RenewableEnergy, Volume 11, Issue 2, June 1997, Page 273.V. S. Raju and M. Ravindran, Wave energy : potential and programme in India, Renewable Energy,Volume 10, Issues 2-3, 3 February 1997, Pages 339-345.P. R. S. White, The European programme to develop the Wells air turbine for applications in waveenergy, Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1207-1212.F. Peter Lockett, Mathematical modelling of wave energy systems, Renewable Energy, Volume 9, Issues1-4, September-December 1996, Pages 1213-1217.Kunal Ghosh, Cascade wind turbines for the oscillating water column wave energy device: Part 1,Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1219-1222.61

Hitoshi Hotta, Yukihisa Washio, Hitoshi Yokozawa and Takeaki Miyazaki, R&D on wave power device"Mighty Whale", Renewable Energy, Volume 9, Issues 1-4, September-December 1996, Pages 1223-1226.Wibisono Hartono, A floating tied platform for generating energy from ocean current, RenewableEnergy, Volume 25, Issue 1, January 2002, Pages 15-20.R. C. McGregor and W. E. R. Desouza, On the analysis of tidal energy schemes with large diurnalvariations with application to Singapore, Renewable Energy, Volume 10, Issues 2-3, 3 February 1997,Pages 331-334.H. Maeda, S. Santhakumar, T. Setoguchi, M. Takao, Y. Kinoue and K. Kaneko, Performance of animpulse turbine with fixed guide vanesfn2 for wave power conversion, Renewable Energy, Volume 17,Issue 4, 1 August 1999, Pages 533-547.V. N. M. R. Lakkoju, Combined power generation with wind and ocean waves, Renewable Energy,Volume 9, Issues 1-4, September-December 1996, Pages 870-874.Renewable and Sustainable Energy Reviews - 1997-2004Alain Clément, Pat McCullen, António Falcão, Antonio Fiorentino, Fred Gardner, Karin Hammarlund,George Lemonis, Tony Lewis, Kim Nielsen, Simona Petroncini et al., Wave energy in Europe: currentstatus and perspectives, Renewable and Sustainable Energy Reviews, Volume 6, Issue 5, October 2002,Pages 405-431.Roger H. Charlier, A "sleeper" awakes: tidal current power, Renewable and Sustainable Energy Reviews,Volume 7, Issue 6, December 2003, Pages 515-529.Roger H. Charlier, Re-invention or aggorniamento? Tidal power at 30 years, Renewable and SustainableEnergy Reviews, Volume 1, Issue 4, December 1997, Pages 271-289.Journal of Energy Engineering - 1995-2004Performance Prediction of Contrarotating Wells Turbines for Wave Energy Converter Design. R. Curran,T. J. T. Whittaker, S. Raghunathan, and W. C. Beattie, J. Energy Engrg. 124, 35 (1998)Productivity of Ocean-Wave-Energy Converters: Turbine Design R. Curran, J. Energy Engrg. 128, 13(2002)62

AppendicesAppendix 1: Mathematical formulation and computation ofwave power/energy in irregular seasThe local behaviour of waves is determined by what is defined as the spectrum of theVHDVWDWH7KLVLVJLYHQPDWKHPDWLFDOO\E\WKHIXQFWLRQ6I WKDWVSHFLILHVKRZWKHZDYHHQHUJ\LVGLVWULEXWHGLQWHUPVRIIUHTXHQF\IDQGGLUHFWLRQ 6HHIRr exampleKinsman (1984) for further details on this topic.The spectrum is usually summarized by a small number of wave parameters. They areWKHZDYHKHLJKW+PWKHSHULRG7VHFWKHIUHTXHQF\I 7+]DQGGLUHFWLRQ (rad).For the wave height the most widely used parameter is the significant wave heightwhich can be computed from the wave directional spectrum as follows:1/ 2H s= 4m 0where m0is the zero-th spectral moment, the n-th moment being defined asm02∫ ∫ ∞= π nf S( f , θ ) df dθ0 0The convention of considering S as the distribution of the variance of the sea surfaceelevation has been used. Thus the total variance is m0(m 2 ) and the actual energy perunit area is ρ gm0. Also the significant wave height is defined (followed convention) as1/ 2H s= 4m 0. (See Mollison 1986 or other references)The energy mean period is defined asm−1T e=m0and the peak period Tpis the inverse of the peak frequency fpwhich corresponds tothe highest spectral density. For sea states with only one wave system Tpis useful inproviding a measure of the period of the waves with the highest energy density. Thesignificant wave period Tswhich is the average period of the highest one third of thewaves is generally less used. The wave power level or flux of energy per unit wavefront can be computed from2π∫0 ∫ ∞0P = ρgcg( f , h)S(f , θ ) dfdθwhere ρ is the water density. In deep water the group velocity which is the velocity atwhich the energy propagates is given byc g=g4πf63

Thus the wave power is given byPρg4πρg4π222π∞−1= ∫ ( , ) =−10 ∫ f S f θ df dθm0which can be expressed in terms ofHsand TeasP ≅20.5 H sT eWhenHsis expressed in meters and Tein seconds the power level is given in kW/m.Appendix 2: Useful wave & tidal power linksUniversities, National and Government sitesAustralian Renewables including Wave Energy :http://www.greenhouse.gov.au/renewable/index.htmlCaddet renewable energy website : http://www.caddet.org/index.phpDanish Wave Energy : http://www.waveenergy.dk/DTI Renewables (UK Government) : http://www.dti.gov.uk/energy/renewables/European Commission ’Thermie’ wave energy site :http://europa.eu.int/comm/energy_transport/atlas/htmlu/wave.htmlEuropean Marine Energy Centre, Orkney (test centre for marine energy) :http://www.emec.org.uk/European Wave Energy Research Network (EWERN) :http://www.ucc.ie/ucc/research/hmrc/ewern.htmEuropean Wave Energy Thematic Network : http://www.wave-energy.net/Japan Marine Science & Technology Center, JAMSTEC :http://www.jamstec.go.jp/jamstec/MTD/Whale/Marine Institute, Cork Ireland: http://www.marine.ie/rnd+projects/index.htmNorwegian Wave Energy Site :http://www.phys.ntnu.no/instdef/prosjekter/bolgeenergi/index-e.htmlOpen University, UK : http://www.openuniversity.edu/Scottish Executive Energy website:http://www.scotland.gov.uk/about/ELLD/EN-CS/00017058/energyhome.aspxWorld Wave Atlas : http://seawatch.mg.uoa.gr/ZZZPHFKHGDFXNUHVHDUFKZDYHSRZHU64

Device Developers and Data ProvidersAquaEnergy Group Ltd (USA) :http://www.aquaenergygroup.com/home.htmDaedalus Informatics Greece: Hybrid wave and wind system :http://www.daedalus.gr/DAEI/PRODUCTS/RET/General/What%20is%20Wave%20PowerEMU Consult, Denmark (Wave Dragon) :http://www.spok.dk/consult/waves.shtmlEnergetech Australia Pty Ltd (includes Denniss-Auld Turbine) :http://www.energetech.com.au/The Engineering Business Ltd (Stingray tidal stream device) :http://www.engb.com/Interproject Service AB (IPS OWEC Buoy): http://www.ips-ab.com/Marine Current Turbines Ltd : http://www.marineturbines.com/home.htmOceanor, Norway (Wave data) : http://www.oceanor.no/Ocean Power Delivery Ltd., Scotland (Pelamis) :http://www.oceanpd.com/Ocean Power Technologies, USA :http://www.oceanpowertechnologies.com/Sea Power International AB, Sweden :http://www.seapower.se/indexeng.htmlSeaVolt Technologies, USA (Wave Rider) under constructionStrom AS (Tidal Stream generator at Hammerfest in Norway) :http://www.tidevannsenergi.com/Teamwork Technology Bv (Archemides Wave Swing):http://www.waveswing.com/Verdant Power (tidal current demonstration in NYC):http://www.verdantpower.com/Initiatives/eastriver.shtmlWave Dragon ApS (Danish) : http://www.wavedragon.net/Wavegen, Scotland (Limpet) : http://www.wavegen.co.uk/Wavemill Energy Corporation (Cape Breton, Nova Scotia) :http://www.wavemill.com/Lobbying, Promotional & Trade OrganisationsPractical Ocean Energy Management Systems (US) :http://www.poemsinc.org/Renewable Power Association (UK) : http://www.r-p-a.org.uk/home.fcmSeapower - Marine Renewable Energy Association (UK):http://www.bwea.com/marine/resource.htmlScottish Coastal Forum :http://www.scotland.gov.uk/environment/coastalforum/Scottish Energy Environment Foundation :http://www.mecheng.strath.ac.uk/feature-seef.htmScottish Renewables Forum: http://www.ipa-scotland.org.uk/65

National Review documentsA brief review of Wave Energy : http://www.researchinnovation.ed.ac.uk/expertise/physical-sciences/energy.pdfAlso see http://www.research-innovation.ed.ac.uk/flashindex.htmlOptions for the development of wave energy in Ireland : http://www.irishenergy.ie/uploads/documents/upload/publications/wave.pdfOther LinksEdinburgh Designs Ltd (Test tanks and absorbing wave maker systems):http://www.edesign.co.uk/http://www.edesign.co.uk/McGraw-Hill Higher Education virtual wave tank :http://www.mhhe.com/physsci/physical/giambattista/wave_tank/wave_tank.htmlAppendix 3: Recent publications from IOT66

Proceedings of OMAE ’ 0423 rd International Conference on Offshore Mechanicsand Arctic EngineeringJune 20-25, 2004, Vancouver, CanadaOMAE2004-51044THE NONLINEAR INTERACTION AND RESONANCE OF STEEP LONG-CRESTEDBICHROMATIC SURFACE WAVES IN A NUMERICAL WAVE TANKR. E. BaddourNational Research Council -CanadaInstitute for Ocean TechnologyP.O.Box 12093, A1B 3T5St John’s, NL, CANADAW. ParsonsCollege of the North AtlanticRidge road,Box 1150, A1C 6L8St John’s, NL, CANADAABSTRACTWe are studying numerically the problem of generation andpropagation of long-crested gravity waves in a tank containingan incompressible inviscid homogeneous fluid initially at restwith a horizontal free surface of finite extent and of infinitedepth. A non-orthogonal curvilinear coordinate system, whichfollows the free surface is constructed which gives a realistic’’continuity condition’’, since it tracks the entire fluid domain atall times. A depth profile is assumed and employed to perform awaveform relaxation algorithm to decouple the discreteLaplacian along dimensional lines, thereby reducing itscomputation over this total fluid domain. In addition, the fullnonlinear kinematic and dynamic free surface conditions areutilized in the algorithm. A bichromatic deterministic wavemaker using a Dirichlet type boundary condition and a suitablytuned numerical beach is utilized. This paper pays specialattention to satisfying the full nonlinear free surface conditionsand presents the nonlinear interaction of the higher ordercomponents, especially near resonance.INTRODUCTIONA two-dimensional rectangular basin containing anincompressible inviscid homogeneous fluid initially at rest witha horizontal free surface of finite extent is considered togenerate and propagate long-crested waves. On the left verticalboundary a wavemaker is positioned while at the right hand sideis the radiation boundary. By this we mean, the right hand sideboundary is designed to avoid reflections, which complicatesthe flow field. This can be accomplished by implementing aradiation condition, hence the name, which allows the waves topass through the boundary, or by placing a numerical beach nearthis boundary. We choose the latter. As in Parsons and Baddour(2002), a depth profile for the potential is assumed, giving us awaveform relaxation method, and thereby drastically reducingthe computational cost of solving Laplace’s equation. Anumerical beach is also used to absorb the wave energy at theradiation boundary. A bichromatic wavemaker is employedusing a Dirichlet type boundary condition, a non-orthogonalcurvilinear coordinate system, which follows the free surface,and the full nonlinear kinematic and dynamic free surfaceboundary conditions are employed; see Parsons and Baddour(2003). Although, these ideas can be extended to finite depthtanks, we presently restrict our attention to the infinite depthcase.This method is a volume-discretization method, which useswaveform relaxation to reduce the “computational dimension”of the problem. As such, it compares favorably with theboundary-discretization method that reduces the dimension ofthe problem by solving it only on the boundary. See the reviewby Tsai and Yue (1996).Concerning the phenomenon of resonance in surface gravitywaves, see Debnath (1994) for some of the classical results andthe references there in. Note that these methods usually involveseries methods where a number of high order terms are includedin the expansion of the equations. We point out that our modelutilizes the full nonlinear model, and therefore should containthese results, as special cases. In fact it should be rememberedthat resonance is a purely nonlinear effect, and thereforerequires a nonlinear model to be seen. Also, the results can bequite dramatic, so it is important to be aware of them whensolving ocean engineering problems.PROBLEM FORMULATIONConsider a two-dimensional rectangular basin containing anincompressible inviscid homogeneous fluid of constantdensity ρ , in a Cartesian coordinate system ( x , z), with the1 Copyright © 2004 by ASME

origin at the still water level, the positive z-axis pointingupwards. The horizontal extent of the basin is 0 ≤ x ≤ L , soL > 0 is the length of the tank. The depth of the basin isinfinite. See Figure 1 for the coordinate system configuration.This is an initial-value problem, since the surface is initially atrest for time, t < 0 and is disturbed at t ≥ 0 giving rise tosurface waves. See Debnath (1994). If the Φ ( x,z,t)is thevelocity potential and η( x,t)is the free surface elevation, theproblem is defined by the following equations, where g is theacceleration due to gravity. The conservation of mass equationis Laplace's equation:22∂ Φ ∂ Φ+ = 02 2∂x∂z−∞ ≤ z ≤η ( x,t),0 ≤ x ≤ L,t ≥ 0 (1)For t ≥ 0,the full nonlinear kinematic and dynamic free surfaceboundary conditions are given, respectively, by:∂η∂η∂Φ ∂Φ+ − = 0∂t∂x∂x∂z(2)and∂Φ 1 ∂Φ 2 ∂Φ+ [( ) + ( )] + gη = 0∂t2 ∂x∂z(3)on z = η( x,t), and∂Φ→ 0 as z → −∞(4)∂zIntroducing the following function:χ( x,t)= Φ(x,η(x,t),t)(5)which represents the potential on the free surface, we have:∂χ∂Φ ∂Φ ∂η= +(6)∂t∂t∂z∂timplying:∂Φ∂χ∂Φ ∂η= −(7)∂t∂t∂z∂tAlso, we have that∂χ∂Φ ∂Φ ∂η= + .(8)∂x∂x∂z∂xSee Wehausen and Laitone (1960).The initial conditions are given by:(••0 00 x•Φ x,0,0)= (1/ ρ ) Φ ( x);Φ(x,0,0)= (1/ ρ)Φ (x); η(x,0)= η ( )where ( 1/ ρ ) Φ 0(x ) and ( 1/ ρ ) Φ 0 ( x)represent the given freesurface impulse and η 0 ( x ) the initial displacement, where the“dot” represents a time derivative. Since we are considering afluid in which the initial velocities are zero and the initial freesurface is at rest at z = 0 , we take•Φ 0 ( x)= 0; Φ 0 ( x)= 0; η 0 ( x)= 0 for 0 ≤ x ≤ L (9)Therefore we are left with two lateral boundary conditions fort ≥ 0 at x = 0 and x = L , which we call the LHS lateralboundary condition and the RHS lateral boundary condition,respectively.The LHS boundary condition involves a wavemaker, which weassume is the sum of two sinusoid waves, each of the generalformη ( x,t)= A cos( k x −ωt),- ∞ < x ≤ 0, - ∞ < t < ∞ (10)iiiiwhere, A i = H i / 2 is the amplitude of each wave, kiis thewavenumber and ωiis the angular frequency of each wave;i = 1,2 . The steepness of the wave is S i = kiAi/ π . Usingstandard water wave mechanics, the associated linear velocitypotential gives the LHS boundary condition, and is the sum oftwo terms of the general formA i g k i zΦ i ( x , z , t ) = e sin( k i x − ω it),(11)ωfor x = 0, t ≥ 0, - ∞ < z ≤ 0,where the linear dispersion relation isiω g , i = 1, 2 . (12)i =k iNote that in the implementation of the wavemaker algorithmwe assume the validity of (11) over the range −∞ < z ≤ η( 0, t)i.e.up to the free surface.It is prudent to point that it is not necessary, but only aconvenience and for completeness to make the assumption ofthe existence of the sinusoids in Equation (10), since we areonly inputting the potential given by Equation (11). Thereforeconditions outside the interval [ 0 ,L]have no effect on thesolution. This progressive wave at x = 0 and t ≥ 0 is the solesource of the disturbance that gives rise to water waves in theinitially calm basin.If we assume a wall at x = L so the RHS lateral boundarycondition becomes∂Φ(L,z,t)= 0, - ∞ < z ≤ η,t ≥ 0,(13)∂xwe will get reflections from waves going from the wavemakerat x = 0 toward the RHS wall. To absorb these waves we placea “numerical beach” on the RHS. This is accomplished byincluding a damping term µ in the dynamic free surfaceboundary condition equation applied at the damping zone. Thisis accomplished by adding a damping term µχ to the righthand side of the dynamic free surface boundary condition; seeEquation (33).However, an abrupt introduction of this damping term causessome reflection at the boundary between the damping zone andthe non-damping zone. The following third-order polynomialdistribution of the damping coefficient ensures a smoothtransition between these two zones:⎛ 2 33ˆ x 2xˆ⎞µ ( xˆ)=ν ⎜ − ⎟ kg , where L⎜ 2 3L1L ⎟1 is the length of the⎝1 ⎠damping zone, k is the representative wave number, which willbe discussed later, and ν is the normalized damping coefficient.Note that xˆ is a local variable where, x ˆ = 0,when x = L − L1,and x ˆ = L1, when x = L .This numerical beach is found to be most effective when,0.8≤ν ≤1.0.Clearly, setting ν = 0 is equivalent to removing the beach.Also, L 1 is usually taken to be about two representativewavelengths, L 1 ≈ 2λ, where λ is the wavelength associated2 Copyright © 2004 by ASME

¡with the representative wavenumber, and is given byλ = 2π / k . See Parsons and Baddour (2002).THE LAPLACIAN ON THE FREE SURFACECOORDINATE SYSTEMWe define the Free Surface Coordinate System (FSCS), ( s , w),where s ≥ 0 is the arclength along the free surfacez = η( x,t)and w ≤ 0 is the vertical depth to any point in thefluid from the free surface. This is a non-orthogonal movingcoordinate system that follows the free surface. Clearly, thecoordinates satisfy the following equations (over the completefluid domain),2⎛ ∂ ( , ) ⎞= ∫ x η τ ts 1 + ⎜ ⎟ dτ, 0 ≤ x ≤ L , (14)0 ⎝ ∂τ⎠w = z −η ( x,t),0 ≤ x ≤ L, w ≤ 0.(15)If we define the unit vector ŝ along the arclength, and thereforetangent to the free surface z = η( x,t), thens ˆ = cosθ î + sinθkˆ(16)i ˆ = secθŝ - tanθkˆ(17)where−tan 1 ⎛ ∂η⎞θ = ⎜ ⎟ .⎝ ∂x⎠(18)The arclength along the coordinate curve s is given by2ds =⎛ ∂η⎞1+⎜ ⎟⎝ ∂x⎠dx(19)ords = secθ dx . (20)Clearly we have the following differential operator formulasthat allow us to transform easily between rectangularcoordinates ( x , z)and free surface coordinate system ( s , w):¢£∂ ∂ ∂= secθ- tanθ∂x∂s∂w(21)∂ ∂ ∂= cosθ+ sinθ∂ s ∂ x ∂z(22)∂ ∂= .∂z ∂w(23)The Laplacian for this non-orthogonal coordinate system can beconstructed from the second-order covariant fundamental metrictensorJ⎛1sinθ⎞=⎜⎟⎝sinθ 1 ⎠(24)where its associated second-order contravariant tensor is givenby⎛2sectan sec ⎞⎜θ − θ θ⎟J =⎜2tan secsec ⎟⎝−θ θθ ⎠(25)where clearly¤¥ = GHWJ) = cos2θJ .The Laplacian is given by the following tensor equation1 ∂ ⎛ αλ ∂Φ ⎞∆Φ = ⎜ JJ ⎟,α,λ = 1,2λαJ ∂x⎝ ∂x⎠12where x = s,x = w.See Kaplan (1993). We get∆Φ = secθ∂ ⎛ ∂Φ ∂Φ ⎞[ ⎜secθ− tanθ⎟ +∂s⎝ ∂s∂w⎠∂ ⎛ ∂Φ ∂Φ ⎞+ ⎜−tanθ+ secθ⎟]∂w⎝ ∂s∂w⎠(26)(27)THE DYNAMIC AND KINEMATIC BOUNDARYCONDITIONS ON THE FREE SURFACE COORDINATESYSTEMUsing equation (8) and equation (18), the kinematic free surfaceboundary condition (KFSBC) given by equation (2), becomes∂η∂χ2 ∂Φ+ tanθ− sec θ = 0 . (28)∂t∂x∂zThe dynamic free surface boundary condition (DFSBC), givenby equation (3), can be rewritten using equations (7) andequation (8). After some algebra and using equation (18) andthe KFSCB, equation (28), we get the following final form:22∂χ1 ⎡⎛∂χ⎞ 2 ⎛ ∂Φ ⎞ ⎤+ ⎢⎜⎟ − sec θ ⎜ ⎟ ⎥ + gη= 0.(29)∂t2 ⎢⎣⎝ ∂x⎠ ⎝ ∂z⎠ ⎥⎦where all values in (28) and (29) are evaluated on the freesurface. Since the free surface coordinate system tracks the freesurface, these values are readily obtained and allow for theintegration in time t .SPATIAL APPROXIMATION AND WAVEFORMRELAXATIONIn this paper, we employ a curvilinear grid that follows the freesurface. In light of equation (18), this grid is taken to be{( s w ),i = 0,1, , M ; j = 0,1, ,N },(30)i , jLwheresi = −si−1 + secθ i 1∆x, and⎡2 ⎤⎢ ⎛ ∂η⎞secθ ⎥i = 1+⎜ ⎟ ,⎢ ⎝ ∂x⎠ ⎥⎣ ⎦where we take s 0 = 0 , and w j = − j ∆ w , where ∆ x = L / M , sox i = i∆x,and ∆ w = h / N L , where h is the depth to which wesolve Laplace’s equation, taken to be greater than or equal toone wavelength of the wavemaker. We allow the possibility that∆ w ≠ ∆x; see equation (23). We must solve Laplace’s equation,(27) over this grid.The semi-discretized approximation of this potentialΦ ( s , w , t)is written as Φ ( ) i = 0,1,,M ;iji, j tj = 0,1,,N L ; t ≥ 0.To implement the wavemaker, we assume the sum of the twosinusoids in (10), for x ≤ 0 , −∞ < t < ∞ , and their associated(linear) velocity potentials, (11) and dispersion relation (12).The LHS boundary condition is implemented as a Dirichletboundary condition.x=x i3 Copyright © 2004 by ASME

The Dirichlet wavemaker:Φ0, j− A2where,( t)= −Ag[ ek21k2wjgk1k1wj[ e ]]sin( ki2sin( k c t)c t)211gc i = is the velocity of each wave, i = 1, 2k(31)., andj = 0,1,,N L ; t ≥ 0 . The horizontal bottom boundarycondition, given by (4), will be automatically satisfied by therelaxation method, as we shall soon see. The boundarycondition at the free surface, z = η( x,t),{ Φ t): i = 1,2, , M −1;t 0}i (32)is given by the kinematic and dynamic boundary conditions,(28) and (29), respectively, and involve integration over time.This will be discussed in the next section. The RHS boundarycondition for the numerical beach is incorporated in thecalculation of the free surface potential over the damping zone,and involves modification of equation (29) to,, 0 (≥∂χ1 ⎡ ∂χ2 2 ∂Φ( ) sec ( )2 ⎤++ + = 0.2⎢ − θ∂⎥ µχ gη(33)t ⎣ ∂x∂z⎦Therefore, the potential should vanish everywhere at the wall,and as a convenience, we may employ equation (13) to get theRHS boundary condition; that is,t)= Φ ( t),j = 0,1, , N ; t 0. (34)Φ M , j ( ( M −1),jL ≥Even with a uniform ∆ x , the curvilinear grid will have nonconstant∆ s.Therefore, to apply finite difference formulas todiscretize Laplace's equation (27), we will need appropriatefinite difference operators. Clearly, applying these discreteoperators to Laplace's equation (27), is best done usingcomputer algebra, and the authors utilized Maple. We refer tothe resulting equation, as the "semi-discretized Laplaceequation". The details are to be given elsewhere.Consider its application over any s -coordinate curve for fixedw in the interior of the domain. That is, we evaluate the semidiscretizedLaplacian at s , w ) , to calculate( i jΦ i, j ( t)i = 1,2, , M −1;j = 1,2, , N L −1(fixed); t ≥ 0.Thiswill involve nine potentials, namely:s j−1 -coordinate curve: Φ i−1,j−1(t)Φi, j−1(t)Φ i+1,j−1 ( t)s -coordinate curve: Φ ( ) Φ ( ) Φ ( )ji−1,j ti, jti+ 1,j ts j+1 -coordinate curve: Φ i −1,j+1( t)Φi , j+1( t)Φ i + 1,j+1 ( t)See Figure 1. The idea behind waveform relaxation is to followthe lead of separation of variables methods and assume that thepotential can be written as,~Φ ( t)χ ( t)φ i = 0,1,,M , j = 0 ,1, , N t 0 (35)i, j =ijL; ≥where χ i (t)is the potential at z = η( x,t)(i.e. the free surfacepotential) and therefore satisfies Equations (28) and (33), andsince we are considering the infinite depth basin,~ kw jφ = e , j = 0,1,2, ,N(36)jLwhere k is a representative wavenumber, which we take“based” of the wavenumbers of the wavemaker given inEquation (11). For example k can be taken to be the average ofk 1, kthe two wavenumbers,2. Therefore the nine potentials thatoccur in the discretized Laplace equation become,s j−1 -coord.curve: χ i−1( t)φ χ ( )~j−1i t φ j −1~χ~i+ 1( t)φ j −1sj-coord.curve: Φ i−1,j ( t)Φi, j( t)Φ i + 1,j ( t )~s -coord.curve: χ i−1( t)φ~~j+1 i ( t)φ j + 1 χ i+ 1 t)φ jj+1χ ( + 1So clearly the only "unknowns" are the three potentials alongthe s -coordinate curve, and we have successfully decoupledjthe system along dimensional lines, by "relaxing" the wdependence. Note that, in light of equation (36), the bottomboundary condition (4) is automatically satisfied. Since the~χ i ( t φ j , i = 0,1, , M ; j = 1,2, ,N L ; t ≥ 0 areknown and become incorporated into the vector b &fs , this givesterms { ), }rise to the iterated matrix equation& q + 1 & qA Φ ( t ) = b ( t )&( ) ( ) + b ( t )j= where ( ) 0fswm, (37)for q 0,1 ,...,qMaxb & fs is obtained from equation (36)and all subsequent ( b & ) qfs , q = 0,1 ,..., are obtained from( Φ &) qj . We refer to this equation as the semi-discretizedLaplace equation, since Φ ( ) is discrete over space buti, j tcontinuous over time. In general, two types of iteration schemesare possible, Gauss-Jacobi in which "new values" are used onlyafter a complete iteration has been completed and Gauss-Seidelin which "new values" are used as soon as they are available.Clearly, the Gauss-Jacobi method is fully parallel, but presentlywe employ the Gauss-Seidel method, since it generallyconverges faster.The matrix A is a square sparse tridiagonal matrix ofdimension ( M −1)× ( M −1)and&TΦ t)= Φ ( t),Φ ( t),,Φ ( t)[ ] ,j ( 1, j 2, jM −1,jwhere j = 1,2, , N L ; t ≥ 0 .Note that the matrix A does not depend on the iterationparameter q , and therefore must be inverted only once per timestep.The terms { Φ ( )},where j = 0,1,2,,N ; t ≥ 0 are known0,j tsince they are given by the wavemaker condition (31)&incorporated into the vector b wm (t).The inversion of the completeM − 1)( N −1)× ( M −1)(N −1)system reduces to ( N −1)( LLinversions of the ( M −1)× ( M −1)linear system given by (37).Furthermore, if ∆wis a constant, then the matrix A is identicalfor any s -coordinate curve, and the Gaussian eliminationLL4 Copyright © 2004 by ASME

operations necessary to invert each system, using LUdecomposition,need be performed only once. This effectivelyreduces the dimension of the problem by one. Such reduction incomputation is essential, since the matrix A changes at eachtime-step, and therefore the system must be inverted at eachtime-step.TIME INTEGRATIONEquation (37) must be fully discretized to complete thenumerical model. For T a positive finite real number, we solveour initial-value problem over the time interval [ 0 ,T]. For anatural number N T , we let ∆ t = T / NT, which gives thesequence of time steps t }, where n∆t,n = 0,1,,N .{ n t n = Ti, j ( n i nThe approximation of Φ t ), χ ( t ) and the free surfaceelevation η x i , t ) is denoted by Φ , χ( nni,jniandnη i ,respectively, for i = 0,1,,M , j = 0,1,,N L , n = 0,1,,N T .To solve this linear system we require the vector b &) whichfs ( t ncontains the terms { χ in }, i = 1,2, , M −1,n = 1,2,,NT,which involve integration over time. The differential Equations(28) and (33) are used to generate the approximations { η in } and{ χ in }, respectively for i = 1,2, , M −1,n = 1,2,,NT. Inboth cases, we use a four-point one-sided finite difference to∂Φnapproximate . To calculate∂z( ∂η∂x ) ni and ( ∂χ∂x ) i , we spline{ ηin } and { χ in }, respectively, with a piecewise polynomial ofdegree three, and then calculate it's derivative in closed form.n −1 ∂ηnThis also allows us to evaluate { θ i = tan ( ) i } .∂xSince the fluid in the basin is initially at rest, we have the initialconditions, η 0 = 0 , 0 0χ = 0,and 0,whereiiΦ i , j =i = 1,2, , M −1,j 1,2, , N −1.A fourth-order (explicit)= LRunge-Kutta method gets the additional starting values { η in } ,{ n iχ }, where n = 1,2, 3 and i = 1,2, , M −1.At each of thesetime-steps, the current value for { χ in } and the fully discretizedn{ i,jversion of equation (37) is used to get Φ } n = 1,2,3,i = 1,2,,M −1and j 1,2, , N −1.After this initial phase, for= Ln = 4,5,6,,N , i = 1,2, , M −1,j = 1,2, , N L −1,an Adams fourth-order predictor-correctormethod is used. Using the starting values from the Runge-Kuttamethod, a fourth-order Adams-Bashford method is used as thepredictor to advance { χ in }. Then, using this current value andthe fully discretized version of equation (37), we advancenΦ } . Next a fourth-order Adams-Moulton corrector method{ i,jis used, with these values for { χ in } and { Φ i,j } , to advance{ η in } . This is followed by another application of the Adams-TnMoulton corrector method to advance { χ in } using this“corrected” value of { η in } . See Baddour and Parsons (2003).TEST PROBLEMS AND CONCLUSIONSMonochromatic WavemakerTo test and illustrate the “high order” of nonlinearity of themodel in the monochromatic case, we consider the followingtest problem:Starting from “calm water”, we input a monochromatic sinusoidat x = 0 , with moderately high steepness, S = 0. 06 . Thehorizontal extent of the tank is L = 1.0(m), and we input a totalof 4 waves including a damping zone of 2 wavelengths. Thissets the wavelength of the wavemaker λ = 0.25(m), withcorresponding wavenumber k = 25.1(1/m), and by using thelinear dispersion relation, Equation (12), the angular frequencyω = 15.7(1/s). The discretization parameters are∆x = λ / 25, ∆z= λ/15, with h λ , q = 10 , and= Max∆ t = T / 40 .In Figure (2), we plot the free surface elevation at a fixedlocation in the tank, namely η ( 0.1L)after 38 periods, afterwhich the model became unstable. Note that the instability isnot visible in the “generation” over time at a fixed location inspace. It is the “propagation” over space that is subject to thisinstability. See Baddour and Parsons (2003). Figure (3) givesthe spectrum of this time series using the Matlab “fft”, andemploying a Gaussian window to minimize leakage. In fact allthe spectrum plots we present in this paper utilizes thiswindowing technique. See Briggs and Henson (1995).Note that the model is generating (at least) the first five ordersin the classical Stokes expansion.Bichromatic WavemakerIn this test, we employ a bichromatic wavemaker, again, startingfrom “calm water”, with steepness, S = .015, S 0. 03 . The1 0 2 =horizontal extent of the tank is L = 1.0(m), and the wavelengthsof the wavemaker are λ = 0.25( ) and λ = 0.125( ) , where1 m2 mthe damping zone is 2λ 2 . This sets the wavenumbers of thewavemaker, to be k 1 = 25.1(1/ m), and k 2 = 50.3(1/ m),respectively, and using the linear dispersion relation Equation(12), the corresponding angular frequencies are ω = 15.7(1/ ) ,1 sand ω = 22.3(1/ ) , respectively. The discretization parameters2 sare ∆x = λ / 40, ∆z= λ/20, with h λ , q = 10 , and= Max∆ t = T / 50 . Figure (4) shows the spectrum after 100 periods ofsimulation, relative to the second frequency, ω 2 . There are “atleast 25 interactions” visible in the plot. This is further evidenceof the nonlinear interactions generated by the model. Since thewavemakers are simple linear sinusoids, all these nonlinearinteractions are the result of nonlinearities in the problem; recallthat we are solving the full nonlinear problem. Clearly, it isthese interactions that can give rise to resonance.5 Copyright © 2004 by ASME

Resonance in the Bichromatic Wavemaker CaseWe are interested in showing resonance in the “generation overtime” at a fixed location in the tank, namely at x = 0. 1L.ω1Therefore, for these tests, we define r = , where ω 1 and ω 2ω2are the angular frequencies of the bichromatic wavemakeremployed, and the corresponding wavenumbers can becalculated using the linear dispersion relation, given byEquation (12). The resonant interactions occur when r ≈ 2 ,since the interactions given by ω + ω = ω r 1)and1 2 2 ( +2ω1 −ω2= ω2 (2 r −1), coincide.For the first test we take r = 1. 5 , and show that the solutiondoes not show resonance. Let L = 1.0(m), and∆x = λ / 20, ∆z= λ/10, with h λ , q = 10 , and ∆ t = T / 30 .= MaxWe choose ω 1 = 15. 7 ω 2 = 10. 5 , and we set the steepnessS1 = 0.03 and S 2 = 0. 009 , respectively. Figure (5) shows thespectrum after 50 periods of simulation, relative to the firstfrequency, ω 1 . Note that the components given byω 1 + ω 2 = 26.2 , and 2ω1 − ω 2 = 20. 9 are generated, but ofcourse, do not coincide, and therefore both are significantlysmaller than ω 1 and ω 2 . This represents a non-resonantsolution.In the second test, we set r = 2. 0 , and show that we getresonance. In particular, the setup is identical to the first test,above, except ω 1 = 15. 7 ω 2 = 7. 9 , and we set the steepnessS1 = 0.03 and S 2 = 0. 015 , respectively. To add emphasis to theresonance phenomenon, we give the spectrum of the freesurface elevation of the wavemaker (at x = 0 ) in Figure (6).The spectrum of the free surface elevation generated by themodel at (at x = 0. 1L), is shown in Figure (7). Note that we geta very significant resonant effect atωr= ω1 + ω2= 23.6= 2ω1−ω2. In fact the amplitude of thisresonant wave is slightly greater than the amplitude generatedfor ω 1 , and slightly less than the amplitude generated for ω 2 .Comparing these amplitudes with the corresponding amplitudesfor the wavemaker, given in Figure (6), it is clear that most ofthe “energy” that goes into creating this resonant componentcomes from the energy associated with ω 2 , since the amplitudegenerated by the model for ω 2 is significantly less than theamplitude corresponding to ω 2 in the wavemaker. Thecorresponding comparison for ω 1 shows that the resonant effecton this component is much smaller, but nonzero, nonetheless.We can calculate an “effective steepness” for this resonantwave, using the linear dispersion relation, given by Equation(12). We calculate the wavenumber of this resonant wave to bek r = 56.8(1/m), and measure its amplitude to beA r = 0.00368(m), giving a steepness of S r = 0. 067 , which ismore than twice the maximum steepness of the componentwaves in the wavemaker, S 1 = 0. 03 . This warns us of aninstability, which we call “resonance induced instability” causedby the increase of steepness of the resonant wave, relative to thesteepness of the waves in the wavemaker. In fact, in the nexttest, it was necessary to choose the steepness of one of thewaves in the wavemaker, very small. An initial attempt at thisrun with a larger steepness wavemaker resulted in almostimmediate overflow.The final test shows that resonance decreases, but is not lost, ifwe increase the value of r to r = 3. 0 . As before, the setup isidentical to the first test, above, except ω 1 = 15. 7 ω 2 = 5. 24 ,and we set the steepnesses S1 = 0. 03 and S 2 = 0. 003 ,respectively. See Figure (8) for the spectrum of the free surfaceelevation (at x = 0. 1L), after 50 periods. Note that we get astrong interaction at ω −ω 26. 05 , which does not coincide2 1 2 =with a weaker interaction at ω +ω 20. 94 .1 2 =We conclude with a general observation of the value of usingthe full nonlinear model over the simpler linear model.Although, the higher order components are smaller than the firstorder linear component, for each wave generated by the model,for example, see Figure (4), the nonlinear interaction of thesecomponents can be quite significant. In this paper, we haveconsidered cases in which the discrepancies between the linearand nonlinear models were extreme. Clearly, in any of theseruns, the difference between a linear and our nonlinear model isquite significant. In fact it should be remembered that thisresonance is a purely nonlinear effect, and does not occur in apurely linear solution. Since these resonance effects can be quitedramatic, it is important to be aware of them when solvingocean engineering problems. They can be a nuisance in somecases, and a hazard in other cases, but they should never beblindly ignored. This requires a fully nonlinear approach to themodeling.REFERENCESBaddour, RE and Parsons, W (2003) A comparison of Dirichletand Neumann wavemakers for the numerical generation andpropagation of transient long-crested surface waves, Proc 22ndInt. Conf. on Offshore Mechanics and Arctic Engineering,OMAE, Cancun, Mexico, paper 37281. See also Jour. OMAE,accepted for publication.Briggs, WL and Henson, VE (1995). The DFT, An owner’smanual for the discrete Fourier transform, SIAM, Philadelphia.Debnath, L. (1994) Nonlinear water waves, AcademicPress, New York.Kaplan, W (1993).Advanced calculus, Fourth Edition, Addison-Wesley, Reading, Massachusetts.Parsons, W. and Baddour, R. E. (2002), The generation andpropagation of transient long-crested surface waves using awaveform relaxation method, Proceedings Advances in FluidMechanics 2002, Wessex Institute of Technology Press.Parsons, W. and Baddour, R. E. (2003), A numerical wave tankfor the generation and propagation of bi-chromatic nonlinearlong-crested surface waves, Proceedings Fluid StructureInteraction 2003, Wessex Institute of Technology Press.6 Copyright © 2004 by ASME

Tsai, W. and Yue, D. K. (1996), Computation of nonlinear freesurfaceflows, Annual Review of Fluid Mechanics, Vol. 28,Annual Reviews Inc., Ca.The MathWorks Inc. (1996), MATLAB 5.2 user’s guide,Natick, Mass.zFSz = η ( x,t)s 0j = 0j − 1xWehausen, J.V. and Laitone, E.V. (1960) Surface Waves. InHandbuch der Physik Ed. S. Flugge, Vol. IX, Fluid DynamicsIII, Springer-Verlag, Berlin.w j −1w ji = 0∆ zj∆ zj +1i − 1i i + 1sjjj + 1w j +1ACKNOWLEDGMENTSTo College of the North Atlantic, Newfoundland, Canada, forgiving Wade Parsons a one year sabbatical.To Natural Sciences and Engineering Research Council ofCanada and the National Research Council’s Institute forMarine Dynamics, Newfoundland, Canada, for awarding WadeParsons a Visiting Fellowship.Nzxi −1∆ x i∆ x i + 1x xi i + 1N LFigure 1. Free surface coordinate systemFigure 2. Free surface elevation at x=0.1L as a function oftime in (s), for a monochromatic wavemaker at x=0.0.Figure 3. Spectrum of the time series of the surfaceelevation given in Figure 2 at x=0.1L7 Copyright © 2004 by ASME

Figure 4. Spectrum of free surface elevation at x=0.1L after100 periods of simulation relative toω 2 , with a bichromaticwavemaker: ω = 15.7(1/ ) and ω = 22.3(1/ )1 s2 sFigure 6. Spectrum of the free surface elevation of thewavemaker at x=0 : ω 1 = 15.7(1/ s)and ω 2 = 7.9(1/ s),r = 2.0Figure 5. Spectrum of the free surface elevation at x=0.1Lafter 50 periods of simulation relative to ω 1, for abichromatic wavemaker: ω = 15.7(1/ ) andω = 10.5(1/ ) , r = 1. 52 s1 sFigure 7. Spectrum of the free surface elevation at x=0.1Lgenerated by the model : ω 1 = 15.7(1/ s)and ω 2 = 7.9(1/ s),r = 2.08 Copyright © 2004 by ASME

Figure 8. Spectrum of the free surface elevation at x=0.1Lafter 50 periods of simulation : ω = 15.7(1/ ) andω = 5.24(1/ ) ; r = 3. 02 s1 s9 Copyright © 2004 by ASME

The Generation and Propagation of Deep Water Multichromatic Nonlinear Long-crested Surface WavesW . Parsons 1 and R. E. Baddour 21College of the North Atlantic, Box 1150,Ridge Road, St John’s, NF, A1C 6L8, Canada.2National research Council - CanadaInstitute for Ocean technologySt. John’s, NL, CanadaABSTRACT.A two dimensional rectangular basin containing an incompressibleinviscid homogeneous fluid, initially at rest with a horizontal freesurface of finite extent is considered to generate and propagatenonlinear, long-crested waves. A depth profile for the potential isassumed, giving us a waveform relaxation method, thereby drasticallyreducing the computational cost of solving Laplace’s equation. Amultichromatic stochastic wavemaker employing a Dirichlet typeboundary condition is applied, with the latter following a standard waveenergy spectrum. Laplace's equation is solved using a non-orthogonalboundary fitted curvilinear coordinate system, which follows the freesurface, and the full nonlinear kinematic and dynamic free surfaceboundary conditions are employed. The behavior of this model isstudied using standard signal processing tools and a discussion of theresults is given. In addition, statistical properties of the output of themodel are related to the corresponding statistical properties of the input.KEY WORDS: Nonlinear-waves; regular; irregular waves; nonlinearfree surface conditions; multichromatic; non-orthogonal coordinates;waveform relaxation.INTRODUCTIONA two-dimensional rectangular basin containing an incompressibleinviscid homogeneous fluid initially at rest with a horizontal freesurface of finite extent is considered to generate and propagate longcrestedwaves. On the left vertical boundary a wavemaker is positionedwhile at the right hand side is the radiation boundary. The samenumerical beach is used as in Parsons and Baddour (2002), and a depthprofile for the potential is assumed, giving us a waveform relaxationmethod, and thereby drastically reducing the computational cost ofsolving Laplace's equation. A multichromatic deterministic andstochastic wavemaker employing a Dirichlet type boundary condition isapplied, see Baddour and Parsons (2003) for a comparison of Dirichletand Neumann monochromatic wavemakers and Parsons and Baddour(2003) for the bichromatic Dirichlet wavemaker model. This methodcan easily be extended to the case of a wavemaker utilizing a Neumanntype boundary condition. Laplace's equation is solved using a nonorthogonalboundary fitted curvilinear coordinate system, whichfollows the free surface. This gives a more realistic "continuitycondition", since it involves the entire fluid domain. Also, the fullnonlinear kinematic and dynamic free surface boundary conditions areemployed. Although, these ideas can be extended to finite depth tanks,this requires a rewriting of the coordinate system, so we restrict ourattention to the infinite depth case. The extension of these ideas to thefinite depth case will be presented elsewhere.To place our work in perspective, see the review by Tsai and Yue(1996) "of the recent advances in computations of incompressible flowsinvolving a fully nonlinear free surface". The present work advancesthe field of volume-discretization methods, using finite differences, byapplying the relatively new method of waveform relaxation to reducethe "computational dimension" of the problem. This puts volumediscretizationmethods on a similar footing with boundarydiscretizationmethods, which reduce the dimension of the problem bysolving it on the boundary.The purpose of the present paper is to show that the method presentedcould be applied to the multichromatic case of excitation with bothdeterministic and statistical wavemakers. A complete review of theliterature concerning sea-states reproduction is outside the scope of thepresent objective.PROBLEM FORMULATIONConsider a two-dimensional rectangular basin containing anincompressible inviscid homogenHRXV IOXLG RI FRQVWDQW GHQVLW\ LQ DCartesian coordinate system (x,z), with the origin at the still waterlevel, the positive z-axis pointing upwards. The horizontal extent of thebasin is 0 ≤ x ≤ L , so L > 0 is the length of the tank. The depth of thebasin is infinite. See Figure 1 for a diagram of the numerical wave tankand the rectangular coordinate system configuration. This is an initialvalueproblem, since the surface is initially at rest for time, t < 0 andis disturbed at t = 0 giving rise to surface waves. See Debnath (1994).If Φ ( x,z,t)is the velocity potential and η ( x,t)is the free surfaceelevation, the problem is defined by the following equations, where g isthe acceleration due to gravity. The conservation of mass equation isLaplace's equation:2 2∂ Φ ∂ Φ+ = 0 − ∞ ≤ z ≤η ( x,t),0 ≤ x ≤ L,t ≥ 0 (1)2 2∂x∂zFor t ≥ 0 , the full nonlinear kinematic and dynamic free surface

oundary conditions are given, respectively, by:∂η∂η∂Φ ∂Φ+ − = 0∂t∂x∂x∂z∂Φand 1 ∂Φ 2 ∂Φ+ [( ) + ( )2 ] + gη = 0∂t2 ∂x∂zon z = η( x,t), and∂Φ→ 0 as z → −∞(4)∂zIntroducing the following function:χ( x,t)= Φ(x,η(x,t),t)(5)which represents the potential on the free surface, we have:∂χ∂Φ ∂Φ ∂η= +(6)∂t∂t∂z∂timplying:∂Φ∂χ∂Φ ∂η= −(7)∂t∂t∂z∂tAlso, we have that∂χ∂Φ ∂Φ ∂η= + .(8)∂x∂x∂z∂xSee Wehausen and Laitone (1960).The initial conditions are given by:••( 0 00 xΦ x,0,0)= (1/ ρ ) Φ ( x);Φ(x,0,0)= (1/ ρ)Φ (x); η(x,0)= η ( )where ( 1/ ρ ) Φ 0(x ) and ( 1/ )•ρ Φ 0 ( x)represent the given freesurface impulse and η 0 ( x ) the initial displacement, where the “dot”represents a time derivative. Since we are considering a fluid in whichthe initial velocities are zero and the initial free surface is at rest atz = 0 , we take•(2)(3)Φ 0 ( x)= 0; Φ 0 ( x)= 0; η 0 ( x)= 0 for 0 ≤ x ≤ L (9)Therefore we are left with two lateral boundary conditions for t ≥ 0 atx = 0 and x = L , which we call the LHS lateral boundary conditionand the RHS lateral boundary condition, respectively.The LHS boundary condition involves a wavemaker, which we assumeis the superposition of a finite number of sinusoid waves as follows:( , )∑ Ω η x t = { Ancos( knx− ωnt)− Bnsin( knx−ωnt)},(10)n=1- ∞ < x ≤ 0, - ∞ < t < ∞ ; where Anand Bnis the amplitude, k thenwave number and ωnthe angukar frequency of each wave, forn = 1,2,...,Ω.. Using standard water wave mechanics, the associatedvelocity linear potential is given by⎧ Ang k⎫∑ Ω nz⎪ e sin( k n x −ωnt)⎪ω nΦ(x,z,t)= ⎨⎬(11)n=1 ⎪ Bng knz+ e cos( k − ) ⎪n x ω nt⎪⎩ ω⎪n⎭for - ∞ < x ≤ 0, t ≥ 0 and −∞ < z ≤ η( 0, t), where the dispersionrelation isω k n g(12)n =The free surface elevation is at the wavemaker is then given by∑ Ω (0, t)=n=1{ A cos( ω t)+ B sin( ω t)}η (13)and its associated potential isnnn∑ Ω ⎧ AΦ(0,, ) = ⎨−ngk z Bsin( ) + ngk zz tenωntencos( ωnt)= ⎩⎭ ⎬⎫ (14)ωn 1nωnIt is prudent to point that it is not necessary, but only a convenience andfor completeness to make the assumption of the existence of thesinusoids in Equation (10), since we are only inputting the potentialgiven by Equation (14). Therefore conditions outside the interval [ 0 ,L]have no effect on the solution. This progressive wave at x = 0 andt ≥ 0 is the sole source of the disturbance that gives rise to waterwaves in the initially calm basin.If we assume a wall at x=L so the RHS lateral boundary conditionbecomes∂Φ(L,z,t)= 0 for −∞ < z ≤ η , t ≥ 0(15)∂xwe will get reflections from waves going from the wavemaker at x = 0toward the wall. This distracts from the objective of the present modeland is to be avoided. To absorb these waves we place a "numericalbeach" on the RHS. This is accomplished by including a damping termLQWKHG\QDPLFIUHHVXUIDFHERXQGDU\FRQGLWLRQHTXDWion. See Parsonsand Baddour (2002) for details.THE LAPLACIAN ON THE FREE SURFACE COORDINATESYSTEMWe define the free surface coordinate system (FSCS), ( s , w), wheres ≥ 0 is the arclength along the free surface z = η( x,t)and w ≤ 0 isthe vertical depth to any point in the fluid from the free surface. This isa non-orthogonal moving coordinate system that follows the freesurface. Clearly, the coordinates satisfy the following equations (overthe complete fluid domain),2⎛ ∂ ( , ) ⎞= ∫ x η τ ts 1 + ⎜ ⎟ dτ, 0 ≤ x ≤ L , (16)0 ⎝ ∂τ⎠w = z −η ( x,t),0 ≤ x ≤ L, w ≤ 0.(17)If we define the unit vector ŝ along the arclength, and therefore tangentto the free surface z = η( x,t), thens ˆ = cosθ î + sinθkˆ(18)i ˆ = secθŝ - tanθkˆ(19)where−tan 1 ⎛ ∂η⎞θ = ⎜ ⎟ .⎝ ∂x⎠(20)The arclength along the coordinate curve s is given by2ds =⎛ ∂η⎞1+⎜ ⎟⎝ ∂x⎠dx(21)ords = secθ dx . (22)Clearly we have the following differential operator formulas that allowus to transform easily between rectangular coordinates ( x , z)and freesurface coordinate system ( s , w):n

¡¢£∂ ∂ ∂= secθ- tanθ∂x∂s∂w(23)∂ ∂ ∂= cosθ+ sinθ∂ s ∂ x ∂z(24)∂ ∂= .∂z ∂w(25)The Laplacian for this non-orthogonal coordinate system can beconstructed from the second-order covariant fundamental metric tensorJ⎛1sinθ⎞=⎜⎟⎝sinθ 1 ⎠(26)where its associated second-order contravariant tensor is given by⎛2sectan sec ⎞⎜θ − θ θ⎟J =⎜2tan secsec ⎟⎝−θ θθ ⎠(27)where clearly¤¥ = GHWJ) = cos2θJ .The Laplacian is given by the following tensor equation1 ∂ ⎛ αλ ∂Φ ⎞∆Φ = ⎜ JJ ⎟,α,λ = 1,2 (28)λαJ ∂x⎝ ∂x⎠12where x = s,x = w.See Kaplan (1993). We get∂ ⎛ ∂Φ ∂Φ ⎞∆Φ = secθ[ ⎜secθ− tanθ⎟ +∂s⎝ ∂s∂w⎠(29)∂ ⎛ ∂Φ ∂Φ ⎞+ ⎜−tanθ+ secθ⎟]∂w⎝ ∂s∂w⎠We note that equation (29) can be derived directly, by carrying out thedifferentiations in (1), using the differential operators (23) and (24) andrecalling that the unit vector (19) is non-constant and must bedifferentiated.THE DYNAMIC AND KINEMATIC BOUNDARY CONDITIONSON THE FREE SURFACE COORDINATE SYSTEMUsing equation (8) and equation (20), the kinematic free surfaceboundary condition (KFSBC) given by equation (2), becomes∂η∂χ2 ∂Φ+ tanθ− sec θ = 0 . (30)∂t∂x∂zThe dynamic free surface boundary condition (DFSBC), given byequation (3), can be rewritten using equations (7) and equation (8).After some algebra and using equation (20) and the KFSCB, equation(30), we get the following final form:22∂χ1 ⎡⎛∂χ⎞ 2 ⎛ ∂Φ ⎞ ⎤+ ⎢⎜⎟ − sec θ ⎜ ⎟ ⎥ + gη= 0.(31)∂t2 ⎢⎣⎝ ∂x⎠ ⎝ ∂z⎠ ⎥⎦where all values in (30) and (31) are evaluated on the free surface.Since the free surface coordinate system tracks the free surface, thesevalues are readily obtained. The above derivatives allow for theintegration in time t .SPATIAL APPROXIMATION AND WAVEFORM RELAXATIONIn this paper, we employ a curvilinear grid that follows the free surface.In light of equation (23), this grid is taken to be{( si, w j ),i = 0,1, , M ; j = 0,1, ,N L},(32)wheresi = −si−1 + secθ i 1∆x, and⎡2 ⎤⎢ ⎛ ∂η⎞secθ ⎥i = 1+⎜ ⎟ ,⎢ ⎝ ∂x⎠ ⎥⎣ ⎦where we take s 0 = 0 , and w j = − j ∆ w , where ∆ x = L / M ,so x i = i∆x,and ∆ w = h / N L , where h is the depth to which wesolve Laplace’s equation, taken to be greater than or equal to onewavelength of the wavemaker. We allow the possibility that ∆ w ≠ ∆x;see equation (25). We must solve Laplace's equation, (29) over thisgrid.The semi-discretized approximation of this potential Φ ( si, w j , t)iswritten as Φ i, j ( t)i = 0,1,,M ; j = 0,1,,N L ; t ≥ 0.To implement the wavemaker, we assume the sinusoid (10), for x ≤ 0 ,−∞ < t < ∞ , and associated (linear) velocity potential, (11) anddispersion relation (12). The LHS boundary condition can beimplemented as a Dirichlet or Neumann condition. We refer to these asthe Dirichlet wavemaker and Neumann wavemaker, respectively; seeBaddour and Parsons (2003).In this paper we restrict our attention to the Dirichlet condition, andassure the reader that the extension to the Neumann wavemaker isstright forward. In light of Equation (14) the Dirichlet wavemaker isthen given by:The Dirichlet wavemaker:⎧ A+⎫∑ Ω ⎪−ng⎡ kn( η(0,t)w j )e ⎤sin(ωnt)⎪ωn⎢⎣⎥⎦Φ0,j ( t)= ⎨⎬(33)+= 1 ⎪ B+ ⎡ k ( (0, t)w )nngn η je ⎤cos(ω ) ⎪⎪nt⎩ ω ⎢⎣⎥⎦ ⎪n⎭for j = 0,1,,N L ; t ≥ 0 and where Anand B n is the amplitude, k nthe wave number and ω the angular frequency of each componentnwave, n = 1,2,...,Ω , and we recall the linear dispersion relation (12).There is no limitation in the present model that precludes modeling aphysical wavemaker. The present choice given in (33) is a convenientway that would allow us to compare with the standard linear theory andto the Stokes expansion expression for a nonlinear free surface. SeeBaddour and Parsons (2003) and Parsons and Baddour (2003). Theconstruction of a coordinate system with a moving left boundary is inprogress. Together with (33) a ramping function over two meanperiods is utilized to satisfy the initial conditions of the problem as wellas to minimize any initial transient impulses in the free surfaceelevation and velocities.The horizontal bottom boundary condition, given by (4), will beautomatically satisfied by the relaxation method, as we shall soon see.The boundary condition at the free surface, z = η( x,t),{ Φ t): i = 1,2, , M −1;t 0}i (34)is given by the kinematic and dynamic boundary conditions, (30) and(31), respectively, and involve integration over time. This will bediscussed in the next section.The RHS boundary condition for the numerical beach is incorporated inthe calculation of the free surface potential over the damping zone, andinvolves modification of equation (31) to,∂χ⎡ ∂χ∂Φ ⎤, 0 (≥1 2 2 2+ ( ) sec ( ) + + = 0.2⎢ − θ ⎥ µχ gη(35)∂t⎣ ∂x∂z⎦see Parsons and Baddour (2002). Therefore, the potential should vanisheverywhere at the wall, and as a convenience, we may employ equation(15) to get the RHS boundary condition; that is,Φ M , j ( ( M −1),jL ≥x=t)= Φ ( t),j = 0,1, , N ; t 0. (36)x i

(YHQZLWKDXQLIRUP [WKHFXUYLOLQHar grid will have non-FRQVWDQW VTherefore, to apply finite difference formulas to discretize Laplace’sequation (29), we will need appropriate finite difference operators.Clearly, applying these discrete operators to Laplace’s equation (29), isbest done using computer algebra, and the authors utilized Maple. Werefer to the resulting equation, as the "semi-discretized Laplaceequation". The details are to be given elsewhere.Consider it’s application over any s -coordinate curve for fixed w inthe interior of the domain. That is, we evaluate the semi-discretizedLaplacian at ( s i , w j ) , to calculate Φ i, j ( t)i = 1,2, , M −1;j = 1,2, , N L −1(fixed); t ≥ 0.This will involve nine potentials,namely:s j−1 -coordinate curve: Φ i−1,j−1(t)Φi, j−1(t)Φ i+1,j−1(t)s -coordinate curve: Φ ( ) Φ ( ) Φ i+ 1,j ( t)ji−1,j ti, jts j+1 -coordinate curve: Φ i −1,j+1( t)Φi , j+1( t)Φ i + 1,j+1( t)See Baddour and Parsons (2003). The idea behind waveform relaxation(WR) is to follow the lead of separation of variables methods andassume that the potential can be written as,~Φ i, j ( t)= χi(t)φi,j ( t¦) i = 1,2,, M , j = 0 ,1, , N L; t ≥ 0 (37)where χ i (t)is the potential at z = η( x,t)(i.e. the free surfacepotential) and therefore satisfies equations (30) and (35), and since weare considering the infinite depth basin,~*k w j§φ = e , i = 1,2,, M , j = 0 ,1, , N t 0 (38)i,jL; ≥*where k is the representative wavenumber, which for themonochromatic case we take as the wavenumber of the wavemaker; seeequation (10) and Baddour and Parsons (2003). In the multichromaticcase, however, there are many wavenumbers to choose from. In this*case we use an iterative approach and choose some ’’reasonable’’ k ,[ ]*k ∈ min{ k , k2,...,kΩ} ,max{ k1,k2,...,kΩ}where1 and use equation (38)as a ’’first guess’’ only. We then iterate to converge on the depth profile~φ ( t)= γ ( x , ), i = 1,2,...,M , j = 0 ,1, , N t 0 (39)i, j j i tL; ≥where γ , t)= 1, i = 1,2,..., M , t 0.Therefore the nine potentialsi , 0 ( x i≥that occur in the discretized Laplace equation become,~~s j−1 -coord.curve: χ i−1( t)φ i −1,j−1χ i( t)φ i , j−1χ i+ 1( t)φ i + 1, j−1sj-coord.curve: Φ i−1,j ( t)Φ i, j ( t)Φ i+ 1,j ( t)~s -coord.curve: χ i−1( t)φ ~~i −1,j+1 i ( t)φ j + 1 χ i 1 t)φ ij+1~χ + ( + 1, j+1Clearly the ’’unknowns’’ are the three potentials along the s j -coordinatecurve and we have successfully decoupled the system alongdimensional lines, by ’’relaxing’’ the w -dependence. Note that, in lightof equations (38) and (39), the bottom boundary condition (4) is~automatically satisfied. Since the terms { χ i( t),φ i , j ( t)}, i = 1,2,...,M ,j = 0 ,1, , N L; t ≥ 0 are known and become incorporated into thevector b &fs this semi-discretized Laplace equation gives rise to theiterated matrix equation& q( ) 1 & q &A Φ + j ( t)= ( bfs) ( t)+ bwm(t), (40)for q = 0,1,...,where ( b &) 0fs is obtained from equation (38) and allsubsequent ( b & ) qfs , q = 0,1,...,are obtained ( Φ &) qj . IN general, twotypes of iteration schemes are possible, Gauss-Jacobi in which "newvalues" are used only after a complete iteration has been completed andGauss-Seidel in which "new values" are used as soon as they areavailable. Clearly, the Gauss-Jacobi method is fully parallel, butpresently we employ the Gauss-Seidel method, since it generallyconverges faster. The matrix A is a square sparse tri-diagonal matrixof dimension ( M −1)× ( M −1), and[ Φ ( t),Φ ( t),...,Φ ( t ] TΦ & t ) , where j 1,2, , N −1;j ( ) = 1,j 2, j M −1,j= Lt ≥ 0. Note that the matrix A does not depend on the iterationparameter q , and therefore must be inverted only once per time step.Φ ( ) where j 1,2, , N −1, t ≥ 0.are knownThe terms { }0,j t= Lsince they are given by the wavemaker condition (33) and are&incorporated into the vector b wm (t).The inversion of the complete ( M −1)(N −1)× ( M −1)(N −1)systemreduces to ( N −1)inversions of the ( M −1)× ( M −1)linear systemLgiven by (40). Furthermore, if ∆wis a constant, then the matrix A isidentical for any s -coordinate curve, and the Gaussian eliminationjoperations necessary to invert each system, using LU-decompositionneed be performed only once. This effectively reduces the dimension ofthe problem by one. Concerning speed our method belongs to the classof very efficient iterative methods for solving large systems. Thisbecomes very significant when we move to 3-D. There is a vastliterature on techniques for accelerating these schemes in the area ofiterative matrix methods and are not discussed here.TIME INTEGRATIONEquation (40) must be fully discretized to complete the numericalmodel. For T a positive finite real number, we solve our initial-valueproblem over the time interval [ 0 ,T ]. For a natural number N T , we lett = Tt , whereN Tt n = n∆t, n 0,1,...,NT∆ , which gives the sequence of time steps { }L= . The approximation of Φ ( t ) , χ ) andi, j nLni ( t nn nthe frees surface η ( x i , tn)is denoted by Φ i , j , χ i ( t n ) and η i ,respectively, for i = 0,1,,M , j = 0,1,,N L , n = 0,1,,N T .To solve this linear system we require the vector b &) whichfs ( t ncontains the terms { χ in } , i = 1,2, , M −1,n = 1,2,,NT, whichinvolve integration over time. The differential equations (30) and (35)are used to generate the approximations { η in } and { χ in } , respectivelyfor i = 1,2, , M −1,n = 1,2,,NT. In both cases, we use a fourpointone-sided finite difference to approximate . To calculate∂Φ∂z( ∂η n∂x ) ni and ( ∂χ∂x ) i , we spline { ηin } and { χ in } , respectively, with apiecewise polynomial of degree three, and then calculate it’s derivativen −1 ∂ηnin closed form. This also allows us to evaluate { θ i = tan ( ) i } .∂xSince the fluid in the basin is initially at rest, we have the initialconditions, η i0 = 0 , 0 0χ i = 0,and Φ i , j = 0,where

= Li = 1,2, , M −1,j 1,2, , N −1.A fourth-order (explicit)Runge-Kutta method gets the additional starting values { η in } , { χ in } ,where n = 1,2, 3 and i = 1,2, , M −1.At each of these time-steps,the current value for { χ in } and the fully discretized version ofn{ i,jequation (41) is used to get Φ } n = 1,2,3,i = 1,2,,M −1and= Lj 1,2, , N −1.After this initial phase, forn = 4,5,6,,N , i = 1,2, , M −1,j = 1,2, , N L −1,an Adams fourth-order predictor-correctormethod is used. Using the starting values from the Runge-Kuttamethod, a fourth-order Adams-Bashford method is used as thepredictor to advance { χ in } . Then, using this current value and thefully discretized version of equation (40), we advance Φ } . Next aTn{ i,jfourth-order Adams-Moulton corrector method is used, with thesevalues for { n nχ i } and { Φ i,j } , to advance { η in } . This is followed byanother application of the Adams-Moulton corrector method to advance{ χ in } using this “corrected” value of { η in } . See Baddour and Parsons(2003) for a flow chart.IRREGULAR WAVES AND OCEAN SPECTRAThe wavemaker (33), can be rewritten in terms of a sum of "phaseshifted cosine functions", as follows:∑ Ω Φ0 , j ( t)= n0cos( ωnt+ εn)n=1φ (41)where the coefficients are2 2−1⎛A ⎞φn0 = A n + Bnand the phase angles are ε = ⎜ nn tan ⎟, (42)⎝ Bn⎠n = 1,2,...,Ω..To analyze the irregular wave generated when n ≥ 2 in (41), we recallthe following results; see Goda (2000). An energy spectrumSη( ω n ) corresponding to any irregular time history can be defined,WKH DUHD RI HDFK UHFWDQJOH RI ZLGWK WKH IUHTXHQF\ LQWHUYDO LVproportional to the energy attributed to that frequency band andrepresented by the corresponding single sinusoid wave component.The amplitude of each component sine wave in (41) is therefore givenbyφ = 2Sη ( ω ) δω (m) (43)n0 nIt is also necessary to specify the phase angles ε n , and we will choosethem randomly. This will will give different time histories for different(otherwise identical) runs, but where the energy spectrum will be thesame. Note that if the spectrum is defined at equally spaced frequenciesit will be necessary to interpolate the spectral ordinates at the randomlyspaced frequencies. It was found that the randomly chosen frequenciescase does not reproduce the spectrum very well. We choose to use theuniform frequencies discretization of the spectrum with random phasesε n in equation (41). Note that the period of the simulation T 1 beforerepetition can be increased by choosing larger number of frequencies,i.e. larger Ω . The advantage of this is that the spectrum is wellpreserved as discussed below. In the test problem to follow, weconsider the multichromatic, uniform frequencies, random phases caseand document how the model performs.Open Ocean Conditions:ITTC has adopted the Bretschneider spectrum as the standard waveenergy spectrum to represent the conditions which occur in the openocean. It is called the ITTC two-parameter spectrum, and is defined by( ) A ⎛ − ⎞= ⎜ ⎟5 exp BS B η ω4ω ⎝ ωm²/(rad/s) (44)⎠21 34Hwhere A = 172.752 / s 4691 −4m and B = s4TTClearly, the two parameters that are input to the wavemaker, (41), arethe significant wave height H 1 3 and the average period T , and we2πdefine ω = , (45)Tand using the linear dispersion relation (12),2ωk = , (46)g2πand so λ = . (47)kIn analogy to regular waves, we can define the significant steepness:H13S 1 3 = . (48)λIn all cases, we choose a real number α > 1 , and setω Ω = αWM* ω(49)For the uniform frequencies case mentioned above, we chooseω − 1*1 = = Ω ω ω= Ω α ωω δω= WM. (50)Ω Ω + 1 Ω + 1We then define the period2π ⎛ Ω + 1 ⎞T1 = = ⎜⎟⎟ T , (51)ω1⎝ αWM ⎠and note that the resulting irregular wave will repeat with this period,since this is the lowest frequency in the spectrum and all thefrequencies are harmonically related. If we also choose the simulationtimeT S = N *T 1 , (51)for some natural number N , then all frequencies will be harmonicallyrelated to the "window" [ 0 ,T S ], used to evaluate the discrete Fouriertransform. One big advantage of this setup is that the problem of"leakage" in the discrete Fourier transform can be eliminated for thewavemaker. Clearly, this can never be strictly attained for the outputfrom the model. However, even this inevitable leakage can be reducedby using special types of windows. This will be discussed in the nextsection concerning the tests. Also, see Briggs and Henson (1995).WMTEST PROBLEM AND CONCLUSIONSWe present the following tests designed to show that our model cangenerate and propagate multichromatic waves with reasonableaccuracy. This will be judged by comparing the mean period andsignificant waveheight, and the spectrum of the free surface elevation ata fixed point inside the tank, with that for the wavemaker.In our first test, we take Ω = 29 uniformly distributed frequencies,with α = 3 , and the parameters, T = 4. 0sand H1 3 = 0. 25minWMthe spectrum (44). Therefore, λ = 25.0m, and we set L = 250. 0m,with "10 waves" in the tank (not counting the beach), at steady-state,which gives the required mean wavelength, and significant steepness,S 0.01. The discretization parameters are ∆x = λ 40 ,1 3 =

∆z = λ 20 , and ∆ t = T 200 , and we plot the numericalapproximation of the free surface elevation { ( x,T S ) : 0 ≤ x ≤ L}T S 100T= 10.0T1η after= . See Figure (2). In Figure (2) the aboveparameters are shown in the figure for convenience. In Figures (3) and(6) we plot the free surface elevation of the wavemaker for this fullsimulation and it’s spectrum, respectively. Note that the wavemaker isperiodic with exactly 10 periods during the simulation and the spectrumgiven by (47) is well reproduced, see Figure (6). Recall the discussionabove on leakage of the Fourier transform. From Figure (3) themeasured mean period is 3.98 s with significant wave height 0.244 m.Again, we choose to generate periodic irregular waves to show that themodel was behaving as expected. To generate longer irregularnonperiodic waves we need to take more frequencies.Similarly, we plot the free surface elevationη ( 0.1* L , t ) : n = 1,2,...,generated by the model for this full{ }n N Tsimulation and it’s spectrum, respectively in Figures (4) and (7). Themeasured mean period is found to be 3.26 s with significant waveheight 0.246 m. Unlike the wavemaker which is exactly periodic, themodel is "almost periodic" with 10 "periods", as expected. Moreinterestingly, the spectrum given by (47) is reasonably well reproducedover most of the frequency range, with a couple of exceptions. We get alittle "overshoot" around the peak frequency, and the lower frequencymodes are lacking. This latter deficiency can be alleviated by taking alonger tank and a longer simulation time. The most importantdifference, however is that we get a bimodal spectrum, with theappearance of a second peak in the spectrum. We believe this is due tothe nonlinear generation of higher order modes and their interactions.See Parsons and Baddour (2003), where this is reported for the case ofa monochromatic and bichromatic wavemaker. Again, recalling thediscussion above on leakage, we point out that a "Gaussian" windowwas employed to reduce leakage. See Figure (5).In the second test, we take Ω = 299 uniformly distributed frequencieschosen for the above spectrum. Conditions are set such that thesimulation will not be periodic for 400 seconds. If a longer nonperiodicsimulation is required more frequencies may be chosen. The freesurface elevation at the wavemaker is shown in Figure (8) with theresulting free surface elevation generated by the model at the fixedlocation in the tank at x = 0. 1Lgiven in Figure (9). The spectrum ofthe wave maker is shown in Figure (10) along with the Bretschneiderspectrum. However, we purposely ran the model for more than anintegral number of periods to show the effect of leakage. The spectrumof the free surface elevation generated by the model again atx = 0. 1L is shown after 740 sec in Figure (11). Note that we still getthe "bimodal" spectrum as in the 29 frequencies case shown in Figure(7). Fig (12) shows the free surface elevation elevation after 1.85T 1 .FUTURE WORKWe showed in this paper that the waveform relaxation method that wesuccessfully applied to the generation and propagation ofmonochromatic transient long-crested nonlinear surface gravity waveson a two-dimensional rectangular basin of finite extent and infinitedepth containing an incompressible inviscid homogeneous fluidinitially at rest with a horizontal free surface can be extended to themultichromatic case. This involved the use of the full nonlineardynamic and kinematic boundary conditions, and the solution ofLaplace’s equation on a non-orthogonal curvilinear coordinate systemthat follows the free surface. Despite the fact that this required theLaplace matrix to be inverted at each time-step, we get an efficientaccurate method. Clearly, this efficiency becomes more significant forlarger problems. In particular for short-crested surface waves, ourproblem becomes fully three dimensional and such efficiency becomesessential. In addition, we can employ the dimensional decoupling ofwaveform relaxation methods to reduce the three dimensional problemto two iterated two dimensional problems, Parsons (1999).This dynamic Laplace solver permits us to generate and propagatetransient multichromatic waves, which are then completely absorbed bythe same numerical beach given in Parsons and Baddour (2002).Finally, we stated in the introduction of this paper, that these ideas canbe extended to the finite depth case as well as to physical wavemakersat the LHS boundary. In Parsons and Baddour (2002) we used theappropriate "hyperbolic profile" to facilitate the extension on aCartesian coordinate system for the linear proble. Using methods fromRiemannian geometry, as extension of the free surface coordinatesystem used in this paper is being developed to construct a curvilinearcoordinate system for a domain with finite depth and irregular shapeddynamic bottom and side boundaries.ACKNOWLEDGEMENTSTo College of the North Atlantic, Newfoundland, Canada, for givingWade Parsons a one year sabbatical.To Natural Sciences and Engineering Research Council of Canada andthe National Research Council’s Institute for Ocean Technology,Newfoundland, Canada, for awarding Wade Parsons a VisitingFellowship.REFERENCESBaddour, RE and Parsons, W (2003). ’’A comparison of Dirichlet andNeumann wavemakers for the numerical generation and propagationof transient long-crested surface waves,’’ Proc 22nd Int Conf onOffshore Mechanics and Arctic Engineering, OMAE, Cancun,Mexico, paper 37281. Also in Jour. of OMAE, accepted forpublication.Briggs, WL and Henson, VE (1995). The DFT, An owner’s manual forthe discrete Fourier transform, SIAM, Philadelphia.Debnath, L (1994). Nonlinear water waves, Academic Press, New YorkGoda, Y (2000). Random seas and design of marine structures, WorldScientific, Singapore.Kaplan, W (1993). Advanced calculus, Fourth Edition, Addison-Wesley, Reading, Massachusetts.Parsons, W (1999). “Waveform relaxation methods for Volterraintegro-differential equations,” Ph.D. Thesis, MUN, St. John's, NL.Parsons, W and Baddour, RE (2003). “A numerical wavetank for thegeneration and propagation of bichromatic nonlinear long-crestedsurface waves”, Fluid Structure Interaction, Proc of Second IntConf, Wessex Institute of Technology Press, pp 407-420 .Parsons, W and Baddour, RE (2002). “The generation and propagationof transient long-crested surface waves using a waveform relaxationmethod, Advances in Fluid Mechanics, Proc Int Conf, WessexInstitute of Technology Press, pp 683-693.Tsai, W and Yue, DK (1996). Computation of free-surface flows,Annual Review of Fluid Mechanics, Vol. 28, Annual Reviews Inc.,Ca.Wehausen, JV and Laitone, EV (1960). Encyclopedia of Physics, Vol.IX, Fluid Dynamics III, Springer-Verlag, Berlin.

Figure 1 System configuration and coordinate systemχ( x,t)= Φ(x,η(x,t),t)∂η∂Φ ∂χ= sec 2 θ − tan θ∂t∂z∂x⎡ 22∂χ1 ⎛ ∂χ⎞ ⎛ ∂Φ ⎞⎥ ⎥ ⎤2= −gη− ⎢⎜⎟ − sec θ ⎜ ⎟∂t2 ⎢⎣⎝ ∂x⎠ ⎝ ∂z⎠ ⎦Figure 4 Free surface elevation in (m) at x=0.1L2∂ Φ2∂x2∂ Φ+2∂z= 0Figure 2 Free surface elevation after 100 periodsFigure5 Gaussian window of free surface elevation at x=0.1 LFigure 3 Free surface elevation in (m) of wavemaker at x=0Figure 6 FFT (eta at wavemaker x=0) with Guassian window

Figure 7 FFT (eta at x=0.1L) with Guassian windowFigure 10 Spectrum of surface elevation at x=0Figure 8 Free surface elevation at x=0Figure 11 Spectrum of Surface elevation at x=0.1LFigure 9 Free surface elevation at x=0.1LFigure 12 Free Surface Elevation after 1.85 T 1