6914.tp.indd 1 8/25/09 2:40:31 PMThis page intentionally left blank This page intentionally left blankedited by Young C KimCalifornia State University, Los Angeles, USAHANDBOOK OF COASTAL AND OCEAN ENGINEERINGNE WJ E RSE Y L ONDON SI NGAP ORE BE I J I NG SHANGHAI HONGKONG TAI P E I CHE NNAI World Scientifc6914.tp.indd 2 8/25/09 2:40:35 PMBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required fromthe publisher.ISBN-13 978-981-281-929-1ISBN-10 981-281-929-0Typeset by Stallion PressEmail:enquiries@stallionpress.comAll rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or tobe invented, without written permission from the Publisher.Copyright 2010 by World Scientific Publishing Co. Pte. Ltd.Published byWorld Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office:27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office:57 Shelton Street, Covent Garden, London WC2H 9HEPrinted in Singapore.HANDBOOKOFCOASTALANDOCEANENGINEERINGYHwa - Hdbk of Coastal & Ocean.pmd 10/21/2009, 6:30 PM 1August 25, 2009 18:3 9.75in x 6.5in b684-fm FAPrefaceAlthough coastal and ocean engineering is a very ancient eld with the constructionofPort A-urnearthemouthoftheNilein3,000 BC,signicant advances inthiseld have been made in the last several decades. The rise of interest in this eld canbeseenfromthenumberofattendeesbyacademicsandpractitionersininterna-tional conferences. The rst International Conference on Coastal Engineering washeld in Long Beach, California in 1950 with less than 100 people. When the sameconference was held in San Diego, California, in 2006, over 1000 delegates attended.Inthelast several decades, theworld has seensignicant coastal andocean engi-neering projects, one of which is the Delta Project in the Netherlands. This projectwas designed to shorten and strengthen the total length of coast and dykes washedby the sea by closing o the sea arms in the Delta region. Other noteworthy coastalengineeringprojectsincludetheKansai AirportProjectinJapanand, inrecentyears, theconstructionofmobilebarriers atinletstoregulate tidesintheVeniceLagoonknownastheVeniceProject. Interestincoastal andoceanengineeringhasariseninrecentyearsashumankindexperiencescoastaldisastersthatderivefromcoastal storm, hurricaneandcoastal oodingandseismicactivitiessuchastsunamis, andtheimpactsof climatechangewhichresultinsea-level rise. Thetsunami activityinSumatrainDecember2004aectedcountriesthroughouttheIndianOcean andresultedintheloss ofthousands oflives.Hurricane Katrina inNew Orleans also claimed many lives with property damage exceeding $63 billion.Global warming and sea-level rise will aect shoreline retreats, inundate low coastalareas, damage coastal structures, and accelerate beach erosion. The need for betterunderstandingof our coastal andoceanenvironment has risenconsiderablyinrecent years.This handbook contains a comprehensive compilation of topics that arethe forefronts of many technical advances in ocean waves, coastal and oceanengineering. Itrepresentsthemostcomprehensivereferenceavailableoncoastaland ocean engineering todate, and italso provides the mostup-to-date technicaladvances and latest research ndings on coastal and ocean engineering. More than70 internationally recognized authorities in the eld of coastal and ocean engineeringcontributedpapersontheir areasof expertisetothis handbook. Theseinterna-tional luminaries are from highly respected universities and renowned research andconsulting organizations from all over the world.vAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAvi PrefaceThis handbook provides a comprehensive overviewof shallow-water waves,water-level uctuations, coastal and oshore structures, ports and harbors, coastalsediment processes, environmental problems, sustainable coastal development,coastal hazards, physical modeling, and coastal engineering practice and education.Thisbookisanessential sourceof referenceforprofessionalsandresearchersinthe areas of coastal engineering, ocean engineering, oceanography, meteorology, andcivil engineering, and as a text for graduate students in these elds. This handbookwill be of immediate, practical use to coastal, ocean, civil, geotechnical, and struc-tural engineers, and coastal planners and managers as well as marine biologists andoceanographers. It will also be an excellent source book for educational and teachingpurposes, and would be a good reference book for any technical library.I would like to express my indebtedness to those who guided me and supportedme as a mentor and a colleague throughout my professional life. They are:Professor Robert L. Wiegel, University of California, BerkeleyProfessor Joe W. Johnson, University of California BerkeleyProfessor Robert G. Dean, University of FloridaProfessor Fredric Raichlen, California Institute of TechnologyProfessor Raymond C. Binder, University of Southern CaliforniaProfessor Shoshichiro Nagai, Osaka City UniversityDr Basil Wilson, Science Engineering AssociatesDr Lars Skjelbreia, Science Engineering AssociatesDr Bernard LeMehaute, University of MiamiProfessor Richard Silvester, University of Western AustraliaMr Orville T. Magoon, Coastal Zone FoundationProfessor Billy L. Edge, Texas A&M UniversityProfessor Michael E. McCormick, US Naval AcademyProfessor Yoshimi Goda, Yokohama National University and ECOH CorporationProfessor Philip L.F. Liu, Cornell UniversityProfessor Forrest M. Holly, The University of IowaDr Etienne Mansard, National Research Council, CanadaProfessor J. Richard Weggel, Drexel UniversityMr Ronald M. Noble, Noble Consultants, Inc.Ialsowishtoexpressmyindebtednesstothosewhonurturedmefrommyearlyteen years and changed my course of life. They are:Dr Helen Miller Bailey, East Los Angeles CollegeMr H. Karl Bouvier, Jet Propulsion LaboratoryI extend my gratitude to my wife, Janet, for her constant support, encouragement,patience, and understanding while I was undertaking this task and to my daughter,Susan Calix, for proofreading some of the materials.August 25, 2009 18:3 9.75in x 6.5in b684-fm FAPreface viiFinally, I wish to express my deep appreciation to Ms Kimberley Chua of WorldScientic Publishing Company who gave me invaluable support and encouragementfrom the inception of this handbook to its realization.Young C. KimLos Angeles, CaliforniaJanuary 2008August 25, 2009 18:3 9.75in x 6.5in b684-fm FAThis page intentionally left blank This page intentionally left blankAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContentsPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiSection1:Shallow-Water Waves1. Wave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1R. G. Dean and T. L. Walton2. Wavemaker Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25R. T. Hudspeth and R. B. Guenther3. Analyses by the Melnikov Method of Damped ParametricallyExcited Cross Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57R. B. Guenther and R. T. Hudspeth4. Random Wave Breaking and Nonlinearity Evolution Acrossthe Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Y. Goda5. Aeration and Bubbles in the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115N. Mori, S. Kakuno and D. T. Cox6. Freak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131N. Mori7. Short-Term Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151A. KimuraSection2:Water-LevelFluctuations8. Generation and Prediction of Seiches in Rotterdam Harbor Basins . . . . . 179M. P. C. de Jong and J. A. Battjes9. Seiches and Harbor Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193A. B. RabinovichixAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAx Contents10. Finite Dierence Model for Practical Simulation ofDistant Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237S. B. YoonSection3:CoastalStructures11. Tsunami-Induced Forces on Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261I. Nistor, D. Palermo, Y. Nouri, T. Murty and M. Saatcioglu12. Nonconventional Wave Damping Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 287H. Oumeraci13. Wave Interaction with Breakwaters Including Perforated Walls . . . . . . . . 317K.-D. Suh14. Prediction of Overtopping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341J. van der Meer, T. Pullen, W. Allsop, T. Bruce,H. Sch uttrumpf and A. Kortenhaus15. Wave Run-Up and Wave Overtopping at Armored RubbleSlopes and Mounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383H. Sch uttrumpf, J. van der Meer, A. Kortenhaus,T. Bruce and L. Franco16. Wave Overtopping at Vertical and Steep Structures . . . . . . . . . . . . . . . . . . . 411T. Bruce, J. van der Meer, T. Pullen and W. Allsop17. Surf Parameters for the Design of Coastal Structures . . . . . . . . . . . . . . . . . . 441D. H. Yoo18. Development of Caisson Breakwater Design Based onFailure Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455S. Takahashi19. Design of Alternative Revetments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479K. Pilarczyk20. Remarks on Coastal Stabilization and Alternative Solutions . . . . . . . . . . . 521K. Pilarczyk21. Geotextile Sand Containers for Shore Protection. . . . . . . . . . . . . . . . . . . . . . . 553H. Oumeraci and J. Recio22. Low Crested Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601A. Lamberti and B. Zanuttigh23. Hydrodynamic Behavior of Net Cages in the Open Sea . . . . . . . . . . . . . . . . 633Y.-C LiAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContents xiSection 4:Oshore Structures24. State of Oshore Structure Development and Design Challenges . . . . . . . 667S. ChakrabartiSection5:PortsandHarbors25. Computer Modeling for Harbor Planning and Design . . . . . . . . . . . . . . . . . . 695J.-J. Lee and X. Xing26. Prediction of Squat for Underkeel Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . 723M. J. Briggs, M. Vantorre, K. Uliczka and P. DebaillonSection6:CoastalSediment Processes27. Wave-Induced Resuspension of Fine Sediment . . . . . . . . . . . . . . . . . . . . . . . . . 775M. Jain and A. J. Metha28. Suspended Sand and Bedload Transport on Beaches . . . . . . . . . . . . . . . . . . . 807N. Kobayashi,A. Payo and B. D. Johnson29. Headland-Bay Beaches for Recreation and Shore Protection . . . . . . . . . . . 825J. R.-C. Hsu, M. M.-J. Yu, F.-C. Lee and R. Silvester30. Beach Nourishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843R. G. Dean and J. D. Rosati31. Engineering of Tidal Inlets and Morphologic Consequences . . . . . . . . . . . . 867N. C. KrausSection 7:EnvironmentalProblems32. Water and Nutrients Flow in the Enclosed Bays . . . . . . . . . . . . . . . . . . . . . . . 901Y. Koibuchiand M. IsobeSection 8:SustainableCoastalDevelopment33. Socioeconomic and Environmental Risk in Coastaland Ocean Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923M. A. Losada, A. Baquerizo, M. Ortega-S anchez,J. M. Santiago and E. Sanchez-Badorrey34. Utilization of the Coastal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953H.-H. HwungAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxii ContentsSection9:CoastalHazards35. Ocean Wave Climates: Trends and Variations Due toEarths Changing Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971P. D. Komar, J. C. Allan and P. Ruggiero36. Sea Level Rise: Major Implications to Coastal Engineeringand Coastal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997L. Ewing37. Sea Level Rise and Coastal Erosion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023M. J. F. Stive, R. Ranasinghe and P. J. Cowell38. Coastal Flooding: Analysis and Assessment of Risk. . . . . . . . . . . . . . . . . . . . 1039P. Prinos and P. GaliatsatouSection 10:PhysicalModeling39. Physical Modeling of Tsunami Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073M. J. Briggs, H. Yeh and D. T. Cox40. Laboratory Simulation of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107E. P. D. Mansard and M. D. MilesSection 11:CoastalEngineering PracticeandEducation41. Perspective on Coastal Engineering Practice and Education . . . . . . . . . . . 1135J. W. KamphuisAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributorsJonathanC.AllanCoastal Field OceOregon Department of Geology and Mineral IndustriesNewport, Oregonjonathan.allan@dogami.state.or.usWilliam AllsopTechnical DirectorHR WallingfordWallingford, UKw.allsop@hrwallingford.co.ukElena SanchezBadorreyAssociate ProfessorCEAMA Universidad de GranadaGranada, Spainelenasb@ugr.esAsuncion BaquerizoAssociate ProfessorCEAMA Universidad de GranadaGranada, SpainJurjenA.BattjesEmeritus ProfessorEnvironmental Fluid Mechanics SectionDelft University of TechnologyDelft, The Netherlandsj.a.battjes@tudelft.nlMichaelJ.BriggsResearch Hydraulic EngineerCoastal and Hydraulics LaboratoryU.S. Army Engineer Research and Development CenterVicksburg, Mississippimichael.j.briggs@erdc.usace.army.milxiiiAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxiv ContributorsTomBruceSchool of Engineering and ElectronicsUniversity of EdinburghEdinburgh, UKtom.bruce@ed.ac.ukSubrataChakrabartiJoint Professor, Civil and Mechanical EngineeringUniversity of Illinois at ChicagoChicago, Illinoischakrab@aol.comPeterJ.CowellAssociate ProfessorSchool of GeosciencesInstitute of Marine ScienceUniversity of SydneySydney, AustraliaDanielT.CoxProfessorSchool of Civil and Construction EngineeringOregon State UniversityCorvallis, Oregondtc@oregonstate.eduRobert G.DeanGraduate Research Professor of Coastal Engineering, EmeritusDepartment of Civil and Coastal EngineeringUniversity of FloridaGainesville, Floridadean@coastal.u.eduPierreDebaillonResearch Hydraulic EngineerCentre dEtudes Techniques Maritimes Et Fluviales (CETMEF)Compiegne, Francepierre.debaillon@equipement.gouv.frMartijnP.C.deJongFormerly at Environmental Fluid Mechanics SectionDelft University of TechnologyPresently at Delft HydraulicsDelft, The NetherlandsAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributors xvLesley EwingSenior Coastal EngineerCalifornia Coastal CommissionSan Francisco, Californialewing@coastal.ca.govLeopoldo FrancoProfessor of Coastal EngineeringDepartment of Civil EngineeringUniversity of Rome 3Rome, Italyleopoldo.franco@uniroma3.itPanagiotaGaliatsatouResearch AssociateDepartment of Civil EngineeringAristotle University of ThessalonikiThessaloniki, Greecepgaliats@civil.auth.grYoshimi GodaProfessor EmeritusYokohama National UniversityAdviser to ECHO CorporationTokyo, Japangoda@ecoh.co.jpRonaldB.GuentherProfessor EmeritusDepartment of MathematicsOregon State UniversityCorvallis, Oregonguenther@math.orst.eduJohnRong-ChungHsuProfessorDepartment of Marine Environment and EngineeringNational Sun Yat-sen UniversityKaohsiung, TaiwanHonorary Research FellowSchool of Civil and Resource EngineeringUniversity of Western AustraliaNedland, Australiajrchsu@mail.nsysu.edu.twAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxvi ContributorsRobert T.HudspethProfessor and Director, EmeritusCoastal and Ocean Engineering ProgramOregon State UniversityCorvallis, Oregonrobert.hudspeth@orst.eduHwung-HwengHwungProfessor of Hydraulic and Ocean EngineeringDirector of Tainan Hydraulics LaboratoryDepartment of Hydraulic and Ocean EngineeringNational Cheng Kung UniversityTainan, Taiwanhhhwung@mail.ncku.edu.twMasahikoIsobeProfessor and Special Adviser to the PresidentDepartment of Sociocultural Environmental StudiesGraduate School of Frontier SciencesThe University of TokyoChiba, Japanisobe@k.u-tokyo.ac.jpMamtaJainCoastal EngineerHalcrow Inc.Tampa, Floridamjain@halcrow.comBradleyD.JohnsonCoastal and Hydraulics LaboratoryU.S. Army Engineer Research and Development CenterVicksburg, MississippiShohachiKakunoProfessor and Vice PresidentDepartment of Civil EngineeringOsaka City UniversityOsaka, Japankakuno@ado.osaka-cu.ac.jpJ.William KamphuisProfessor of Civil Engineering, EmeritusDepartment of Civil EngineeringQueens UniversityKingston, Ontario, Canadakamphuis@civil.queensu.caAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributors xviiAkiraKimuraProfessorDepartment of Social Systems EngineeringTottori UniversityTottori, Japankimura@sse.tottori-u.ac.jpNobuhisa KobayashiProfessor and DirectorCenter for Applied Coastal ResearchUniversity of DelawareNewark, Delawarenk@coastal.udel.eduYukioKoibuchiAssistant ProfessorDepartment of Sociocultural Environmental StudiesGraduate School of Frontier SciencesThe University of TokyoChiba, Japankoi@k.u-tokyo.ac.jpPaulD.KomarProfessor of OceanographyCollege of Oceanic and Atmospheric SciencesOregon State UniversityCorvallis, Oregonpkoma@coas.oregonstate.eduAndreasKortenhausLeichtweiss-Institute for HydraulicsTechnical University of BraunschweigBraunschweig, Germanykortenhaus@tu-bs.deNicholasC.KrausSenior ScientistCoastal and Hydraulics LaboratoryU.S. Army Engineer Research and Development CenterVicksburg, Mississippinicholas.c.kraus@erdc.usace.army.milAlbertoLambertiProfessorDepartment of Civil EngineeringUniversity of BolognaBologna, Italyalberto.lamberti@unibo.itAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxviii ContributorsFang-ChunLeeDepartment of Marine Environment and EngineeringNational Sun Yat-sen UniversityKaohsiung, TaiwanJiin-JenLeeProfessor of Civil and Environmental EngineeringSonny Astani Department of Civil andEnvironmental EngineeringUniversity of Southern CaliforniaLos Angeles, Californiajjlee@usc.eduYu-ChengLiProfessorSchool of Civil EngineeringDalian University of TechnologyDalian, Chinaliyuch@dlut.edu.cnMiguel A.LosadaProfessorResearch Group on Environmental Flux DynamicsCEAMA Universidad de GranadaGranada, Spainmlosada@ugr.esEtienneP.D.MansardExecutive DirectorCanadian Hydraulics CentreNational Research Council CanadaOttawa, Ontario, Canadaetienne.mansard@nrc-cnrc.gc.caAshish J.MehtaProfessor of Coastal EngineeringDepartment of Civil and Coastal EngineeringUniversity of FloridaGainesville, Floridamehta@coastal.u.eduMichaelD.MilesCanadian Hydraulics CentreNational Research Council CanadaOttawa, Ontario, CanadaAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributors xixNobuhitoMoriAssociate ProfessorDisaster Prevention Research InstituteKyoto UniversityKyoto, Japannobuhito.mori@hy5.ecs.kyoto-u.ac.jpTadS.MurtyAdjunct ProfessorDepartment of Civil EngineeringUniversity of OttawaOttawa, Ontario, Canadatadmurty@gmail.comIoanNistorAssistant ProfessorDepartment of Civil EngineeringUniversity of OttawaOttawa, Ontario, Canadainistor@uottawa.caYounesNouriDepartment of Civil EngineeringUniversity of OttawaOttawa, Ontario, CanadaMiquelOrtegaAssociate ProfessorCEAMA Universidad de GranadaGranada, Spainmiguelos@ugr.esHocineOumeraciUniversity ProfessorLeichtweiss-Institute for Hydraulic Engineeringand Water ResourcesTechnical University of BraunschweigBraunschweig, Germanyh.oumeraci@tu-bs.deDanPalermoAssistant ProfessorDepartment of Civil EngineeringUniversity of OttawaOttawa, Ontario, Canadapalermo@eng.uottawa.caAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxx ContributorsAndresPayoGraduate School of Science and TechnologyUniversity of KumamotoKumamoto, JapanKrystianW.Pilarczyk(Former) Manager, Research and DevelopmentHydraulic Engineering InstituteRykswaterstaatDelft, The NetherlandsHYDROpil ConsultancyZoetermeer, The Netherlandskrystian.pilarczyk@gmail.comPanayotisPrinosProfessor of Hydraulic EngineeringDepartment of Civil EngineeringAristotle University of ThessalonikiThessaloniki, Greeceprinosp@civil.auth.grTimPullenSenior EngineerHR WallingfordWallingford, UKtap@hrwallingford.co.ukAlexanderB.RabinovichP.P. Shirshov Institute of OceanologyRussian Academy of SciencesMoscow, RussiaDepartment of Fisheries and OceansInstitute of Ocean SciencesSidney, B.C., Canadaabr@iki.rssi.ru.RoshankaRanasingheAssociate ProfessorUNESCO-IHE/Delft University of TechnologyDelft, The Netherlandsr.ranasinghe@unesco-ihe.orgJuanRecioLeichweiss-Institute for Hydraulic Engineeringand Water ResourcesTechnical University of BraunschweigBraunschweig, GermanyAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributors xxiJulieD.RosatiResearch Hydraulic EngineerCoastal and Hydraulics LaboratoryU.S. Army Corps of EngineersMobile, Alabamajulie.d.rosati@erdc.usace.army.milPeterRuggieroAssistant ProfessorDepartment of GeosciencesOregon State UniversityCorvallis, Oregonruggierp@science.oregonstate.eduMuratSaatciogluProfessorDepartment of Civil EngineeringUniversity of OttawaOttawa, Ontario, Canadamurat.saatcioglu@uottawa.caJuanM.SantiagoAssociate ProfessorCEAMA Universidad de GranadaGranada, Spainsanti@ugr.esHolger SchuttrumpfProfessor and DirectorInstitute of Hydraulic Engineeringand Water Resources ManagementRWTH Aachen UniversityAaachen, Germanyschuettiumpf@iww.rwth-aachen.deRichardSilvesterProfessor EmeritusSchool of Civil and Resource EngineeringUniversity of Western AustraliaNedland, Australiarsilvest@bigpond.net.auAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAxxii ContributorsMarcelJ.F.StiveProfessor and DirectorDelft Water Research CentreDepartment of Hydraulic EngineeringDelft University of TechnologyDelft, The Netherlandsm.j.f.stive@tudelft.nlKyung-DuckSuhProfessorDepartment of Civil and Environmental EngineeringSeoul National UniversitySeoul, Koreakdsuh@snu.ac.krShigeoTakahashiExecutive Researcher and DirectorTsunami Research CenterPort and Airport Research InstituteYokosuka, Japantakahashi s@pari.go.jpKlemens UliczkaResearch Hydraulic EngineerFederal Waterways Engineering and Research Institute (BAW)Hamburg, Germanyklemens.uliczka@baw.deJentsjevanderMeerPrincipalVan der Meer ConsultingHeerenveen, The Netherlandsjm@vandermeerconsulting.nlMarcVantorreProfessorDivision of Maritime TechnologyGhent UniversityGhent, Belgiummarc.vantorre@ugent.beAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAContributors xxiiiToddL.WaltonDirectorBeaches and Shore Resource CenterFlorida State UniversityTallahassee, Floridatwalton@mailer.fsu.eduXiuyingXingGraduate Research AssistantSonny Astani Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos Angeles, CaliforniaHarryYehProfessorSchool of Civil and Construction EngineeringOregon State UniversityCorvallis, Oregonharry@engr.orst.eduDongHoon YooProfessorDepartment of Civil EngineeringAjou UniversitySuwon, Koreadhyoo@ajou.ac.krSungBumYoonProfessorDepartment of Civil and Environmental EngineeringHanyang UniversityAnsan, Koreasbyoon@hanyang.ac.krMelissa Meng-Jiuan YuDepartment of Marine Environment and EngineeringNational Sun Yat-sen UniversityKaohsiung, TaiwanBarbaraZanuttighAssistant ProfessorDepartment of Civil EngineeringUniversity of BolognaBologna, Italybarbara.zanuttigh@mail.ing.unibo.itAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FAThis page intentionally left blank This page intentionally left blankAugust 25, 2009 18:3 9.75in x 6.5in b684-fm FATheEditorYoungC. Kim, PhD, iscurrentlyaProfessorof Civil Engineering, Emeritusat California State University, Los Angeles. Other academic positions heldbyhimincludeaVisitingScholarof Coastal EngineeringattheUniversityof Cali-fornia, Berkeley (1971); a NATO Senior Fellow in Science at the Delft University ofTechnology intheNetherlands(1975); andaVisitingScientistattheOsaka CityUniversityfortheNational ScienceFoundations USJapanCooperativeScienceProgram (1976). For more than a decade, he served as Chair of the Department ofCivil Engineering and recently he was Associate Dean of the College of Engineering.For his dedicated teaching and outstanding professional activities, he was awardedthe university-wide Outstanding Professor Award in 1994.Dr Kim was a consultant to the US Naval Civil Engineering Laboratory in PortHueneme and became a resident consultant to the Science Engineering AssociateswhereheinvestigatedwaveforcesontheHoward-Dorisplatformstructure, nowbeing placed in Ninian Field, North Sea.DrKimisthepast ChairoftheExecutive CommitteeoftheWaterway, Port,Coastal and Ocean Division of the American Society of Civil Engineering (ASCE).Recently, heservedasChairof theNominatingCommitteeof theInternationalAssociation of Hydraulic Engineering and Research (IAHR). Since 1998, he servedon the International Board of Directors of the Pacic Congress on Marine ScienceandTechnology(PACON). HecurrentlyservesasthePresidentof PACON. DrKim has been involved in organizing 10 national and international conferences, hasauthored three books, and has published 52 technical papers in various engineeringjournals.xxvThis page intentionally left blank This page intentionally left blankJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAChapter1WaveSetupRobert G. DeanDepartment of Civil and Coastal EngineeringUniversity of Florida, Gainesville,FL, USAdean@coastal.u.eduTodd L. WaltonBeaches and Shores Resource CenterFlorida State University, Tallahassee,FL, USAtwalton@fsu.eduWave setup is the increase of water level within the surf zone due to the transfer ofwave-relatedmomentumtothewatercolumnduringwave-breaking.Wavesetuphas beeninvestigatedtheoreticallyandunder laboratoryandeldconditions,anditincludes bothstaticanddynamiccomponents. Engineeringapplicationsinclude a signicant ooding component due to severe storms and oscillating waterlevels that canincrease hazards to recreationalbeachgoers and can contribute toundesirable oscillations of both constructed and natural systems including harborsand moored ships. This chapter provides a review of the knowledge regarding wavesetup and presents preliminary recommendations for design. It will be shown thatwavesetupis not adequatelyunderstoodquantitativelyfor engineeringdesignpurposes.1.1. IntroductionWave setup was brought to the attention of coastal engineers and scientists in the1960s (i.e., see Ref. 1, p. 245) afterthe initial theoretic developments of Longuet-Higgins2and Longuet-Higgins and Stewart3,4along with limited eld observationsandlaboratorystudiessupportedtheexistenceof wavesetup, themagnitudeofwhichwasobservedtobeintheorderof1020% oftheincidentwaveheight. Itwasnotedinearlyeldobservationsthatwaterlevelsonthebeachwerehigherthanthose recorded by a tidegauge attheendof apiersuggesting awave setupphysically forced by wind waves and swell.1July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA2 R.G. DeanandT.L.WaltonFig. 1.1. Denitionsketch. Energyandmomentumaretransferredfromwindstowavesinthegeneratingarea. The wavesconveyenergyandmomentumto the surf zone where the wavesbreak.Uponbreaking, theenergyisdissipatedandthemomentumistransferredtothewater columnresultinginlongshoreandonshoreforcesexertedonthewatercolumn.Wave setup is the additional water level that is due to the transfer ofwave-relatedmomentumtothewatercolumnduringthewave-breakingprocess.As waves approach theshoreline, they convey both energy and momentum inthewave direction. Upon breaking, the wave energy is dissipated, as is evident from theturbulence generated; however, momentum is never dissipated but rather is trans-ferred to the water column resulting in a slope of the water surface to balance theonshore component of the ux of momentum (see Fig. 1.1). If waves are irregular,in addition to a steady wave setup, the setup includes a dynamic component thatoscillateswiththewave groupperiodandtheremaybeaweak resonance withinthe nearshore amplifying this oscillating component. These have been termed infra-gravity waves and are more dominant for narrow banded spectra both in frequencyand in directional spreading. The oscillatory component is denoted dynamic wavesetup in this chapter.This chapter discusses the signicanceof wavesetuptocoastal engineeringdesign, provides a review of the classical linear wave theory of wave setup, reviewsresults from laboratory and eld studies, summarizes results and recommends pre-liminarydesignapproaches forthestaticcomponent.Toprovidealookahead,wewillseethatthephenomenonofwave setupisnotyetadequately understoodfor satisfactory engineering calculations and that the eects of prole slope are veryJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 3signicant. The interested reader is also referred to an earlier review article on wavesetup by Holman.51.2. Engineering Signicanceof WaveSetupWave setup (both static and dynamic components) is relevant to a number of engi-neering applications. The contributions of wave setup under extreme storm eventscan be substantial, adding several feet to the elevated water levels. The interactionof wave setup withvegetation diers from windsurge and thus it is important todierentiatethetwo components,for exampleinascertaining thebenetsof wet-landsinreducingwave setup. Finally,theoscillatingcomponentofwavesetupisrelevanttobeachsafetyinsomelocationsandtomanynatural andconstructedcoastal systems that have the capability to resonate including harbors and mooredships.1.3. Terminology andRelatedConsiderationsStandard terminology denes the water level in the absence of wave eects as stillwaterlevel,whereas wave setupwillcauseadeparture fromthestillwaterlevelandthiswaterlevelincludingtheeectsofthewavesisthemeanwaterlevel.Asimplied, themeanwaterlevelisdeterminedastheaverageoftheuctuatingwaterlevel overasuitabletimeframeusuallytakenasanumberofmultiplesofthe short wave period, say the spectral peak. In considering wave setup, often thelocation of interest is that of the maximum wave setup at the shoreline. This raisesthequestionof whetherwavesetupisdenedatelevationsabovethemaximumrundown, say on the beach face where the wateris present over only a portion ofthewaveperiod. Sincewavesetupisdenedasthemeanwaterlevel, overwhatperiodshouldthewatersurfacebeaveraged onthebeachfacewhichiswettedover only a portion of the wave period? If the time average is over only the portionof the period that water is present, in the upper limit, the maximum setup will bethe maximum runup. For purposes here, wave setup will usually be dened only forconditions where water is present over a full wave period.When calculating wave runup on a structure such as a levee or revetment, thequestion arises whether it is appropriate to rst calculate wave setup and then addthewave runupwhichisusuallyempiricallybasedonmodelresults.Inthemorerecent empirical results (e.g., the TAW method, see Ref. 6), the runup is expressedas a proportion of the signicant wave height at the base of the steeper slope (e.g.,atarevetmentorlevee).Thewaverunupdeterminedinthemodelonwhichthemethod was based generally included some wave setup (or setdown) seaward of thetoe of the slope and included wave setup landward of the toe of the slope. Thus, inthe application of interest, the most appropriate approach is to calculate and includewave setup at the toe of the slope; however, recognizing that the measured landwardrunupincludessetup,noadditionalsetupshouldbeaddedexplicitlylandward ofthe toe of the slope.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA4 R.G. DeanandT.L.Walton1.4. ABrief ReviewofWaveSetupMechanics1.4.1. StaticwavesetupformonochromaticwavesLonguet-Higgins2and Longuet-Higgins and Stewart3,4were the rst to formalize thenotion of wave momentum ux and its relationship to wave setup. The momentumux,Sij, is a second-order tensor given bySxx = E

n(cos2 + 1) 12

,Syy = E

n(sin2 + 1) 12

,Sxy = Syx =E2sin 2,(1.1)whereEis the wave energy density, n is the ratio of wave group velocity to wavecelerityandistheanglebetweenthewavedirectionandthex-axis. ThetermSxyreads the uxper unit width, inthe x-direction, of the y-component ofmomentum, etc.The steady-state equations of motion obtained by time averaging over the shortwave period are, including the eects of wind stress and bottom friction: (wind +wave)x= 1g(h + )

Sxxx+Sxyysx +bx

and (wind +wave)y= 1g(h +)

Syyy+Syxxsy +by

.(1.2)In the above, wind is the surge component due to the wind stress, wave is the wavesetup, is the mass density of water,g is the gravitational constant,h is the localwaterdepth, sxandbxarethesurfaceandbottomshearstresses, respectively,and similarly for they-direction. The coordinate direction,x is oriented shorewardand a right-handed coordinate system is considered.The most simple solution is for waves propagating directly shoreward (Sxy = 0)in which the surface and bottom stresses are considered negligible, and all variablesare considered uniform in they-direction. The resulting equation iswavex= 1g(h +)

Sxxx

. (1.3)To proceed, we need to determine a boundary condition forwave,aat the seawardendof thesurf zone. Longuet-Higgins7hasshownthatintheabsenceof energydissipation, the following general relationship for applies = C 12g(u2w2)=0, (1.4)aForpurposesofconvenience,hereafterthesubscriptonwavewillbeomittedsuchthatthewavesetupissimplydenotedas.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 5whereu2andw2represent the time averages of the square of the rst-order hori-zontal and vertical wave velocities evaluated at the mean water surface, respectively.Equation (1.4) is a type of a Bernoulli equation for unsteady ows which, when eval-uated at the break point and considering no wave setup in deep water to evaluatethe constant,C = 0, the setup is negative (setdown) and given byb = H2bkb8 sinh 2kbhb, (1.5)whereHbis the breaking wave height andkbis the wave number at breaking. Forshallow water conditions and depthlimited breaking (Hb=(hb + b)), Eq. (1.5)yieldsb = Hb16. (1.6)As an example, for a value of 0.78, the wave setdown is approximately 5% of thebreaking wave height.With theseaward boundary condition now established, forthe case of shallowwater wave-breaking and the consideration of depth limited breaking across the surfzone, the wave setup is = Hb16+32/8(1 + (32/8))(hbh) . (1.7)Itis noted that inthe above equation, the bottomshear stress has been taken aszeroandthatashorewarddirectedbottomshearstressonthewatercolumnaswould occur due to undertow would increase the wave setup. As examples, the ratioof wave setup to breaking height at the still water line (h = 0) and at the locationof maximum wave setup ( = h) for a value of 0.78 areF0|=0.78 (h = 0, = 0.78)Hb=

516(1 + (32/8))

=0.78= 0.198 (1.8)andFmax|=0.78 (h = , = 0.78)Hb= F0

1 + 328

=0.78=516

=0.78= 0.244 .(1.9)It is seen that the wave setup is strongly dependent on the value of the breaking ratio which will be shown to decrease with decreasing beach slope. Figure 1.2 presentsthe ratios, F0 and Fmax versus . It is useful to relate in an approximate manner tobeach slope. Although there is not a one-to-one correspondence, Fig. 1.3 is based onthe Dally et al.8wave-breaking model and provides an approximate correspondencebetweenuniformproleslopeandtheassociatedvalue. ItisevidentthattheDallyetal.modelprovidesreasonablevaluesforsmallerbeachslopes(saylessthan about 0.06), but the values are too large for steeper slopes.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA6 R.G. DeanandT.L.Walton0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.4KappaFoFmaxFoandFmaxFig.1.2. ValuesofF0andFmaxversuswave-breakingindex,(kappa).0.00 0.02 0.04 0.06 0.08 0.10Profile Slope0.00.51.01.52.0KappaFig. 1.3. Relationshipbetweenproleslopeand(kappa)value. BasedonDallyet al.8wave-breakingmodel.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 71.4.2. EectsofwavenonlinearityWave nonlinearitydepends onthe followingparameters: H/L0andh/L0. Thenonlinearityisexhibitedinthewaveprolebypeakedcrestsandattertroughsandincreases withwave height andshallowwater. Somewhat surprisingly, themomentum uxes in shallow water are less for nonlinear waves than for linear wavesof the same height. This is primarily because the momentum uxes are proportionaltowave energy (Eq.(1.1))andthewave energy isproportional totheroot-meansquare of the water surface displacement that is less for nonlinear waves with longtroughs and peaked wave crests. Figure 1.4 presents the ratio of nonlinear to linearmomentum uxes as determined by Stream Function wave theory.911The reasonthatthequantitiesfornonlinearwavesaregreaterindeepwaterthanforlinearwaves is that the nonlinear calculations extend up to the actual free surface whereasthe linear quantities only extend up to the mean free surface.1.4.3. RoleofwavedirectionalityEquation (1.1) demonstrates that for a given wave height, the maximum shorewarduxofonshoremomentumoccursfornormallyincidentwaves(=0). Thusasexpected for directional waves, theSxx term is reduced. However, this reduction isrelatively smallascanbedemonstratedbyconsidering abreaking wave directionof 30 relative to a beach normal (this represents a reasonably large wave obliquity10-3.00010-2.00010-1.000100.000101.000 2 3 4 5 6 2 3 4 5 6 2 3 4 5 5 2 3 4 5 6h/Lo0.00.20.40.60.81.01.2Ratio of Nonlinear to Linear Momentum FluxH/Hb=0.25H/Hb=0.75H/Hb=1.0H/Hb=0.50Fig.1.4. Ratioofnonlineartolinearwavemomentumux, Sxx,forfortystreamfunctionwavecombinations.12July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA8 R.G. DeanandT.L.Waltonat breaking). The reduction in Sxx for a wave of given height and for shallow waterconditions is 16.7%.1.4.4. EectsofvegetationThe eects of vegetation have been shown to result in a reduced setup and, in somecases, may cause a setdown.12For linear waves, vegetation protruding through thewater surface experiences a net drag force (quadratically related to velocity) on thevegetationinthedirectionofwavepropagationand,ofcourse,theremustbeanequal and opposing force exerted on the water column. This opposing force actingonthewatercolumnpartiallycounteractstheforceduetomomentumtransferand thus reduces the wave setup (similar to an oshore directed wind stress). Forlinear waves and vegetation which is submerged during the entire wave passage, nonet vegetation-related force exists on the water column and thus there is no eectonthewave setup.However, duetothecharacter ofnonlinear waves withhigherandshortershorewardvelocitiesunderthewavecrests, evenif thevegetationisfully submerged during the passage of the wave, a net drag force is induced on thevegetation in the wave propagation direction again resulting in a reduction in thewave setup and, for some cases, a wave setdown.1.4.5. DynamicwavesetupItisnotedthattheoreticalformulationsofthedynamicwave setupmustincludethe time dependent terms in the counterparts of Eq. (1.2). The dynamic wave setupor surf beat was rst identied through eld observations and measurements byMunk13and Tucker.14A number of theoretical treatments of dynamic wave setupbased on various hypotheses have been developed with each focusing on a dierentmechanism. TheseincludeSymondset al.15(time-varyingbreakpoint), Symondsand Bowen16(trapping of long waves by longshore bars), Schaer and Svendsen17(reinforcementof incomingandreectedlongwaves), etc. Kostense18conductedlaboratory experiments to investigate the dynamic setup component and found thatthe results were in qualitative agreement with the theory of Symonds et al.15Wecanapplytheresultsformonochromatic wavestoinvestigate theapprox-imate dynamic wave setup for a simple irregular wave case. Consider a bichromaticwave system with wave heightsH1andH2(H1>H2) and a small frequency dif-ferencebetweenthetwo components.Iftheresulting wave group variesso slowlythat static conditions occur within the surf zone, Eq. (1.7) applies and is written as = FHb, (1.10)whereHbis the breaking wave height andFis a proportionality factor dependingon whether the referenced setup is at the still water shoreline or the maximum wavesetup (see Eqs. (1.8) and (1.9)). The maximum and minimum wave setup values are:max = F(H1 + H2) ,min = F(H1 H2) .(1.11)July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 9Table1.1. Staticanddynamicwavesetupcharacteristics for abiharmonicwavesystem.H2/H1max/FH1avg/FH1(maxavg)/avg0.2 1.2 1.01 0.190.4 1.4 1.04 0.350.6 1.6 1.09 0.470.8 1.8 1.16 0.551.0 2.0 1.27 0.57It can be shown that the average wave setup depends on the ratio H2/H1 as shownin Table 1.1. The fourth column presents the ratios of the maximum dynamic waveset amplitude to the average wave setup component.In the case with H2 = H1, the dynamic wave setup displacement from the meansetup equals 57% of the average wave setup (Table 1.1, Column 4).In the above, we have examined the dynamic wave setup for the case of a simplebichromaticwavesysteminwhichthedierenceinfrequenciesof thetwocom-ponentswasfairlysmall.Forthecaseofawavespectrum, thesituationismuchmore complex with, for the case of a narrow spectrum, the group envelope varyingaccording to the Rayleigh distribution. For the case of a wide spectrum, the dynamiccomponent is reduced considerably.1.5. LaboratoryandField MeasurementsofWaveSetupHavingreviewedthetheoryofwavesetupanditsrelationshiptovariousfactors,the two sources available for evaluation are laboratory and eld data.1.5.1. LaboratoryexperimentsonwavesetupManylaboratory investigations ofstaticanddynamicwave setuphave beencon-ducted. The results of an early laboratory investigation with monochromatic wavesby Bowen et al.19are shown in Fig. 1.5. For this study, the ratio of maximum wavesetdown and wave setup on the beach face to breaking wave height are 0.035 and+0.316, respectively,compared with 0.049 and+0.244 onthebeachfacefora value of 0.78. The eect of beach slope has been noted earlier and the relativelylargebeachslopeof 0.082intheseexperimentsisundoubtedlyacontributortothelargesetupvalue.Later,laboratoryinvestigations haveincludedexaminationof irregular waves including measurements of water particle velocities and pressureswhich form the basis of theSxx momentum ux component.Battjes20conducted one of the earliest laboratory studies of wave setup due toirregular waves. Setup was measured through bottom mounted manometers and itwasfoundthatthewave setupwaslessthanpredicted.Itwas hypothesizedthatthisdierencewaspossiblyduetoairinthewatercolumnof themanometers.The entire setup was shifted landward relative to the theoretical and this delay waslater attributed to a roller that is transported along with the wave crest region andJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA10 R. G.DeanandT.L.WaltonFig.1.5. Measuredwavesetupandsetdowninthelaboratory.19conveys wave energy and momentum landward prior to transfer of the momentumto the water column and the associated wave setup.21Later, Stive and Wind22con-ducted a very detailed laboratory investigation in which they demonstrated the roleof wave nonlinearity. In this study, the momentum ux components (velocities andpressures) were measuredtothedegree possibleanditwasfoundthatthecalcu-latedwave setupbased onnonlinear wave theories was inmuchbetteragreementwith measured wave setup than calculations based on linear wave theories. In thesecomparisons, it was not necessary to introduce the roller concept.Thetwolaboratorystudiesreviewedabovehavefocusedonstaticsetupandithasbeennotedthatirregular waves alsoproducedynamicwavesetup. HedgesandMase23havepresentedaninterestingreanalysisof earlierrunuplaboratorymeasurements by Mase24in which irregular waves provided the forcing.bThe planarslopesrepresentedinthedatawere:1:5,1:10,1:20, and1:30.Figure1.6presentsan example of the form in which the data were plotted where the horizontal axis isbWalton25wasthersttoanalyzetheMasedatatoextractthestaticwavesetup.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 11Fig.1.6. VariationofnondimensionalrunupwithIribarrennumber.23the Iribarren number,0dened as0 =tan

H/L(1.12)in which tan is the prole slope. The interpretation of Fig. 1.6 is that for a zeroslope (zero Iribarren number), there would be no short wave runup; therefore, theintercept represents the sum of the static and dynamic components of wave runup.Equations of the following form were t to plots of the type of Fig. 1.6:RcharH1/3=ScharH1/3+cchar0, (1.13)wherethesubscriptchar referstothepercentassociatedwiththevariable;forexample, the 2% runup is dened as R2%. It was found that both Schar and cchar wereRayleigh distributed withS1/3 andc1/3 equal to 0.27 and 1.04, respectively, whereonly the rst term represents wave setup and is of interest here. The results for Scharcan be interpreted in terms of the static and dynamic wave setup components. As anexample,Smean = 0.17 andS2% = 0.37. Thus, the 2% value of the nondimensionaldynamic setup dened here as S2%isS2% =dyn,2%H1/3= (S2%Smean) = (0.37 0.17) = 0.20 . (1.14)Thus, themeansetupatthestill waterlineis17%ofthesignicantwaveheightmeasured at the toe of the slope and the 2% dynamic component at the still waterlineis 20% of the signicant wave height at the toe of the slope or slightly larger thanthe mean wave setup. These results are interesting and of reasonable magnitudes;however, there are two problems with recommending them for universal application.First, we know that the mean setup depends on the slope (through the dependencyJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA12 R. G.DeanandT.L.Waltonas discussed earlier), and the second is that the oscillating wave setup componentshould depend on the width of the input spectrum. Referring to Fig. 1.6, which isone of several similar plots presented in the Hedges and Mase paper, since each plotmay include a mix of beach slopes, the slope dependency is not resolved in the Smeanresults which of course are derived from they-intercept of these graphs. Secondly,these experiments were not designed to evaluate the eect of spectral width and thespectral characteristics included in the experiments are not known. However, it is ofinterest to identify the representative value and beach slope associated with aSmean = avg/H1/3 of 0.17. Referring to Fig. 1.2, we see that the associated valueis approximately 0.63 forF0. Based on the Dally etal.8breaking wave model, theassociatedbeachslopefromFig. 1.3is1:29comparedtothebeachslopesintheMase experiments ranging from 1:30 to 1:5.1.5.2. FieldexperimentsonwavesetupThe paragraphs below describe several eld experiments and observations of wavesetup.Anearlystudyofwavesetupcomprisedapairofobservationsatanexposedcoastal site (Narragansett Pier, RI) and a calmer water site (Newport, RI) where,atthelatter, waveactionwasassumednotafactorandwasfoundtoshowanapproximate3footwaterlevel dierenceduringthepeakof the1938hurricanestorm surge.26In a second early eld experiment on wave setup at Fernandina Beach, Florida,Dorrestein27placedtransparentplastictubeswithlightweightoatstotrackthewater surface on the beach in the zone of wave setup. To obtain the setup records,16 mm movie lm recorded the tracked surface of the oats. A oat type tide gageon the end of a shing pier provided oshore water level records. Through analysisof the tide gage records and the beach placed setup gages, Dorrestein27evaluatedthe setup (with respect to the end of the pier) and compared observational results toexisting setup theory. He found the measured setup in four of ve experiments to belarger than the computed setup. One shortcoming of Dorresteins work is that thewater level records were only 72 s in length and thus subject to considerable scatterand large standard deviation as later noted by Holman and Sallenger.28AlthoughrationalewasprovidedbyDorrestein27forpossibledierencesbetweenmeasuredand computed setup in this early experiment, large discrepancies between measuredand analytically or numerically computed setup still exist today.ANorth Sea eldwave setupexperimentwas conducted ontheIsland ofSyltby Hansen.29Utilizing a combination of ultrasonic wave gages and pressure sensorwave gages outtoadistanceof1280 mfromshore (10 mdepth),Hansen29foundgood correspondence of data to an empirical expression provided by: = 0.3Hos = 0.42Horms. (1.15)Hansen also noted the maximum wave setup to be approximately 50% of the signif-icant breaking wave height. It is not clear as to the methodology utilized to obtainmaxin this eld experiment.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 13Awavesetupeldexperimentwasconductedaspartof theNearshoreSed-iment Transport Study at Torrey Pines Beach, San Diego, California by Guza andThornton.30The Torrey Pines Beach face was gently sloping (beach slope 0.02)andthebeachmaterial wasamoderatelysortednegrainsediment(0.1 mm).Adualwireresistance runupmeterwasusedfortherecordingandestimationofthe wave setup. It should be noted that the measurements of the wave setup wereconsideredtobetheaveragerunupdeterminedfromwiresplacedapproximately3 cmabovethebeachlevel ratherthananactual waterlevel atonelocationinthese experiments. Oshore pressure sensors outside the surf zone at mean depthsof 7 to 10.5 m were used for estimating wave height with recording lengths of 4096 s.Guza andThornton30notespecic problems inthedataset,which are typical ofeldmeasurements, i.e., thedicultyinobtainingacommondatumfortheo-shore wave measurements and the beach wave setup measurements. Results of theirmeasurement program suggest an empirical relationship as follows: = 0.17Hos = 0.24Horms(1.16)with scatter that suggests/Hosranging approximately from 0.05 to 0.50 for indi-vidual experiments.Holman and Sallenger28conducted a eld experiment for measuring wave setupaswell asothersurfzoneparametersattheU.S. ArmyCorpsofEngineerseldresearchpierinDuck, NC, USA. Dataonwaterlevel attheshorelinewerecol-lected using longshore looking time lapse photography from Super-8 movie camerasmounted on the research pier scaolding. The beach at the experiment site had avery steep foreshore (1 on 10) while the oshore prole slope is much milder (1on 100). Beach material was bimodal in size with a median sand size of 0.25 mm anda coarse fraction of 0.75 mm. Results of the experiments showed considerable scatterand dependence on tide level. Regression lines were t to the data (segmented bytide levels) with results as follows for high tide and mid-tide data:Hs= 0.350 + 0.14 (high tide) , (1.17)Hs= 0.460 + 0.06 (mid-tide) . (1.18)As most of the data fell in a range of0= 1 to 2, the maximum setup was notedto be of the same order as the signicant wave height in many of the experiments,much higher than theoretically suggested values. Note that in terms of Horms (basedon consideration of monochromatic theory results) the setup would be much higherthan most other studies show or suggest.AlthoughHolmanandSallenger28concludefromtheir experiments that thesetup is dependent on the Iribarren number, it is not entirely clear from their data,especially for higher waves (i.e., see Fig. 1.4, Ref. 28). An additional problem thatmustbeconsideredwhencomputingtheIribarrennumberfor real beachesandirregular waves is how to dene beach slope. It should be noted that video camera(visual)approaches estimatesetupviathemeasurementofthewatersurfaceele-vation on the beach (similar to the Guza and Thornton measurements) rather thananactual verticallyuctuatingwaterlevel. TheanomalybetweendependenceofJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA14 R. G.DeanandT.L.Waltonsetup on Iribarren number as noted by Holman and Sallenger28is likely due to theaforementionedrelationshipbetweenthewave-breakingcoecient, , andbeachslope.Nielsen31and Davis and Nielsen32conducted a novel setup experiment on DeeWhy Beach, in New South Wales, Australia using a set of manometer tubes as shownin Figs. 1.7 and 1.8 from Davis and Nielsen.32The tubes were deployed throughoutthe beach face and surf zone. A total of 120 setup proles were measured in 11 days.Wave heights Horms ranged from 0.6 to 2.6 m in height and signicant wave periods(Ts) ranged from 5.8 to 12.1 seconds. A shoreline setup of about 40% of Horms wasfound although Davis and Nielsen32point out that there is reason to believe thatthesurf zonecharacteristicsinuencetherelationshipbetweenwaveheightandsetup magnitude, and also note a problem of dening beach slope via the Iribarrennumber. Nielsen31and Davis and Nielsen32also observe that a major portion of thesetup occurred on the beach face as shown in Fig. 1.9.Nielsen31points outthatprevious eld investigations have typically measuredthe mean water level elevation on the beach as opposed to the average uctuatingmean water level in the vertical plane (i.e., the wave setup as usually dened), andthat the two measurements are often dierent in part due to the beach permeability,which in turn is related to beach material size. The issue of extracting wave setupfrom runup and rundown on the beach is illustrated in Fig. 1.10.King et al.33collected wave setup data at Woolacombe Beach in North Devon,U.K.whichfacestheNorthAtlanticOcean. Thebeachfaceslopevaried between1on40athightideand1on70atmid-tidelevel withatidal rangeof 3 matneap and as much as 9 m at springs. Beach face material consisted of ne sand withFig.1.7. ManometersetupofDavisandNielsen32formeasuringsetup.July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 15Fig.1.8. Schematicdiagramofapparatus(fromRef.32).Fig. 1.9. Dimensionless setup versus total depth where much of the setup occurs on the beach face(fromRef.32).Inthisgure, BandDareequaltoandhasusedinthischapter,respectively.90% in the 0.125 mm to 0.25 mm size range. Pressure transducers were utilized tocollectwaveandsetupinformationatvariousstationsacrossthebeachandalsoinalongshoredirectiontoassessthespatial variabilityofthemeansetup. Bothtripod mounted and buried pressure transducers were utilized. The buried pressuretransducerswere50to80 cmbelowthebeachsurfaceandwereprotectedbyaporous cover. Instruments collected pressure data which were then transformed towater level data over 4096 second intervals. Data did not include sampling in veryshallowwaterandthemaximumwavesetupwasestimatedbyextrapolatingtheJuly 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA16 R. G.DeanandT.L.WaltonFig.1.10. Illustrationofdierencesbetweenmeanwaterlevel(MWL)shorelineandmeanwaterlineonbeach.water surface from the most shoreward water stations. Wave setup estimated fromthe data showed the wave setup to be roughly: = 0.10Hos = 0.14Horms(1.19)withmostof thevaluesof /Hormsbetween0.11and0.15. Theauthorsdonotspeculate as to why such low values of setup (compared to analytical results) werefound in this measurement program.Yanagishima and Katoh34discuss eld measurements of mean water level nearthe shoreline on the Pacic Coast of Japan as measured by an ultrasonic wave gagemountedonapierwherethemeandepthof waterwas 0.4 m. Thesetupwasdetermined via a multiple regression approach on 1305 sets of (20 minute records)data taking into account astronomical tide, wind setup, and atmospheric pressurehead components of mean water level. Their data included 91 records in which theoshore wave height was above 3 m. Yanagishima and Katohs34regression analysissuggested the following relationship: = 0.0520Hos

LosHos

0.2, (1.20)which can be formulated in terms of Iribarren number for theirbeach slope (1 on60) to the following: = 0.27Hos(0)0.4= 0.38Horms(0)0.4. (1.21)Yanagishima and Katoh34noted reasonable agreement with thetheory of Goda35(to be discussed later). Even higher values of setup would be expected on the beachface in accord with theory and ndings of other researchers.GreenwoodandOsborne36conductedeldmeasurementsonaGeorgianBayBeach, inLake Huron, Ontario, Canada. Lake Huron has no measurable tideandthebeachproleatthesitehadaslopeof0.015withasteepersloped(0.031to0.047) inshore bar. Setup was measured using surface piercing resistance wire wavestas with the shoreward most gage being in approximately 0.4 m of water depth.Measured setup values were found as follows: = 0.19Hos = 0.27Horms. (1.22)July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FAWaveSetup 17It is again noted that even higher values of setup would be expected on the beachface in accord with theory and experience of other researchers.Further work by Hanslow and Nielsen37,38utilized the manometer tubedeployment shown in Fig. 1.7 on three additional beaches (Seven Mile, Palm, andBrunswick)inNewSouthWales,Australia.Withbeachfaceslopesranging from0.03 to 0.16 and mean grain sizes of swash zone beach material ranging from 0.18to0.5 mm, shorelinebeachsetupwasmeasuredusing20minuterecordaverages.Usingthedatafromthesethreebeachesaswell asearliermeasurementsatDeeWhyBeach (seeRefs.31 and32), linearleast square relationships were ttothedata as follows: = 0.27Hos = 0.38HormswithR = 0.65 (1.23)or = 0.040

HosL0 = 0.048

HormsL0withR = 0.77 , (1.24)where a somewhat higher value of explained regression was noted using wave heightandwaveperiod.Dataandregression linesforthesetworelationshipsareshowninFigs. 1.11 and 1.12. Theimprovement in tdueto inclusion ofthedeep waterwavelength is not evident visually.Asignicantndingof thesestudies wasthat amajorportionof thesetupoccurred on the beach face (see Fig. 1.9). Further measurements on wave setup attworiverentrancesisalsodiscussedinHanslowandNielsen37andDunnetal.39with the result that the wave setup at river entrances was found to be (somewhatsurprisingly) negligible.Lentz and Raubenheimer40report on a eld experiment at the U.S. Army FieldResearchPierinDuck, NC, USAwhere11pressuresensorgagesand10sonaraltimeters extendedacross the surf zone from2to 8 mof water depth. Closeagreement withLonguet-Higgins radiation stress theory for wave setup was notedFig.1.11. Empiricalrelationshipbetweensetupandwaveheight(fromRef.38).July 31, 2009 8:18 9.75in x 6.5in b684-ch01 FA18 R. G.DeanandT.L.WaltonFig.1.12. Empiricalrelationshipbetweensetupandwaveparameters(fromRef.38).althoughthelackof setupmeasurementsinshallowwater( 2. (2.10b)The eigenseries (2.10a) maybe separatedinto apropagatingp(x, z, t; k) andevanescent eigenmodes e(x, z, t; n) or local wave components3according to(x, z, t; Kn) = p(x, z, t; k) +e(x, z, t; n)=_C1 coshk(z +h) +

n=2Cn cos n(z +h)_exp+i(Knx t +).(2.10c)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 31The wave numberk = 2/ where = wavelength. Because the numerical value ofkh must be computed from an eigenvalue problem in the vertical z coordinate, equiv-alenceoftheeigenvaluektothewave number2/requires apseudo-horizontalboundary condition of periodicity given byk = 2/ and(x + , z) =(x, z). Itis computationally ecient to normalize the eigenseries in (2.10a) according ton(Kn, z/h) =coshKnh(1 +z/h)Nn; n = 1, 2, . . . , (2.11a)where the nondimensional normalizing constantNnisN2n =_01cosh2Knh(1 +z/h)d(z/h) =___2kh + sinh 2kh4kh; n = 1, (2.11b)2nh + sin 2nh4nh; n 2. (2.11c)The eigenseries in (2.10a) may be written as an orthonormal eigenseries by(x, z, t; Kn) =

n=1Cnn(Kn, z/h) expi(Knx t ), (2.12)where the orthonormal eigenfunction n(,) is dimensionless.2.2.2. EvaluationofCnbyWMvertical displacement(z/h)Thefollowing dimensionlesscoecientcomputedfrom(2.5e)willreplace integralcalculus with algebraic substitution for the coecientsCn in the eigenseries (2.12):In(, , b, d, Kn) =_b/h1+d/h[(z/h) +]n(Kn, z/h)d(z/h)=(Knh)2Nn___Knh_1 dh_sinh Knd Knb sinhKnh_1 bh_coshKnh_1 bh_+ coshKnd___+(Knh)Nn_sinh Knh)_1 bh_ sinhKnd_(2.13)thatisdimensionlesswhenandaregivenby(2.7d)and(2.7e)or(2.8c)and(2.8d). The coecients Cnmaybe computedalgebraicallyby(2.13) fromtheKWMBC (2.5e) to obtainCn = iShKnIn(, , b, d, Kn), (2.14)and the orthonormal eigenseries (2.12) is given by(x, z, t; Kn)=

n=1iShKnIn(, , b, d, Kn)n(Kn, z/h) expi(Knx t ). (2.15)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA32 R. T.HudspethandR. B.Guenther2.2.3. Decaydistanceofevanescenteigenmodesn 2Numerical solutions and experimental measurements of ocean and coastal designsrequire thattheKRBC (2.5d)beapplied farenough away so thatonly theprop-agatingeigenmodeforn=1in(2.12)ismeasurable. Theevanescenteigenseriesin(2.12)forn 2willdecay spatially atleastasfastas thesmallestevanescenteigenvalue2. Thiseigenvaluemustbe2h>(n 3/2)=/2. Ifthesmallestvalue for2h>/2, then2>/2h and(x, z) exp (x/2h). For the valuesof the evanescent eigenseries to be less than 1% of their values at the wavemaker,(x, z) exp (xd/2h)=0.01andxd/(2h)=4.6 3/2,andtheminimumdecay distance isxd 3h.2.2.4. TransferfunctionforwaveamplitudefromwavemakerstrokeThe average rate of work or power done by a wavemaker of widthBis1

W= P= B_+1h_01p(x, z, )u(x, z, )d(z/h)d, (2.16a)where the temporal averaging operator is dened by=_+1()d, (2.16b)and

W= P=_3S2Bh422kh_I21(, , b, d, k), (2.16c)sothat all of the average power fromawavemaker is transferredtoonlythepropagating eigenmode. The average energy ux in a linear wave is given by1

E=_gBA22_CG, (2.16d)where the group velocityCGis given by1CG =C2_1 +2khsinh 2kh_. (2.16e)Equating (2.16c) to (2.16d) gives the following transfer function for a planarwavemaker:AS=_kohkh_1(k, 0)I1(, , b, d, k). (2.16f)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 332.2.5. Hydrodynamicpressureloads(addedmassandradiationdamping)The wave loads on a planar wavemaker may be estimated by integrating the totalpressure over the wetted surface of the wavemaker, i.e.,_ FM_ =__0SP_nr n_dS, (2.17a,b)where the outward pointing unit normal n points from the wavemaker into the uid,and the pseudo-unit normal n

for the rotational modes is given byn

= r n = (z +h d)nxey = n

yey. (2.17c)Force. For the Type I piston wavemaker of total width B, the horizontal componentof the pressure force on the uid side only may be computed from the real part ofF1(t) = Re_iBh

n=1Cn_b/h1+d/hn(Kn, z/h)d(z/h) expi(t +)_= F1 cos(t + 1), (2.18a)where the static component of the pressure force on the uid side only isFs = gBh22[1 2(d/h) + (d/h)2(b/h)2]. (2.18b)The hydrodynamic component of F1(t) may be separated linearly into a propagatingand an evanescent component that are related to the piston wavemaker translationalvelocity and acceleration, respectively, from the real part ofF1(t) = Re{[11(S) +11(iS2)] exp i(t +)}= 11(S2sin(t +)) 11(S cos(t +)) (2.18c)= Re{11 X1(t) 11X1(t)}, (2.18d)where the added mass coecient 11may be computed fromthe evanescenteigenmodes only, and the radiation damping coecient 11 may be computed fromthe propagating eigenmode only. The average power may be computed fromF1X1

t =11(S)22. (2.19a)Equating (2.19a) to (2.16d) yields11 =_A1S1_2Bhkoh CG, (2.19b)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA34 R. T.HudspethandR. B.Guentherthat relates theradiationdampingcoecient tothesquareof theratioof theradiated wave amplitude to the amplitude of the wavemaker displacement.Moment. For the Type I wavemaker of widthB, the dynamic pressure momenton one side only of the wavemaker may be computed from the real part ofM5(t) = Re_iBh2

n=1Cn__b/h1+d/h_1 +zh dh_n(Kn, z/h)d_zh__exp (t +)_= M5 cos(t + 5), (2.20a)and the static component of the pressure moment on the uid side only isMs =gBh36_1_dh_3+ 2_bh_3+ 3__dh_2_bh_2_3_dh__1_bh_2__.(2.20b)The pressure moment M5(t) in (2.20a) may be separated linearly into a propagatingand an evanescent component that are related to the rotational velocity and accel-eration from the real part of1M5(t) = Re__55_iS2(1 +b/)_+55_iS(1 +b/)__exp i(t +)_,M5(t) = 55 5(t) 55 5(t), (2.21a)where55 = Bh4

n=2I2n(, , b, d, n)nh, (2.21b)55 = Bh4I21(, , b, d, k)kh. (2.21c)2.3. CircularWavemakerHavelock5appliedFourierintegralstodevelopatheoryforsurfacegravitywavesforced by circular wavemakers in water of both innite and nite depth. The uidmotionmaybeobtainedfromthenegativegradientofascalarvelocitypotential(r, , z, t) according toq(r, , z, t) = (r, , z, t), (2.22a)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 35where the 3D gradient operator () in polar coordinates is() =_rer +_1r_e +ze3_(). (2.22b)The total pressure eldP(r, , z, t) may be computed from the unsteady Bernoulliequation in polar coordinates according toP(r, , z, t) = p(r, , z, t) +pS(z)= _(r, , z, t)t 12|(r, , z, t)|2+Q(t)_gz, (2.22c)where Q(t) = the Bernoulli constant, and the free surface elevation (r, , t) for zeroatmospheric pressure according to(r, , t) =1g_(r, , , t)t 12|(r, , , t)|2+Q(t)_;r b +(, , t); z = (r, , t). (2.22d)The scalar velocitypotential (r, , z, t) must be a solutionto the continuityequation2 =1rr_rr_+1r222+2z2= 0,r b +(, z, t); 0 2; h z (r, , t), (2.23a)with the following boundary conditions:Kinematic Bottom Boundary Condition (KBBC):z= 0; r b +(, h, t); 0 2; z = h. (2.23b)Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):2t2+gz_ t 12 _||2+dQdt= 0;r b +(, , t); 0 2; z = (r, , t). (2.23c)Kinematic WaveMaker Boundary Condition (KWMBC):r +t 1r2zz= 0; r = (, z, t); h z (b, , t). (2.23d)Twotypesofcircularcylindrical wavemakerdisplacements(, z, t)maybeana-lyzed, viz., amplitude-modulated(AM) andphase-modulated(PM)wavemakers.July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA36 R. T.HudspethandR. B.GuentherThedistinctionbetweenthesetwotypesisintheazimuthal dependencyofthewavemaker displacement(, z, t) from its mean positionr = b, given by(, z, t) =_mS(/h)_(z/h)_cos m sin(t +)sin(t + +m)_.(2.23e)(2.23f)Kinematic Radiation Boundary Condition (KRBC):lim|Knr|+_r iKn_(r, , z, t) = 0; r . (2.23g)Finally, physically realizable solutions to (2.23a) must be periodic in; i.e.,(r, , z, t) = (r, + 2, z, t). (2.23h)The dimensional WMBVP may be scaled and linearized1by expanding the variablesin perturbation series with a dimensionless perturbation parameter = kA. A scalarradiated velocity potential(r, , z) may be dened by the real part of(r, , z, t) = Re{(r, , z) expi(t +)}. (2.24)AlinearizedWMBVPmaybeobtainedbysettingthedimensionlessparameterkA = = 0 and by requiring that kh = O(1). This linearized WMBVP is2(r, , z) = 0; b r < +; 0 2; h z 0, (2.25a)(r, , z)z= 0; b r < +; 0 2; z = h, (2.25b)(r, , z)zko(r, , z) = 0; b r < +; 0 2; z = 0, (2.25c)lim|Knr|+_r iKn_(r, , z) = 0, (2.25d)(r, , z)rexpi(t +) = (, z, t)t; r = b; 0 2; h z 0,(2.25e)(r, , t) = Re_i(r, , z)gexp i(t +)_; b r < ; 0 2; z = 0,(2.25f)P(r, , z, t) = {p(r, , z, t)} +ps(z)= Re{i(r, , z) expi(t +)} gz, (2.25g)(r, , z) = (r +, , z); (r, , z) = (r, + 2, z). (2.25h,i)The specied wavemaker shape function (z/h) is valid for either a double-articulated piston or hingedcircular AM orPM wavemaker of variable draft thatJuly 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 37Fig.2.2. Denitionsketchforcircularwavemaker.is showninFig. 2.2is identical to(2.7) for a2Dplanar wavemaker withthedimensionbreplaced withaandthestrokeSreplaced withtheazimuthalstrokemS.Thesolution totheWMBVP(2.25) may becompactly expressed by thefol-lowing orthonormal eigenseries:m(r, , z) =

n=1Cmnn(Kn, z/h)H(1)m(Knr)MA(P)(m), (2.26a,b)where the azimuthal mode function isMA(P)(m) =_cos mexp im_; m 0 and integer, (2.26c,d)and where (2.26a) represents an AMwavemaker; (2.26b) represents a PMwavemaker; n(Kn, z/h) = the orthonormal eigenseries dened in (2.11);H(1)m(Knr) = the Hankle function of the rst kind. When K1 = k and Kn = in forn 2 and integer,H(1)m(inr) = Jm(inr) +iYm(inr) =2i(m+1)Km(nr), (2.26e)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA38 R. T.HudspethandR. B.GuentherwhereKm() =theModiedBessel (orKelvin)functionof thesecondkindoforder m. The coecientsCmn may be computed by expanding the KWMBC in aneigenseries following the procedure in (2.14) and obtainingCmn = mSjhKnIn(, , a, d, Kn)Ln(H(1)m(Knb)); n 1 and integer, (2.26f)Ln(Zm(n)) =dZm(n)dn=12{Zm1(n) Zm+1(n)};Zm(n) = Jm(n), Ym(n), Km(n), H(1)m(n). (2.26g)The solution to (2.25) is given by the real part of the following eigenseries expansion:mj(r, , z, t)[mSjh]=mpj(r, , z, t) +mej(r, , z, t)[mSjh]= Re_____I1(, , a, d, k)k1(k, z/h)H(1)m(kr)L1(H(1)m(kb))+

n=2In(, , a, d, n)nn(n, z/h)Km(nr)Ln(Km(nb))__MA(P)(m)ei(t+)___. (2.27a,b)BecausetheasymptoticbehavioroftheevanescenteigenseriesKm(nr)dependson the modem(1), it is not possible to specify a minimum distance from the wave-makerequilibriumboundaryat r =bwheretheevanescenteigenvaluesarelessthan 1% of their value at the circular wavemaker boundary. The wave eld must becomputed faraway from the wavemaker, and it is understood that far away mustbecomputeduniquelyforeachradial modemforeitheranAMorPMcircularwavemaker. The evaluation of the power, forces, and moments, and added mass andradiation damping coecients for both AM and PM circular wavemakers are givenby Hudspeth.12.4. DirectionalWavemakersDirectional wavemakers areverticallysegmentedwavemakers thatundulatesinu-ouslyand,consequently,arealso calledsnake wavemakers. Segmenteddirectionalwavemakers may be driven either in the middle of each vertical segment or at thejoint between vertical segments. Because of these two methods of wave generation,parasitic waves are formed along the wavemaker due to either the discontinuity ofJuly 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 39thewavemaker surface(middlesegmentdriven)orofthederivativeofthewave-maker surface (joint driven). A dimensional scalar spatial velocity potential (x, y, z)may be dened by the real part of(x, y, z, t) = Re{(x, y, z) expi(t +)}. (2.28a)The dimensional linear uid dynamic pressure eld p(x, y, z, t) and 3D uid velocityvector eld may be computed fromp(x, y, z, t) = (x, y, z, t)t, (2.28b)q(x, y, z, t) = 3(x, y, z, t), (2.28c)3() =()xex +()yey +()zez. (2.28d)The dimensional WMBVP for directional waves is given by23(x, y, z) = 0; x 0; B y +B; h(x, y) z 0, (2.29a)(x, y, 0)zko = 0; x 0; B y +B; z = 0, (2.29b)y= 0; x 0; y = B; h(x, y) z 0, (2.29c)(x, y, z, t)x= (y, z, t)t;___x = 0,B y +B,h(0, y) z 0,(2.29d)limx+_x iKn_(x, y, z) = 0, (2.29e)(x, y, z)z= 2(x, y, z) 2h(x, y); z = h(x, y), (2.29f)22() =_2x2,2y2_(), (2.29g)(y, z, t) = Re_iU(y, z)[U(y, a)][U(z, b, d)] expi(t +)_, (2.29h)U(y, z) =_S/ho_(y)(z/ho), (2.29i)U(y, a) = U(y +a) U(y a+),U(z, b, d) = U(z +h d) U(z +b),(2.29j)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA40 R. T.HudspethandR. B.GuentherFig.2.3. Denitionsketchforrectangulardirectionalwavebasin.where ko = 2/g and where a denotes the (possibly nonsymmetric) transverse endsof the directional wavemaker in the transverse y-direction in Fig. 2.3. The solutionto the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions:(x, y, z) = ig

n=1n(x, y)n(Kn, z/h), (2.30a)n(Kn, z/h) =___1(k, z/h)1(k, 0); n = 1, (2.30b)1(n, z/h); n 2, (2.30c)koh = KnhtanhKnh = 0; n = 1, 2, 3, . . . , (2.30d)whereK1=kandKn=+infor n 2andn()isdenedin(2.11). Theorthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for con-stant depth wave basins; however, they may be applied to slowly varying depth wavebasins if (2.30b) and (2.30c) are considered to be evaluated only locally over rela-tively small horizontal length scales (e.g., several wavelengths), where the depthmay be considered to be locally equal to a constant by a Taylor series expansion ofthe depth.1Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation:22(x, y) +K2n(x, y) = 0; x 0; B y +B. (2.31)Alternatively, for wave basins with mildly sloping bottoms, the mild slope equationmay be applied according to12 (CCG2(x, y)) +K2nCCG(x, y) = 0, (2.32)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 41wherethewavegroupvelocityCGisgivenby(2.16e). If theproductCCGisaconstant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJapproximation1for the x-dependent solution in the method of separation of variablesto (2.32) yields the following solution1:(x, y, z, t) = Re_ig(x, y)n(Kn, z/h) expi(t +)_= Re___igM

m=0

n=1Amn[C(x)CG(x)]x=0C(x)CG(x)m(m, y/B)n(Kn, z/h) expi__xQmnd_expi(t +)___(2.33a)m(m, y/B) =cos mB(y/B 1)Mm; Mm = 1 +m0; m =m2B(2.33bd)Qmn = _K2n2m; m > 0; K1 = k > 2m. (2.33e)ThecoecientsAmmay becomputedfrom (2.29d) by expanding thewavemakershape function in orthonormal eigenfunctions1and are given byAmnkQmnS(/hx0)= n(Kn, 0)In(, , b, d, k)_+a+/Ba/Bj(qj, y/B)m(m, y/B)d(y/B). (2.34)2.4.1. Full-draftpistonwavemakerTheprescribedtransversey-componentof thesnakedisplacementof afull-draft(b = d = 0) piston ( = 0 and = 1) wavemaker may be expressed asj(qj, y/B) =+

j= cj exp i[qjB(y/B +y)], (2.35a)where the coecients cjmay be computed from the integral in (2.34) bycmj=_+a+/Ba/Bj(qj, y/B)m(m, y/B)d(y/B)=Ra+,a +iIa+,amB(q2j 2m)(1 +m0). (2.35b)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA42 R. T.HudspethandR. B.GuentherIf the full-draft piston snake wavemaker spans the entire width of the wave basinso thata = B, then (2.35b) reduces to the integral in (2.34) andcmj =4qjB_sin[qjB(j 1)] + (1)msin[qjB(j + 1)]i{cos[qjB(j 1)] + (1)mcos[qjB(j + 1)]}_((qjB (m)2)2)(1 +m0). (2.35c)Values forcmjfor (possibly nonsymmetric) values foraare given by Hudspeth.12.5. Sloshing Wavesina2DWaveChannelAlong rectangular wave channelwithahorizontal atbottom,tworigid verticalsidewalls, and a wavemaker may generate either 2D, long-crested progressive wavesor two types of transverse waves, viz.,(1) sloshing waves that are excited directly by transverse motion of the wavemakeror(2) cross waves that are excitedparametricallybythe progressive waves at asub-harmonic of the wavemaker frequency.TheWMBVPfor3Dsloshingwavesisidentical to(2.5)forplanar2Dwave-makersexceptfortheKWMBCatx=0andanadditionalkinematicboundaryconditiononthesidewallsof the2Dwavechannel at y= BinFig. 2.4. Thekinematic and dynamic wave elds may be computed from a dimensional 3D scalarFig.2.4. Denitionsketchforasloshingwavechannel.July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 43velocity potential (x, y, z, t). The uid velocity q(x, y, z, t) may be computed fromthe negative directional derivative of a scalar velocity potential byq(x, y, z, t) = (x, y, z, t). (2.36a)The dimensional uid dynamic pressure eldp(x, y, z, t) may be computed fromp(x, y, z, t) = (x, y, z, t)t. (2.36b)A spatial velocity potential(x, y, z) may be dened by the real part of(x, y, z, t) = Re{(x, y, z) expi(t +)}. (2.36c)The WMBVP for sloshing waves is given by the following:2(x, y, z) = 0; x 0; B y +B; h z 0, (2.37a)(x, y, h)z= 0; x 0; B y +B; z = h, (2.37b)(x, y, 0)zko = 0; x 0; B y +B; z = 0, (2.37c)(x, y, z, t)x= (y, z, t)t; x = 0; B y +B; h z 0, (2.37d)y= 0; x 0; y = B; h z 0, (2.37e)limx+_x iKn_(x, y, z) = 0, (2.37f)(x, y, t) = Re_ig(x, y, 0, t)_;___x 0,B y +B,z = 0.(2.37g)A solution to (2.37) is given by the following eigenfunction expansions1:(x, y, z, t) = Re_

n=1n(x, y)n(Kn, z/h) expi(t +)_, (2.38a)(x, y, z) =n(x, y)n(Kn, z/h), (2.38b)(x, y, t) = Re_

n=1n(x, y) exp i(t +)_, (2.38c)n(x, y) = ign(x, y)n(Kn, 0), (2.38d)n(x, y) = ign(x, y)n(Kn, 0), (2.38e)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA44 R. T.HudspethandR. B.Guentherwhere n(x, y) is sometimes referredtoas adisplacement potential. The scalarpotential (2.38a) may be expressed from (2.38d) and (2.38e) as(x, y, z, t) = Re_

n=1ign(x, y)n(Kn, z/h)n(Kn, 0)exp i(t +)_. (2.39)The WMBVP may be expressed in terms of a displacement potentialn(x, y) by2n(x, y)x2+2n(x, y)y2+ K2nn(x, y) = 0;___x 0,B y +B,h z 0.(2.40a)

n=1n(x, y)xn(Kn, z/h)n(Kn, 0)= igU(y, z)___x = 0,B y +B,h z 0.(2.40b)limx+_x iKn_n(x, y) = 0. (2.40c)n(x, y)y= 0; x 0; y = B; h z 0, (2.40d)where (2.40a) is the 2D Helmholtz equation.9,10Because the boundary conditions are prescribed on boundaries that are constantvalues of (y,z), a solution to the WMBVP (2.40) may be computed by the methodof separation of variables and is given by1(x, y, z, t)= Re___ig___M

m=0Cm1m(y/B)1(k, z/h)1(k, 0)expiPm1x+ m=M+1Cm1m(y/B)1(k, z/h)1(k, 0)expm1x+

m=0

n=2Cmnm(y/B)n(n, z/h)n(n, 0)expQmnx___expi(t +)___,(2.41a)wherem(m, y/B) =cos mB(y/B 1)Mm; M2m = 1 +m0; (2.41b)m =m2B ,July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 45Pm1 =_K21 2m = _k2 2m; k > m; m M, (2.41c)m1 = _2mk2; k < m; m > M, (2.41d)Q2mn = 2mK2n; m 0; n 1, (2.41e)n 1:K1 = k > m : Qm1 = i_k22m = iPm; m M, (2.41f)n = 1:k = m : Qm1 = 0k < m : Qm1 = _2mk2= m1> 0; m > M, (2.41g)n 2 : Kn = in : Qmn = _2m +2n> 0, (2.41h)whereMisthemaximumintegervalueforminorderform M + 1 (2.42b)Cmn = ign(Kn, 0)Qmn_+11d_ yB__01d_zh_U_y,zh_n_Kn,zh_m_m,yB_;m 0, n 2. (2.42c)The rst three transverse eigenmodes are illustrated in Fig. 2.5.2.6. ConformalMappingsConformal and domain mappings are applications of complex variables to solve 2Dboundary value problems. Conformal mapping is an angle preserving transformationthatwill computeexactnonlinearsolutionsforsurfacegravitywavesofconstantFig.2.5. Firstthreetransverseeigenmodesina2Dwavechannel.July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA46 R. T.HudspethandR. B.Guentherform that may be treated as a steady ow following a Galilean transformation from axed inertial coordinate system to a noninertial moving coordinate system. Domainmapping is a transformation of the wavemaker geometry into a xed computationaldomain where a solution may be computed eciently.2.6.1. Conformal mapping1Conformal mappingof the WMBVPprovides aglobal solutionthat accuratelyaccountsforthesingularbehavioratall irregularpoints. Theirregularpointsinthe physical wavemaker domain are transformed into both weak and strong singularkernels in a Fredholm integral equation. The two irregular points on the WMBVPboundary are located at (1) the intersection between the free-surface and the wave-makerboundaryand(2)theintersectionbetweenthehorizontalbottomandthewavemaker boundary. These two irregular points exhibit integrable weakly singularkernels. The far-eld radiation boundary exhibits a strongly singular kernel and sig-nicantly aects the solution. The irregular frequencies3,4are included in the globalsolution by the Fredholm alternative. A theory for the planar WMBVP computesaglobal solutionbyaconformal mappingof thephysical wavemakerboundarytoaunitdiskthatincludesthemotionofaninvisciduidnearirregularpointsthatareillustratedinFig. 2.6. Anumerical solutiontoLaplacesequationinatransformedunitdiskmaybecomputedfromaFredholmintegralequation.TheWMBVP dened by (2.5) is transformed into complex-valued analytical functionswhere thecomplex-valued coordinates are dened asz=x + iy. The coordinatesfor the semi-innite wave channel in Fig. 2.1(a) must betransformed to complex-valuedcoordinatesz. Theuidvelocityq(x, y, t)anddynamicpressurep(x, y, t)may be computed from a scalar velocity potential (x, y, t) according toq(x, y, t) = (x, y, t); p(x, y, t) = (x, y, t)t. (2.43a,b)The WMBVP and Type I wavemaker shape function are given by (2.5)(2.7).There are both Irregular (I) and Regular (R) points at the intersections betweenthe Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustratedin Fig. 2.6 where these two boundary intersection points are identied as P1 and P2.The classication of the boundary pointsP1andP2in Fig. 2.6 depends on (1) theboundary conditionsi(Pj) and (2) the continuity of the boundaries Bm and theirderivatives wherei, j,andm = 1or2.Aconformal mapping ofthesemi-innitewave channel strip in the physical plane will yield a Fredholm integral equation,6,7where these critical points may be transformed to singular points that are integrableover a smooth continuous mapped boundary.2.6.1.1. Conformalmapping to the unit disk 1Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-planeshown in Fig. 2.7; and (ii) a semi-circle in the upperZ-plane to a unit disk in theJuly 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 47Fig. 2.6. Combinations of Irregular (I) andRegular (R) boundarypoints P1andP2betweenSmooth(S)andCritical(C)boundariesB1andB2intersectionsintheWMBVP.8Fig.2.7. Mappingofthesemi-innitestripinthelowerhalfxy-planeinthephysicalz-plane totheupperhalfXY -planeintheZ-plane.8July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA48 R. T.HudspethandR. B.GuentherFig.2.8. Mappingoftheupperhalf-planeintheZ-planetotheunitdiskintheQ-plane.Q-plane shown in Fig. 2.8. The SchwarzChristoel transformationdzdZ=C1Z + 1Z 1(2.44a)may be integrated to obtainz = x +iy= hLog[Z _Z21], (2.44b)wheretheLog[] denotestheprincipal valueof Log[] andtheargumentof theLog[] is arg < . Inverting (2.44b) yieldsZ = X +iY= cosh(z/h), (2.44c)whereX= cosh_xh_cos_yh_, (2.44d)Y= sinh_xh_sin_yh_. (2.44e)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 49The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane byR2= X2+Y2=12_cos_2yh_+ cosh_2xh__, (2.44f)tan =YX= tanh_xh_tan_yh_. (2.44g)Details of the transformation of the WMBVP are given by Hudspeth.12.6.1.2. MappingZ-plane to a unit diskTheupper-half-planeof the Z-planemaybemappedintoaunitdiskshowninFig. 2.8 by the following bilinear transformation:Q = +i = exp(i0)_i Zi +Z_, (2.45a)that may be inverted to obtainZ = X +iY= i_1 Q1 +Q_, (2.45b)and the mapping function coordinates are =1 X2Y2X2+ (Y+ 1)2, =2XX2+ (Y= 1)2,(2.45c,d)that maybe transformedinto the cylindrical coordinates for the unit disk inFig. 2.8 byr2= 2+2=(X2+Y21)24X2(X2+ (Y+ 1)2)2;= arctan__= arctan_2X1 X2Y2_.(2.45e,f)Details of thetransformationaregivenbyHudspeth.1Anumerical solutiontothe transformed WMBVP may be computed from the following Fredholm integralequation1:_2_ (r, ) = _+_( r, )G(r, r, , ) r+G(r, r, , ) ( r, ) r_ rd ,(2.46a)whereG(r, r, , ) = ln[(r, r, , )]; 2(r, r, , ) = r2 2r r cos( ) + r2.(2.46b,c)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA50 R. T.HudspethandR. B.GuentherFig. 2.9. Nodal pointsontheunit disk intheQ-plane andthecorresponding nodalpointsonthewavemakerinthephysical z-plane.8Numerical solutions to(2.46a) maybe computedbydiscretizingthe unit diskboundaryas showninFig. 2.9. Thenumerical details regardingthe evaluation(2.46a)atthetwoweaklysingularirregularpoints atBandDinthephysicalz-plane inFig. 2.7andthe stronglysingular point at that is the verticalradiationboundaryAEat+inthephysical z-planeinFig. 2.7aretedious.8Global numerical solutions may be computed for both the linear and the nonlinearWMBVPs.2.6.1.3. Conformalmapping to the unit disk 2ThewavemakergeometryshowninFig. 2.10ismappedtotheunitdiskbytwotransformations. The WMBVP is given by (2.5) and the WM shape function is(y/h) = [(y/h) +][U(y/h + 1 b0/h) U(y/h +a0/h)]. (2.47)In order to transform the wavemaker geometry to a Jacobian elliptic function,itmustberotatedandtranslatedasshowninFig.2.11.The90rotationtothez

-plane is given byz

= x

+iy

= iz = y +ix. (2.48a)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 51Fig.2.10. WMBVP11withthesixcriticalboundarypointsataa0bb0cd.Fig. 2.11. Rotationandtranslationof thephysicalwavemakerrectangularstripinthez-plane tothew-plane.11The horizontal shift to the left in thez

-plane is given byz

= x

+iy

= w

h/2 = y h/2 +ix. (2.48b)In order to map the WM geometry in thez-plane to the semi-circle in theZ-planeshown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated stripin the z

-plane must have the dimensions of K +K and 0 K

, whereK = h/2 andK

= 3h = 6K. This requires a coordinate amplication given byw =2Kh(x

+iy

)=2Kh_y h2 +ix_. (2.48c)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA52 R. T.HudspethandR. B.GuentherFig. 2.12. Mapping of the wavemakersemi-circle in the Z-plane to the unit disk in the Q-plane.11The SchwarzChristoel transformation from thew-plane to theZ-plane isdw =CkdZ_(a Z)(b Z)(c Z)(d Z). (2.48d)The following change of variables:Z = aZ; dZ = adZ : = a/c;C = c,modies (2.48d) to the following Jacobian elliptic function:w =_Z0_dZ[(1 Z2)(1 2 Z2)]12_ = sn1[Z, ], (2.48e)wheresn[Z, ] =theJacobianellipticfunctionof modulus orsineamplitudefunction.9Denek = sn1[1, ], (2.48f)and the mapping of the rectangle {x1, x2; y1, y2} = {0, 3h; 0, h} is given byZ = X +iY=___sn_2Kh_y +h2_, dn_2Kxh,k_1 dn2_2Kh_y +h2_, sn2_2Kxh,k____+i___cn_2Kh_y +h2_, dn_2Kh_y +h2_, , sn_2Kxh,k_cn_2Kxh,k_1 dn2_2Kh_y +h2_, sn2_2Kxh,k____,(2.48g)wheresn[, ]inthecopolarhalf-periodtrioin(2.48g)isdenedin(2.48e),andcn[, ] and dn[, ] are dened by cn2[, ] = 1sn2[, ], dn2[, ] = 12sn2[, ].Themappingfromthesemi-circleintheZ-planetotheunitdiskintheQ-planeJuly 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 53Fig.2.13. Transformedboundaryconditionsmappedtoarcsontheperimeteroftheunitdisk.11is shown in Fig. 2.12; and the mapping to the unit disk in theQ-plane is shown inFig. 2.13. The mapping of theZ-plane to theQ-plane is given byQ =i Z (i +Z)i +Z (i Z)=(1 )2Z2(1 + )2+ 2iZ(1 )2(1 2) +Z2(1 + )2, (2.49a)where 1 < < +1. Changing variables to circular cylindrical coordinates byR2(X, Y ) =(1 2) 2Y (1 2) + (X2+Y2)(1 + )2(1 )22Y (1 2) + (X2+Y2)(1 + 2), (2.49b)(X, Y ) = arctan_2X(1 2)(1 )2(X2+Y2)(1 + )2_, (2.49c)theunitdiskmaybetransformedintofunctionsof thecopolartrioof Jacobianelliptic functions. The transformedWMBVPincircular cylindrical coordinatesisgivenbyHudspeth.1Ageneral solutiontothetransformedWMBVPmaybewritten as10(R, ) =N

n=0Rn_ an1 + n0cos n +bn sinn_, (2.50)where ijis the Kronecker deltafunction. Substituting (2.50) intothe genericboundary conditions on each of the six arcs on the perimeter of the unit disk illus-trated in Fig. 2.13, multiplying each of these six boundary conditions by a memberof the set of the orthogonal series in (2.50), integrating over the interval of orthogo-nality + yields the following matrix equation for each of the coecients an and bnAB = H. (2.51)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA54 R. T.HudspethandR. B.GuentherFig.2.14. Physicaluiddomain.122.7. DomainMappingDomainmappingoftheWMBVP12follows thetheorybyJoseph.13ThephysicaluiddomainshowninFig.2.14forthefullynonlinearWMBVPismappedtoaxed computational uid domain, and the discretized coupled free-surface boundaryconditionsarecomputedbyanimplicitCrankNicholson(CN)method.14,15Ateach iteration of the CN method, the potential eld is computed by the conjugategradient method.15The wavemaker motion (y/h, t) is assumed to be periodic withperiodT= 2/, and the WMBVP with the surface tensionTis given by2(x, y, t) = (x, y, t) = 0;_0 y (x, t),(y/h, t) x L.(2.52a)(x, y, t)t+ 12|(x, y, t)|2T2(x, t)x2_1 +_(x, t)x_2_3/2+g(x, t) = 0. (2.52b)(x, y, t)y(x, t)x(x, y, t)x+(x, t)t= 0;((x, t), t) x L; y = (x, t). (2.52c)(x, y, t)y= 0; ((x, t), t) x L; y = 0. (2.52d)(x, y, t)x= 0; x = L; 0 y (L, t). (2.52e)(x, y, t)x= (y/h, t)t+(y/h, t)y(x, y, t)y;_x = (y/h, t),0 y (0, t).(2.52f)July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FAWavemakerTheories 55The initial conditions fort = 0 are(x, 0) = H(x); (2.52g)(x, t) = 0;_(x, 0) x L. (2.52h)(x, (x, 0), 0) = 0; (2.52i)The physical uid domain shown in Fig. 2.14 may be mapped into a dimensionlessxed rectangle of dimensions 0 1 by 0 1 by the transforms =xL; =y(x, t); = t; (, ) =(x, t)h, (2.53ad)and dimensionless variables byq(, , ) = (x, y, t)A; p(, , ) =P(x, y, t)A22, (2.53e,f)(, , ) =(x, y, t)Ah; w(, ) =(y/h, t)S;T =TALh2. (2.53gi)Because is a function of bothx andy in (2.53b), transforming partial derivativeswith respect tox must be done with some care.12Details of these lengthy transfor-mationsandthetransformedWMBVPinthexedmappeddomainare given byHudspeth.1References1. R. T. Hudspeth, Waves andWaveForces onCoastal andOceanStructures (WorldScientic,Singapore,2006).2. R. T.Hudspeth, J. M.Grassa,J. R. Medina and J. Lozano,J.HydraulicRes.32, 387(1994).3. F.John,Commun.PureAppl.Math.2,13(1949).4. F.John,Commun.PureAppl.Math.3,45(1950).5. T.H.Havelock,Phil.Mag.8,569(1929).6. P.R.Garabedian,Partial Dierential Equations(Wiley,Inc.,NewYork,1964).7. P. M. MorseandH. Feshbach, Methods of Theoretical Physics (McGraw-Hill BookCompany,NewYork,1953).8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhDthesis,OregonStateUniversity(1988).9. G. F.Carrier,M.Krookand C.E. Pearson,FunctionsofaComplexVariable,TheoryandTechnique(McGraw-HillBookCo.Inc.,NewYork,1966).10. R. B. GuentherandJ. Lee, Partial Dierential Equations of Mathematical PhysicsandIntegral Equations(DoverPublications, Inc.,NewYork,1996).11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker,MSthesis,OregonStateUniversity(1993).12. S.J.DeSilva,R.B.GuentherandR.T.Hudspeth,Appl.OceanRes.18,293(1996).July 31, 2009 8:18 9.75in x 6.5in b684-ch02 FA56 R. T.HudspethandR. B.Guenther13. D.D.Joseph,Arch.Rational Mech.Anal.51,295(1973).14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied NumericalMethods (John WileyandSons,NewYork,1965).15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (Springer-Verlag,1984).July 31, 2009 8:18 9.75in x 6.5in b684-ch03 FAChapter3AnalysesbytheMelnikovMethodofDampedParametricallyExcitedCrossWavesRonald B. GuentherDepartment of MathematicsOregon State University, Corvallis, OR 97331,USAguenther@math.orst.eduRobert T. HudspethSchool of Civil and Construction EngineeringOregon State University, Corvallis, OR 97331,USArobert.hudspeth@orst.eduThe WigginsHolmes ex