On the laboratory generation of surface waves of n

J.FluidMech.(2006),vol.559,pp.413–427.c
2006CambridgeUniversityPressdoi:10.1017/S0022112006000103PrintedintheUnitedKingdom413Onthelaboratorygenerationoftwo-dimensional,progressive,surfacewavesofnearlypermanentformondeepwaterByDIANEM.HENDERSON1,MATTHEWS.PATTERSON2ANDHARVEYSEGUR31WilliamG.PritchardFluidMechanicsLaboratory,DepartmentofMathematics,PennStateUniversity,UniversityPark,PA16802,USA2DepartmentofAppliedMathematics,UniversityofWashington,Seattle,WA98195-2420,USA3DepartmentofAppliedMathematics,UniversityofColorado,Boulder,CO80309-0526,USA(Received21September2005andinrevisedform31January2006)RecentexperimentsbyHammacketal.(J.FluidMech.,vol.532,2005,p.1)ondeep-waterwaveswithtwo-dimensional,periodicsurfacepatternsshowedseveralfeatures,someofwhichwereunsteady.Fuhrman&Madsen(J.FluidMech.,vol.559,2006,p.391)explainedthreeoftheseasbeingtheconsequenceofsinusoidalforcingbythewavemakerarraythatdidnotincludeforcedharmonics,eitherintimeorinthedirectionparalleltothewavemaker.Theypredictedthatneglectedthird-harmonictermscausemoreseriousconsequencesthanneglectedsecond-harmonictermswhenthegeneratedwavefieldshavetwo-dimensionalsurfacepatterns.Thispaperpresentsexperimentsthatprovidestrongevidencethattheirexplanationiscorrect:includingthethird-harmonictermsinthewavemakerforcingresultsinwavepatternsthatpropagatewithnearlypermanentform.1.IntroductionInthispaperweconsiderthelaboratorygenerationofpatternsofsurfacewavesthatpropagatewithnearlypermanentformondeepwaterandthathaveperiodicitiesintwohorizontaldimensions.SuchpatternshavebeenobservedinexperimentsbyKimmoun,Branger&Kharif(1999a),whogeneratedtrainsofplanewavesthatreflectedfromawalltocreatetwo-dimensionalpatterns,andbyHammack&Henderson(2003)andHammack,Henderson&Segur(2005,hereafterreferredtoasHHS),whouseda32-plungerwavemakerarraytogeneratepatternsresultingfromtheinteractionoftwotrainsofplanewavesthatweresymmetricaboutthedirectionofpatternpropagation.Figure1showsaphotographofsuchapatternfromHHS.Similarpatternsofdeep-watersurfacewaveswithtwo-dimensionalsurfacestructurehavealsobeengeneratedthroughbifurcationsofone-(horizontal)dimensionalwavetrains(Su1982;Suetal.1982).Othertypesoftwo-dimensionalsurfacepatterns,termed‘crescentwaves’or‘horseshoe’patterns,havebeenobservedinexperimentsonwind-generatedwavesbyCaulliez,Ricci&Dupont(1998)andCollard&Caulliez(1999).AdditionaltypesofwavepatternsduetowindhavebeenobservedinexperimentsbyCaulliez&Collard(1999).Two-dimensionalsurfacepatternsarebecomingmoreandmoreimportantinourunderstandingofdeep-watersurfacewaves,soitisimportantthatwebeabletogeneratethemwithquantitativeaccuracyandcontrol;thispaperispartofthateffort.414D.M.Henderson,M.S.PattersonandH.SegurFigure1.Atypicalbi-periodicpatternofprogressivesurfacewavesindeepwater.Hereweconcentrateonthegenerationoftwo-dimensionalsurfacepatternsbyanarrayofmechanicalwavemakersthatareprogrammedtoproducetwotrainsofwavesinteractingatsymmetric,obliqueanglesfromthedirectionofpatternpropagation.HHScataloguedtwelvefeaturesoftheirpatterns,someofwhichwereunsteady.Examplesoftheunsteadyfeatureswere(i)slowmodulations,or‘beats’ofthewaveamplitudesinthex-direction,(ii)curvingofcrestlinesinthey-direction,and(iii)dipsandpeaksinthecrestlinesinthey-direction.AnexplanationfortheseunsteadyfeatureswasproposedbyFuhrman&Madsen(2006,hereafterreferredtoasFM),whoconductednumericalexperimentsusingaBoussinesq-typemodelthatishighorderindispersion.Theyarguedthatthethreeunsteadyfeatureslistedaboveoccurredbecausetheforcingforthewavemakerarrayusedonlythefirsttermfromtheinfiniteseriesrepresentingawaveofpermanentform.EvenbeforeFM,itwasknownthatwhenattemptingtogeneratesteadilypropagat-ingwaveswithawavemaker,theomissionofhigherharmonicsfromtheforcingofthewavemakercouldleadtospurious‘free’wavemodesthatmakethewavefieldunsteady.SeeSch¨affer&Steenberg(2003)forareviewofworkthataddressestheneglectofsecond-harmonictermswhengeneratingeitherplanarornon-planarwavefields.ThenewinsightofFMwasthatforwavefieldswithtwo-dimensionalsurfacepatterns,theneglectofthird-harmonictermscancauseunsteadyfeaturesthatareevenmoresignificantthantheunsteadyfeaturesresultingfromtheneglectofthesecond-harmonicterms.Theyarguedthatitistheneglectofthesethird-harmonictermsintheStokes-typesolutionoftwo-dimensionalsurfacepatternsthatcausedtheunsteadyfeaturesobservedbyHHSlistedabove.FMprovidednumericalevidencethatsupportstheirhypothesis.Herein,wepresentexperimentalevidencethatverifiestheirhypothesisforthethreeunsteadyfeatureslistedabove.Theremainderofthepaperproceedsasfollows.In§2weoutlinethetheoreticalframeworkrequiredtodiscusstheFMpredictions.TheFMexplanationofunsteadi-nessisindependentofadditionalexperimentalcomplicationsthatarecausedbythepresenceofthewavemakerinthelaboratoryexperimentsandthatcouldpotentiallyLaboratorygenerationofprogressivesurfacewaves415contributetounsteadiness.Theseissuesarepartoftheoverallproblemofdeterminingawavemakerforcingthatwillproduceapermanent-formwavefield,whichisgenerallyreferredtoas‘thewavemakerproblem’.Thewavemakerproblemandadditionalexperimentalcomplicati