Xavier Bertin
email: xbertin@univ-lr.fr

Anouk De Bakker
email: anouk.de_bakker@univ-lr.fr

Ap Van Dongeren
email: ap.vandongeren@deltares.nl

Giovanni Coco
email: g.coco@auckland.ac.nz

Gaël André

Fabrice Ardhuin
email: ardhuin@ifremer.fr

Philippe Bonneton
email: philippe.bonneton@u-bordeaux.fr

Frederic Bouchette
email: frederic.bouchette@gm.univ-montp2.fr

Bertin Xavier

De Bakker Anouk

Van Dongeren Ap

Coco Giovanni

Andre Gael
email: gael.andre@shom.fr

Ardhuin Fabrice

Bonneton Philippe

Bouchette Frédéric

Castelle Bruno
email: bruno.castelle@u-bordeaux.fr

Davidson Mark
email: m.davidson@plymouth.ac.uk

Deen Martha

Dodet Guillaume
email: guillaume.dodet@univ-brest.fr

Guerin Thomas
email: thomas.guerin@univ-lr.fr

Inch Kris
email: kris.inch@plymouth.ac.uk

Leckler Fabien
email: fabien.leckler@shom.fr

Mccall Robert
email: robert.mccall@deltares.nl

Heloise Muller
email: heloise.muller@gmail.com

Olabarrieta Maitane
email: maitane.olabarrieta@essie.ufl.edu

Roelvink Dano
email: d.roelvink@un-ihe.org

Ruessink Gerben

Sous Damien
email: damien.sous@mio.osupytheas.fr

Stutzmann Eleonore

Tissier Marion

Infragravity waves: From driving mechanisms to impacts

Keywords: Infragravity waves, Bound wave, Dissipation, Reflection, Sediment transport, Barrier breaching, Seiche, Earth Hum

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1. Introduction snfrgrvity @herefter sqA wves re surfe oen wves with frequenies elow those of windEgenerted 4short wves4F ypil shortEwve frequenies re etween HFHR nd I rz wheres sq wve frequenies re generlly de(ned s eing etween HFHHR nd HFHR rzF por given wter depthD sq wves hve longer wvelengths thn short wvesX for exmpleD t wter depth of IH mD sq wves hve wvelengths from few hundred meters to kilometres wheres shortEwves hve wvelengths from few meters to hundreds of metersF hile the (rst oservtions of sq wves dte k only to the middle of the PHth entury @wunkD IWRWAD it is now well reognized tht sq wves ontriute onsiderly to nershore hydrodynmis @eFgF quz nd horntonD IWVPY ilE gr et lFD IWWPY eniers et lFD PHHPY uessink et lFD IWWVY omeroy et lFD PHIPY quedes et lFD PHIQAD sediment trnsport @eFgF ussellD IWWQY egrd nd qreenwoodD PHHVY he fkker et lFD PHITAD dune nd rrier rehing @eFgF oelvink et lFD PHHWAD development of seihe in hrours @eFgF ykihiro et lFD IWWQA nd they re onsidered to e the soure of kground free osilltions of the solid erthD lso referred to s 4the hum of the irth4 @eFgF eD PHHUY hie nd omnowizD PHHTAF his lrge rnge of implitions proly exE plins the growing interest in sq wves over the lst dedeD s it is ttested y the inresing volume of literture on the topi @pigF IAF wunk @IWRWA ws the (rst to identify reltion etween low frequeny moE tions long the shoreline nd the presene of groups in the inident short wvesF re lso found tht the mplitude of these long wves ws pproximtely proporE T 1950 1960 1970 1980 1990 2000 tionl to tht of inident short wves ut independent of their periodF uker @IWSHA performed rossEorreltion etween the short wve energy envelope nd the sq wve mplitude nd on(rmed the existene of liner reltion eE tween the twoF e few yers lterD fiesel @IWSPA provided the (rst mthemtil demonstrtion of the existene of seondry long wves fored y inident short wvesF epplying their onept of rdition stress @the momentum )ux ssoE ited with the short wvesA to Ih ihromti wve (eldD vonguetEriggins nd tewrt @IWTPA demonstrted tht groups in the inident short wves n fore soElled ound wveD IVH degrees out of phse with the mplitude of the shortEwve groupF rsselmnn @IWTPA extended this (nding to Ph rndom wve (eldD deriving n nlytil solution to ompute the ound wve from the diretionl spetr of the inident short wvesF sn the IWUHsD qllgher @IWUIA showed tht sq wves re)eted t the shoreline n e refrtively trpped in the nershoreD so tht qusiEsttionry longEshore ptternsD referred to s edge U wvesD n developF his (nding ws lter orroorted y fowen nd quz @IWUVA nd mny othersF he soElled pttern theory ws developed in the sme period nd proposes tht suh edge wves n imprint the morphology nd explin the development of periodi Qh ptterns on sndy ehes @fowen nd snmnD IWUIY rolmn nd fowenD IWVPAF elthough this theory ws hlE lenged y selfEorgniztion theories over the lst IS yers @eFgF plque± et lFD PHHHY goo nd wurryD PHHVAD it fostered muh reserh on sq wves in the period IWUHEPHHHF sn tht periodD ymonds et lF @IWVPA proposed n dditionl genertion mehnism for sq wvesD where the vrition of the rekpoint @the lotion where short wves rekA on the timesle of wve groups releses free long wves oth shorewrd nd sewrd of the reking zoneF e dede lterD hä'er @IWWQA omined the two min genertion mehnisms of sq wves @iFeF ound wve nd vrying rekpointA into one semiEnlytil modelF pield experiments in the VH9s nd WH9s provided importnt new insights into sqEwve dynmisF pollowing wunk @IWRWA nd uker @IWSHAD trends etween o'shore shortEwve height nd sqEwve height were identi(ed in the (eld @eFgF rolmnD IWVIY quz nd horntonD IWVPD IWVSY runtley et lFD IWWQAF he (rst method for seprting the inoming nd outgoing @freeA sqEwve signls ws developed y quz et lF @IWVRAD nd triggered the study of re)etion ptterns of sq wvesF et the sme timeD ispetrl nlysis @rsselmnn et lFD IWTQA o'ered more detiled understnding of the mehnisms responsile for sqE wve genertion @eFgF ilgr nd quzD IWVSY rerers et lFD IWWRY rerers nd furtonD IWWUY xorheim et lFD IWWVY uessinkD IWWVA whih demonstrtedD for V instneD the importne of diretionl spreding nd the spetrl shpe of inE ident short wves on the resulting sqEwve energyF purthermoreD @eFgFD quz nd horntonD IWVPY quz et lFD IWVRY rolmn nd llengerD IWVSY uenE heimer et lFD IWWTY uessink et lFD IWWVY uggiero et lFD PHHRA estlished the importne of sq wves in runup @the mximum vertil extent of wve upE rush on ehAD with prtiulrly lrge in)uene on mildly sloping ehes under energeti wve onditionsF sn dditionD mny (eld studies investigted the suspension nd rossEshore trnsport of snd y sq wves @eFgF edelrhE mn nd horntonD IWVUY feh nd ternergD IWVVY oelvink nd tiveD IWVWY hiym et lFD IWWIY ysorne nd qreenwoodD IWWPY ussellD IWWQAF rowE everD ontrsting onlusions were rehed in terms of trnsport diretion nd the respetive ontriution of sq wvesD suggesting tht prmeters suh s the eh pro(le or inident shortEwve onditions re lso importntF prom the lte WH9s onwrdsD lortory experiments o'ered more deE tiled view of sq wvesF he foers @IWWTA experiment ndD more reentlyD the qvyfi experiment @uessink et lFD PHIQAD nd the susequent nlyE ses resulted in strong improvements onerning the understnding of genertion mehnismsD propgtion @suh s sq wve height growth nd phse orrelE tion with the shortEwve envelopeA nd dissiption trends nd mehnisms @eFgF tnssen et lFD PHHQY fttjes et lFD PHHRY n hongeren et lFD PHHUY he fkker et lFD PHISY snh et lFD PHIUAF yver the lst IS yersD the development of numeril models ple of simE ulting the genertion nd propgtion of sq wves in the nershore strted W to emerge D following two distint pprohesF he (rst pproh ouples irultion model with spetrlD phseEvergedD model representing energy )ututions t the sle of shortEwve groups @eFgF n hongeren et lFD PHHQY eniers et lFD PHHRY oelvink et lFD PHHWAF he seond pprohD known s phseEresolvingD expliitly represents the short wves nd their indued iruE ltionD inluding the detiled intertions etween short wves nd sq wves @eFgF uennedy et lFD PHHHY ijlem et lFD PHIIY fonneton et lFD PHIIY heremet et lFD PHITAF yver the pst dedeD mny studies hve omined numeril modelling with (eld or lortory experiments to improve understnding of the proesses ontrolling the genertion mehnismsD the propgtion nd the trnsformtion of sq wves @eFgF he fkker et lFD PHISY ijnsdorp et lFD PHISY fertin nd ylrrietD PHITAD s well s their impts on sediment dynmis nd rrier rehing @eFgF oelvink et lFD PHHWAD nd on the hydrodynmi irultion in orl reefs @eFgF omeroy et lFD PHIPY n hongeren et lFD PHIQAF sn dditionD over the pst yers the importne of sq wves in other environments hs een identi(edD vrying from their role in the retion of ousti wve tivity in the thermosphere @qodin et lFD PHISY otin et lFD PHITAD their impt on li' top shking @oung et lFD PHITY irlie et lFD PHISAD to their in)uene on ieshelf ollpse in the entrti @eFgF fromirski et lFD PHISAF his pper reviews the stteEofEtheErt in knowledge out sq wves nd gives n overview of the lrge rnge of impts ssoited with this phenomenonF his inititive follows workshop tht took ple in v ohelle @prneA on the IH IUEIVth wrh PHIT nd gthered sustntil prt of the iuropen ommunity working on sq wvesF etion P reviews the min genertion mehnisms for sq wvesF etion Q summrizes the min trnsformtions tht sq wves experiene in the nershoreF he next setion presents lrge rnge of e'ets sq wves hve in vrious ostl environmentsD spnning from sediment trnsport to the development of seihes in hrours or the irth9s humF he lst setion provides onlusion nd disusses the future hllenges onerning reserh on sq wvesF 2. Generation mechanisms 2.1. Bound wave fiesel @IWSPAD followed y vonguetEriggins nd tewrt @IWTPAD demonE strted tht the presene of groups in inident short wves n fore seondry wve of similr frequeny s the groupD soElled ound wveF found sq wves re lredy generted in deep wterD ndD lthough they re smll @on the order of I mAD they undergo signi(nt trnsformtion nd growth in height when propgting from deep wter to the shorelineF sn oneptul desripE tionD onsider two short wves t disrete frequenies @ soElled ihromti wve (eldA propgting over horizontl ed @pigF PEeAF ine the two wves trvel t slightly di'erent eleritiesD the mplitudes of the wves lolly dd up or nel outF his pttern retes wve groups whose frequeny is equl to the di'erene etween the frequenies of the two onsidered short wves @pigF PEfAF sn generlD this frequeny di'erene is out one order of mgnitude lower thn the frequeny of the short wvesF hrough nonliner @seondEorder tokesA II intertionsD the wves fore slight depression nd rise in the men se level t the wve group lengthF his undultion n e seen s wve itself nd it is in ntiEphse with the wve groupsF sn other wordsD wves with higher mplitudes trnsport more momentum thn smll wvesD therey pushing the men wter level down under higher wves nd reting reltive wter level setup t the lotion of the smller wves @pigF PEfAF his long wve trvels phseEloked to the wve groupD nd is therefore lled ound wve @fieselD IWSPY vonguetEriggins nd tewrtD IWTPAF vonguetEriggins nd tewrt @IWTPA derived n equilirium solution tht reltes the ound wve mplitude to the energy of the short wvesX η (x, t) = -S xx (x, t) ρ gh -c 2 g + K @IA sn this equtionD S xx is the wve rdition stressD whih orresponds to the momentum )ux ssoited with short wvesD h is the men wter depthD ρ is the wter densityD g is the grvittionl elertionD c g is the short wves group veloity nd K is onstntF he theoretil results were vlidted with oservtionl dt in the lortory @eFgF uostenseD IWVRY fldok et lFD PHHHAF sn ntureD the short wve (eld is omposed of lrge numer of rndom omponents nd through suEhrmoni intertionsD spetrum of ound wve omponents is fored ording to mehnisms desried y rsselmnn @IWTPA nd rerers et lF @IWWRAD whih in e'et is twoEdimensionl generliztion of vonguetEriggins nd tewrt @IWTPAF hese ound sq wves typilly hve frequenies in the rnge of HFHHREHFHR rzD nd in deeper wter hve mplitudes and bound wave (red) as computed according to Longuet-Higgins and Stewart (1962). of mere entimetersF roweverD the equilirium solution proposed y vonguetE riggins nd tewrt @IWTPA is only vlid for )t ottomF por sloping ottomD the ound wve is no longer in equilirium with the energy envelope of the short wvesF es the wve groups propgte into shllower wterD the phse di'erene etween the wve groups nd the ound wve shifts wy from IVH degreesD so tht the long wves lg ehind the wve groupsF his phenomenon ws oserved in the (eld @eFgF wsselinkD IWWSY snh et lFD PHIUAD ws reprodued numerE illy y vist @IWWPAD ws oserved during lortory experiments y fttjes et lF @PHHRA nd he fkker et lF @PHIQA nd ws explined theoretilly y tnssen et lF @PHHQAF his phse shift llows energy trnsfer from short wves to the ound long wve @eFgF n hongeren et lFD PHHUA nd therefore growth of sq wvesD the rte of whih lies etween onservtive sholing @the growth IQ in height due to the onservtion of the energy )ux when sq wves slow down in deresing wter depthsD lso referred to s qreen' s vwAD h -1/4 nd the equilirium solution of vonguetEriggins nd tewrt @IWTPA @h -5/2 AF vorE tory oservtions show this phse shift to e frequeny dependentD with either lrger phse lg nd onsequent growth of the lowest @he fkker et lFD PHIQA or highest @fttjes et lFD PHHRA sq frequeniesD the reson for these ontrditory (ndings eing s yet unlerF fttjes et lF @PHHRA estlished tht this ound wve sholing mehnism is dominnt under mild slope regimeD whih ours when the normlized ed slope prmeter β b @see eqF PA is typilly elow HFQF

β b = h x ω g h @PA
sn this equtionD h x is the ed slopeD ω is the ngulr frequenyD g is the grvittionl elertionD nd h is the depth t the men rekpoint positionF por typilly β b ≥ 1D steepEslope regime previls nd the growth of sq wves due to this (rst mehnism is wek while the rekpoint mehnism eomes dominnt @see setion PFPAF he depthElimited reking of the individul short wves leves to shorwrd redution in the wveEgroup envelopeF es they re no longer ound to the groupD sq wves re relesed nd propgte s free wves @eFgF wsselinkD IWWSY tnssen during the derese in sqEwve energy @pigF SAF yne is the negtive nd rnging from out B@HFPPDHFPPA to B@HFRRDHAD where energy is trnsferred from the spetrl pek to frequenies lower thn the spetrl pekD inluding sq frequeniesF he other is positive nd rnging from B@HFRRDHA to B@HDRRDHFRRA where energy is trnsferred to frequenies higher thn the spetrl pek y intertions etween the spetrl pek nd frequenE ies lower thn the spetrl pekD inluding sq frequeniesF he positive nd rnging from B@HFRRDHA to B@HDRRDHFRRA in the ispetr @pigF SA domintes PQ over the negtive nd rnging from out B@HFPPDHFPPA to B@HFRRDHAD leding to sq wve energy dereseF yn the ontrryD in the surf zone of gently sloping ehesD or in the inner surf zone of mild sloping ehesD intertions involving short wve frequenies hve lredy disppered entirely nd the wve (eld is dominted y sq wve energyF rereD the ispetrum is dominted y sqEsq intertions @pigF SAD leding to the development of sq hrmonisD nd the steepening of the sq wve lose to shore @see setion QFPAF ttempted to explin the soure of sq wve energy loss proposed ottom frition s the dominnt dissiption mehnism @renderson nd fowenD PHHPAF vter studies showed tht ottom frition is too low on sndy ehes to ount for lrge sq wve energy lossesD nd is therefore now onsidered only s seondry dissiption mehnism @renderson et lFD PHHTY n hongeren et lFD PHHUY vin PR nd rwungD PHIPY n hongeren et lFD PHIQY he fkker et lFD PHIRAF yn the ontrryD for orl reefsD where the frition oe0ient is one order of mgniE tude lrger @c f ≈ HFHPEHFHSA thn on sndy ehesD ottom frition is urrently onsidered s the min use for sqEwve energy loss @omeroy et lFD PHIPY n hongeren et lFD PHIQAF st is now ler thtD on sndy ehesD prt of the energy loss t sq frequenies is not diretly dissiptedD ut is trnsferred wy from the sq ndF yn rther steep slopesD it is trnsferred k to short wve frequeniesD wheres on gentle slopes it is trnsferred to higher sq hrmonisD of whih prt my reh into the shortEwve nd @see setion QFIAF he developE ment of sq hrmonis leds to wve shpe hnge to skewed @peked wve rests nd longer wve troughsA nd symmetri @swEtooth shpedAD nd leds to the steepening of the sq wveF vortory nd (eld oservtions on gently sloping ehes @IXQSEIXVHA oserve this wve shpe hnge nd identify prtiulrly strong dissiption lose to the shorelineD suggesting reking s the dominnt dissiption mehnism @fttjes et lFD PHHRY n hongeren et lFD PHHUY vin nd rwungD PHIPY he fkker et lFD PHIRD PHISAF his phenomenon is supported y oservtions of oreElike sq wves in lortory experiments of xzk nd rino @IWWIA nd n hongeren et lF @PHHUAD nd numeril modelling y uju et lF @PHIPA nd he fkker et lF @PHISAF 3.3. Reection eny vrition of the properties of the medium in whih wves propgteD suh s the urrents or the wter depthD will use prtil wve re)etionsF hese prtil re)etions eome signi(nt when the vritions of the medium PS properties re lrge for the representtive wvelengthD whih is prtiulrly true for sq wves in ostl wtersF he intertion etween the inident nd the re)eted wves give rise to stnding @or sttionryA wve ptternD whihD for normlly inident monohromti wveD n e expressed sX

η(x, t) = η i (x, t) + η r (x, t) = a i sin(ωt -kx) + a r sin(ωt + kx) @QA
where a i nd a r orrespond to the mplitudes of the inident nd re)eted wvesD respetivelyF sf IHH7 of the inoming wve energy is re)eted @suh s ginst vertil wllAD a i = a r = a nd η(x, t) = 2a sin(kx) cos(ωt) @RA whih orresponds to stnding wve of mplitude 2a with nodes t lotions x = (1/4)λ + (n/2)λ @where n is n integerA nd ntinodes t lotions x = (n/2)λF hen the re)etion is not totl @R = a r /a i < 1AD the resulting wve is prtilly stnding wveD whih n e written s the sum of progressive wve nd fully stnding wveX η(x, t) = (a i -a r ) sin(ωt -kx) + 2a r cos(kx) sin(ωt) @SA he pioneering work of uker @IWSHA on sq wves @whih they referred to s 4surfet4 t tht timeA ws proly the (rst to mention the re)etion of sq wves t the ostF everl studies followed @eFgF uhydD IWURY runtley QH the shorelineD sine mesurements on the edge of the swsh zone re lmost impossile to ollet in the (eldD espeilly when using n rry of pressure sensors to estimte re)etionF elsoD n hongeren et lF @PHHUA used lorE tory ihromti wve experimentsD where the trnsition region is expeted to e stleD wheres he fkker et lF @PHIRA nd snh et lF @PHIUA investigted nturl ehes under irregulr wve (eldsD where the trnsition region is exE peted to vry in spe nd timeF he fkker et lF @PHIQD PHIRA oserved ulk squred sq re)etion oe0ients R 2 ≈ 0.1 t the shoreline of lowEsloping lE ortory eh with rndom wve onditionsF iyp nlysis of the sq seEsurfe elevtions demonstrted wellEdeveloped stnding wve pttern t low sq freE queniesD orresponding to R 2 ≈ 0.5D wheres high frequeny sq wves were predominntly onshore progressiveF ijnsdorp et lF @PHISA pplied the phseEresolving wve model er @iE jlem et lFD PHIIA to nturl eh nd showed tht er tends to underE predit the sq re)etion oe0ientD ut still reveled its frequeny dependent ehviour with nerEzero @respetively nerEperfetA re)etion for high @respeE tively lowA frequeny sq wvesF pinllyD snh et lF @PHIUA showed tht ny unorrelted signl in sensor rrys @see eppendix eA n led to n overestimte of the orreltion oe0E ientF elthough orretleD this potentil is in hs not een ounted for in mny prior studiesF QI 3.4. Edge waves he re)etion of infrgrvity wves t the shorelineD desried in setion QFQD n led to refrtively trpped motions whih ould turn into periodi longshoreEtrvelling ptternsD lso referred to s edge wvesF idge wves re freely propgtingD longshore periodi wves tht re trpped to the ost on one side y re)etion from the shoreline nd on the other side y refrtion over sloping thymetryF veky wves represent the ompnion to edge wves ut do not return to the ost y refrtionY instedD they espe to deep wterF hile edge wves re stnding in the rossEshore diretionD their longshore ehviour is usully progressiveF irly models of edge wve formtion dte k to tokes @IVRTA nd ikrt @IWSIAD with severl further re(nements provided in more reent yersF he mehnism leding to the pperne of edge wves is relted to wekly resonnt trnsfer of energy from re)eted inident wves on plnr eh to edge wve perturtionsF yn plnr eh with slope β disrete numer of edge wve modes n existD stisfying the dispersion reltion @ikrtD IWSIAX σ e = gk 2 e sin (2n + 1) tan β, @UA where σ e nd k e re the rdin frequeny nd longshore wve numer of the edge wvesD g is grvittionl elertionD nd n is the edge modl numerD whih orresponds to the numer of zeroErossings @nodesA in the rossEshore diretion efore the (nl mplitude dey t the turning pointF wode H edge wves @n = 0A hveD for given T e D the lrgest longshore wve numerF ith n inrese in nD the wve numer dereses @iFeFD the longshore wve length QP inresesA up to point where k e = σ 2 e /gD the deepEwter wve numerF he edge wve mode tht stis(es this reltion is lled the utEo' modeF por k < σ 2 /g ontinuum of leky wves existsF idge wves hve their lrgest mplitude t the shoreline nd rossEshore mplitude funtion Φ n (x) tht on plnr eh with slope β reds

Φ n (x) = e -kex L n (k e x) , @VA
where L n is the vguerre polynomil of order nF enlytil expressions for Φ n (x) lso exist for n exponentil eh pro(leD ut numeril pprohes must e used for more omplex eh pro(les @rolmn nd fowenD IWUWAF olutions should lso e tken into ountF por the ske of simpliityD only the former se will e onsidered in this setionF sn the permnently sumerged zoneD extending from deep wter to the end of the inner surf zoneD sq wves impt the groundwter dynmis minly throughX @iA the )ututions of pressure t the se ed interfe driven y the sq onE tentD nd @iiA the struture of the porous soil @possily evolving through timeAF he former n e omputed usingD eFgFD liner wve theory while the ltter n e estimted from some soil properties @porosityD sturtionD omptionD depth of the porous lyerD etA using models of poreEpressure trnsmission in QW homogeneous soils @ki et lFD IWWPAF sn the ontext of sedimentry ehesD poroEelsti soil theories revel the importne of enpsulted gs in the sndD even for the smll gs ontents @ few perentA oserved in the super(il lyer @0.5 -1mA of intertidl snds @fonjen et lFD PHHRAF ressure wves pplied on the se ed re oth ttenuted nd phse shifted when propgting within the soilD whih indues the development of vertil pressure grdients @wihllet et lFD PHHWAF hese proesses re strongly dependent on wve frequenyX the longer the wveD the stronger the groundwter pressure osilltionF his efE fet is qunti(edD mong othersD in poroEelsti soil models @ki et lFD IWWPAF events under lowEfreeord @iFeF the height di'erene etween the men wter level nd the rest of the rrierA onditions were relted to sq wve motionsF hespite the srity of dtD sine overwsh is n extension of the swsh zoneD it is generlly expeted tht sq wves ply signi(nt role in overwsh onditions y inresing wve runEup nd lowering overwsh thresholdsD nd inresing the mgnitude of the overwsh dishrge @eFgFD oelvink et lFD PHHWY wgll et lFD dtD edelrhmn nd hornton @IWVUA nd oelvink nd tive @IWVWA proE posed di'erent hypothesis where the presene of the lrgest short wves t either the sqEwve trough @ound wveA or rest @free wveA ould explin the timing of sediment suspension reltive to sq wve phseD nd the susequent trnsport diretionF his timing of suspension n e desried y the orrelE tion r 0 etween the short wve envelope nd sq motionF sn the sholing zone nd outer prt of the surf zoneD lrger @rekingA short wves re present t the sq wve trough @the sq wve n still e onsidered to e oundAD nd r 0 is negtiveF gonsequentlyD net trnsports re direted sewrdD s oserved y for exmple vrsen @IWVPAY runtley nd rnes @IWVUAY ysorne nd qreenwood @IWWPAY uessink @IWWVAY mith nd woke @PHHPAF sn dditionD s wter depths re lowered lolly in the sq troughD short wve oritl veloities re lrger lose RT to the edD y whih the lrger ed sher stresses n suspend even more snd thereF sn the inner surf zoneD the lrger short wves re present on the sq wve rest @the sq wves n e onsidered free wveAD nd the orreltion is posE itiveF hue to the lolly rised wter levelsD short wves n persist longer t the sqEwve rest thn in the troughF gonsequentlyD they suspend more snd t the sqEwve restD providing net shorewrd trnsport t sqEwve timesleD s oserved yD for exmpleD ysorne nd qreenwood @IWWPAF e seond theory hs een proposed y egrd nd qreenwood @PHHVA who studied snd trnsport diretions t two rred ehesD nd oserved trnsport diretions to e relted to the position with respet to suspension mximumF hese suspension mxim our t positions with reltive short wve height mxim @whih t their sites typilly ours on upper sewrd slopes or r restsA where short wves suspend lrge mounts of snd while rekingF hey oserved shorewrd trnsport of snd y sq wves t the lndwrd side of suh suspension mximD nd sewrd trnsport on their sewrd side of suh mximF heir theory my e relted to the ove desried orreltion theoryF yther oservtions ontrdit these theoriesD s for onditions where sq wves dominted the wter motion in the inner surf zoneD the sq wve ws seen to suspend snd s well @eFgF feh nd ternergD IWVVY ussellD IWWQAF xo preferentil lotion of short wves ould e distinguished for this dt @r 0 would e nerly zeroAD nd snd ws mostly suspended under the sq troughD reting sewrd direted snd trnsportF RU eentlyD sed on mesurements on oth gently @β ≈ IXVHA nd intermeE ditely @β ≈ IXQSA sloping eh he fkker et lF @PHITA proposed tht the rtio Four simulations are run in which the barrier island is exposed to four dierent water level, wave height and wave period forcing conditions for a duration of three hours to characterise storms in the swash, collision, overwash and inundation regimes, as dened by Sallenger (2000).

Dierences between the models are particularly apparent in the collision and overwash regime. (2015). Light blue to red vertical stripes correspond to earthquakes (not modeled). The dashed line separates the low frequencies where the primary mechanism dominates from the higher frequencies explained by the secondary or 'double-frequency' mechanism. The Johanna storm, on March 10, is conspicuous with powerful and lower frequency microseisms.

e pourier nlysis of motions in the solid irth nd oen lyer shows tht the trnsfer of energy from oen wves to seismi wves is signi(nt only if oen wve motions mth oth the wvelength nd periodsD nd hene the speedD of seismi wves @rsselmnnD IWTQAF his neessry mthing of the speeds of di'erent wve trins pplies to ll sorts of wve motions @rsselmnnD IWTTAD for exmple the genertion of tmospheri wves y tsunmis @eFgF ertru et lFD PHHSAF wo types of intertions n e the mthEmkers etween sq wvesD whih trvel t few hundreds of meters per seondD nd muh fster seismi wvesD whih reh S kmGs t the se)oorF he primry mehnism is n TR interferene of surfe wves of wvenumer k w nd frequeny f w with ottom topogrphy k b D whih produe seismi wves of wvenumer k s = k w + k b nd frequeny f s = f w F he seondry mehnism is the interferene of two wve trins of wvenumer nd frequeny

(k 1 , f 1 )D (k 2 , f 2 )D produing seismi wve of wvenumer k s = k 1 + k 2 nd frequeny f s = f 1 ± f 2 F feuse |k s |/(2πf s )
must e equl to the seismi phse speedD these resonne onditions impose

k 2 -k 1 nd thus f 1 f 2 nd f s 2f 1 F hese onditions mke it impossile
for the intertion f s = f 1 -f 2 to produe signi(nt seismi wve mplitude @rsselmnnD IWTQY eD PHHVAD ontrry to the propositions y hiym nd willims @PHHVA nd rer nd qerstoft @PHIRAF ithout ny of these two mehnismsD oen wves propgting over )t ottom only produe pressure osilltions in the wter nd deformtion of the ottom tht is proportionl to the lol oen wve mplitude ut whih nnot propgte s seismi wvesF his e'et is known s ompline nd is used in geophysil studies of oeni rustl struture @eFgF grwford et lFD IWWIAF erdhuin et lF @PHISA pplied the wveEwve nd wveEottom intertion theories of rsselmnn @IWTQAD tking into ount the neessry orretion for (nite depth given y erdhuin nd rerers @PHIQAF he result is shown in pigF RFSFIF he seondry mehnism is too wek y IH orders of mgnitude to explin the reorded humD wheres resonle guess of n e'etive ottom slope of T7D omined with the glol sq wve model of erdhuin et lF @PHIRA gives good greement etween oth the men oserved hum level @pigF RFSFIA nd its temporl vriility @erdhuin et lFD PHISAF

η ± = η ± h g 1/2 u 2 , @WA
where h is wter depth nd g is grvittionl elertionF

A.1.2. Spectral-domain method by Sheremet et al 2002 he spetrlEdomin method of heremet et lF @PHHPA lso uses oEloted wve guges nd veloity metersD ut hs inoming nd outgoing energy )uxes F ± f s output for seprte frequenies followingD

F ± f = N ± f gh, @IHA N ± f = 1 4 C ηη,f + h g C uu,f ± 2 h g C ηu,f , @IIA
where N is the energyD C ηu is the η -u oEspetrum nd C ηη nd C uu re η nd sponsile for the trnsfer of energy @pressure fore times veloityA from the power spetrl pek to suE nd superhrmonisD whih led to hnge in wveshpe to skewed nd symmetri lose to shore @eFgF ilgr nd quzD IWVSAF hese energy trnsfers our grdully to djust to the derese in wter depthD nd re stronger when wter depths re shllower s wves re loser to resonneF he disrete ispetrum is de(ned sD

B f1,f2 = E[A f1 A f2 A * f1+f2 ], @ISA
where E is the ensemle verge of the triple produt of omplex pourier oe0ients A t the frequenies f 1 D f 2 nd their sum f 1 + f 2 D nd the sterisk indites omplex onjugtionF imilrlyD the power spetrum is de(ned sD

P f1 = 1 2 E[A f1 A * f1 ].
@ITA he vrine of the ispetrl estimtes is dependent on the power spetrl propE IIU erties of the signlD if not enough verging is performed @y using loks nd frequeny mergingAD the ispetrl estimtes might pper lrge just euse it is highly vrileF e normlized mesure of the strength of the oupling of the interting wve omponentsD whih removes this vrineD the ioherene b 2 f1,f2 D is here de(ned sD b 2 f1,f2 = |B f1,f2 | 2 P f1 P f2 P f1+f2 , @IUA following gollis et lF @IWWVAD iqF PUD nd is the most ommonly used norE mliztion method for sttistil testsD lthough it ontins no upper oundF everl other wys exist in whih ioherene n e qunti(edD see for instne rurih @IWTSA nd uim nd owers @IWUWAD the ltter method hving n upE per ound of ID when there is not verged over frequenyF por omprison etween the normliztion methodsD see for exmple ilgr nd quz @IWVVA nd gollis et lF @IWWVAF he WS7 signi(ne level on zero ioherene is de(ned s

TGdFoFfFD where dFoFfF is the degrees of freedomF he normlized phse of the ispetrumD lled the iphse β f1,f2 gives normlized mesure of the phse reltionship nd is de(ned sD β f1,f2 = arctan I{B f1,f2 } R{B f1,f2 } , @IVA following uim nd owers @IWUWAD where I nd R re the imginry nd rel prt of the ispetrumD respetivelyF he stility of the iphse estimtes is highly dependent on ioherene vluesY for low iohereneD the iphse is IIV rndomly distriuted etween π nd EπF he iphse n e relted to the wve shpeD with vlues lose to zero @imginry prt is lose to zeroA inditing skewed wveD nd vlues lose to EWH @rel prt is lose to zeroA inditing n symmetri @eFgF ilgr nd quzD IWVSAF he iphse n e integrted over seprte frequeny ndsD to single out the wve shpe of the ound infrgrvity wve @iphse ≈ IVH degreesA @eFgF he fkker et lFD PHISAF es the ispetrum hs symmetril propertiesD it is only neessry to evlE ute the ispetrum in the prinipl domin where f 1 D f 2 > HD f 2 < f 1 nd a HFWH rz to oth f 1 a HFVT rz nd f 2 a HFHR rzF hese two intertions re responsile for the growth of the ound infrgrvity wveD while t the sme time using the energy round the spetrl pek nd its hrmonis to shift to slightly lower frequeniesD see lso he fkker et lF @PHISAF fy integrting over the imginry prt of the ispetrum nd herey ominE ing ll the seprte intertionsD nd multiplying tht with oupling oe0ientD net nonliner energy trnsfers etween frequenies S nl n e lulted @eFgF rerers et lFD PHHHY he fkker et lFD PHISAF he urrently de(ned oupling oe0ients vry from the foussinesq sling @eFgF rerers nd furtonD IWWUA vlid for resonnt wves in shllow wterD up to the more generlized theory @eFgF tnssenD PHHTAD whih inludes full liner dispersive e'etsF sn shllow wE ter during oth sholing nd reking seEswellEwve onditionsD the foussinesq pproximtion yields the most relile results @rerers nd furtonD IWWUY rerE ers et lFD PHHHY mit et lFD PHIRAD lthough the strong nonlinerities ourring in the surf zone re somewht unrepresentedD nd thirdE nd higherEorder inE IPH tertions would need to e inorported for more relile estimtes @homson et lFD PHHTY he fkker et lFD PHISAF sing the stohsti formultion of the seondEorder nonliner wve intertion theory of rerers et lF @PHHHA @their iqF@RAA S nl n e determined disretely yX

f 1 + f 2 < f N GPD
S nl,f = 3πf h I      f f =0 ∆f B f ,f -f -2 f N -f f =0 ∆f B f ,f      . @IWA he term f f =0 B f ,

Figure 1 :

 1 Figure 1: Number of paper per year found in Scopus and Web of Science where the title includes infragravity waves or surf-beat.

  Figure 2: (A) Time series of two sinusoidal waves with periods of 14 s (blue) and 15 s (pink) travelling over a at bottom by 20 m water depth. (B) Resulting free surface elevation (blue) and bound wave (red) as computed according to Longuet-Higgins and Stewart (1962).

Figure 4 :Figure 5 :

 45 Figure 3: Schematic representation of the cross-shore variation of the minimum, mean and maximum short wave height, with the associated steady state setup through the surf zone. x 1 and x 2 are the minimum and maximum positions of the breakpoint, and h is the water depth. (Figure reproduced from Symonds et al. (1982).)

  3.2. Dissipationine the lte IWWHsD severl (eld @eFgFD uessinkD IWWVY heremet et lFD PHHPY renderson et lFD PHHTY quedes et lFD PHIQY he fkker et lFD PHIRY snh et lFD PHIUY piedler et lFD PHISAD lortory @eFgFD fttjes et lFD PHHRY n hongeren et lFD PHHUY he fkker et lFD PHISA nd numeril modeling @eFgFD n hongeren et lFD PHHUY uju et lFD PHIPY he fkker et lFD PHISA studies hve oserved tht sqEwve energy my derese onsiderly ner the shorelineF eserh onduted in the swsh zone lso oserved energy dissipE tion t sq frequeniesD s swsh exursion did not inrese with more energeti o'shore wve onditionsD inditing sturtion @eFgFD uessink et lFD IWWVY ugE giero et lFD PHHRY énéhl et lFD PHIIY piedler et lFD PHISAF he (rst study tht

Figure 6 :

 6 Figure 6: Relations between the reection coecient R and the normalized bed slope parameter β H , modied from Van Dongeren et al. (2007)

Figure 7 :

 7 Figure 7: Numerical modelling of wave-driven currents and passive drifter evolution at the ripchannelled beach of Biscarrosse, SW France (Bruneau et al., 2014) for summer wave conditions (Hs = 1m, Tp = 10 s, shore-normal incidence) at a tidal level maximizing rip current activity (see for more detail on the modelling strategy in Castelle et al., 2016a). (a,c) Mean ow patterns with the colour bar and the blue box indicate velocity in m/s and the location of the initially seeded passive drifters, respectively. (b,d) Drifters (black bubbles) after 20 minutes of simulations. Left-hand and right-hand panels are without and with wave-group-forced IG motions, respectively.

Figure 8 :Figure 9 :

 89 Figure 8: Groundwater pressure head and velocity elds during uprush (top plot) and backwash (bottom plot) phases of an IG-driven swash event measured during the Rousty1412 eld campaign in the Rousty beach (Camargue, France). The blue solid line indicates the free surface elevation measured by terrestrial LiDAR while the solid black line indicates the sand bed. Each white circle within the sand soil corresponds to a buried pressure sensor (5Hz acquisition frequency). Color contours indicate the groundwater head eld, interpolated from measurements at each sensor. Vectors are groundwater velocities calculated from the head eld using the Darcy's law.

  PHIHAF 4.1.4. Sediment transport hue to their importnt ontriution to the surfE nd swsh zone hydrodyE nmisD sq wves hve long een thought to ply n importnt role in the sediment dynmis nd susequent morphologil hnges of sndy osts @eFgFD fowen nd runtleyD IWVRY right nd hortD IWVRY llenger et lFD IWVSY vippE mnn nd rolmnD IWWHAF he suspension nd rossEshore trnsport of snd y sq wves hveD thereforeD een investigted in mny studies @eFgF edelrhE mn nd horntonD IWVUY feh nd ternergD IWVVY oelvink nd tiveD IWVWY hiym et lFD IWWIY ysorne nd qreenwoodD IWWPY ussellD IWWQY egrd RS nd qreenwoodD IWWRD IWWSY uessink et lFD PHHHY mith nd wokeD PHHPY gonE ley nd fehD PHHQY rouser nd qreenwoodD PHHSY egrd nd qreenwoodD PHHVY fldok et lFD PHIHY elsin nd gáeresD PHIIY egrd et lFD PHIQY uuE lrtne nd ttirthiD PHIRY omeroy et lFD PHISY he fkker et lFD PHITAF roweverD the diversity in eh pro(les nd o'shore wve onditions hs reted ontrsting oservtions in diretion nd reltive importne of rossEshore sq snd trnsport s well s the suspension mehnismF ome studies oserved tht shortEwves were the min snd stirring mehE nismD wheres sq wves only dvet this suspended snd either onshore or o'shore @eFgF vrsenD IWVPY runtley nd rnesD IWVUY ysorne nd qreenwoodD IWWPY uessink et lFD IWWVY mith nd wokeD PHHPAF fsed on hydrodynmi

Figure 10 :Figure 11 :

 1011 Figure10: Example timeseries of a positive correlation r 0 (a,c,e) in the inner-surf zone of the steeply sloping Zandmotor beach and a negative r 0 (b,d,f) in the inner-surf zone of the gently sloping Ameland beach. Please note the dierent y-axis scales.(a,b) sea-surface elevation (m) and (c,d) cross-shore velocity with in black the total velocity, and in blue the IG velocity, and (e,f) sand concentration. For Ameland location P11, h = 1.00 m, r 0 = -0.04, H total /h = 0.67, ū = -0.15 m/s, H IG /H SW = 1.00. For the Zandmotor location P10, h = 0.5 m, r 0 = 0.17, H total /h = 0.66, ū = -0.16 m/s, H IG /H SW = 0.39. After DeBakker et al. (2016).

Figure 12 :Figure 13 Figure 14 Figure 15 :

 12131415 Figure 12: Time-series of observed water depths (top), H m0 in the gravity (blue) and in the (red) IG bands (middle) and frequence repartition of the energy density inside the Albufeira Lagoon (Portugal) in September 2010

Figure 16 :

 16 Figure 16: Mean values of H IG over (a) January and February 2008, (b) June and July 2008. Small square with numbers correspond to the location of DART stations used here for model validation. Taken from Ardhuin et al. (2014).

u

  utoEspetrD respetivelyF ummtion over infrgrvity frequenies gives the ulk infrgrvity energy )uxD F ± b F he ulk re)etion oe0ientD R 2 b D is de(ned s the rtio of the o'shore to onshore propgting ulk infrgrvity energy )uxD R 2 b = F - b /F + b F IIQ A.2. Array methods erry methods deompose wve signl into its shorewrd nd sewrd propgting omponents y using rossEshore rry of sptilly seprted presE sure sensors only @iFeFD without veloity dtA nd rely on the phse di'erene of wves etween individul sensorsF erry methods provide n estimte of the inident nd re)eted infrgrvity omponents t the entre of the instrument rry used in the nlysisF e numer of these methods existD operting in oth the time @eFgF prigrd nd frorsenD IWWSA nd frequeny @eFgF qod nd uzukiD IWUTY wnsrd nd punkeD IWVHY n hongeren et lFD PHHUA dominsF ressure sensors re typilly more eonomil thn veloity sensors llowing wider rnge of sptil mesurements with lower ostD thus otining etter ppreE ition of the sptil vriility in infrgrvity wve dynmisF purthermoreD pressure sensors less otrusive nd more roust so n e deployed for longer time periodsD essentil when investigting the limtology of infrgrvity wvesF wost rry methods re intended for the study of two dimensionl wves propgting over horizontl ed nd re not designed for use on slopingD nturl ehF fldok nd immonds @IWWWAD using the method of prigrd nd frorsen @IWWSAD showed tht ignoring the e'ets of wve sholing n led to errors of up to WH7 in the estimtion on inident nd re)eted wve mplitudesF roweverD they lso demonstrte tht reltively simple modi(tions re needed to dpt rry seprtion methods for using on sloping edsF sndeedD the method of n hongeren et lF @PHHUA is dpted from tht of fttjes et lF @PHHRA with modi(tions for sholing nd phse speed e'etsF IIR en importnt onsidertion in the study of infrgrvity wves is tht rry methods require strtegi pproh to the seprtion distne etween presE sure sensors to void singulrities ourring t disrete frequenies where the sensor sping is typilly equl to n integer numer of hlf wvelengthsF por this resonD rry methods perform est with three or more pressure sensors sped unevenly to provide greter rnge of seprtion distnes for use in the nlysisF elthoughD s more pressure sensors re usedD the susequent wve re)etion estimtes re verged over lrger rossEshore rngeF e study y snh et lF @PHIUA showed tht n dditionl soure of error tht n 'et rry methodsD prtiulrly in the (eldD is rndom signl noiseF sing numeril simultionsD they show tht the presene of noise introdues signi(nt positive is to inident nd re)eted spetr estimtesD nd orreE sponding re)etion oe0ientsF e tehnique is introdued tht n e pplied to ny rry method to investigte the impt of noise nd develop orretion funtion for suh noiseF epplying this tehnique to the rry method of qillrd et lF @IWVHAD orretion funtion is developed thtD when pplied to (eld dt from dissiptive ehD suggests tht infrgrvity re)etion oe0ients my e overestimted y s muh s SH7 euse of noiseF A.3. Radon transform e fourth method is the reently revisited don trnsform @elmr et lFD PHIQD PHIRAF o fr it hs only een tested on syntheti ses nd lortory dtD ut it shows good potentilF st projets the twoEdimensionl wve (eld IIS into polr spe followingD R(ρ, θ) = ‹ η(x, y) d(xcosθ + ysinθ -ρ) dx dy @IPA where δ is the hir delt funtionD θ nd ρ re the ngle nd distne from origin of the integrtion line de(ned s ρ a xcosθ C ysinθF he origin is the enter of the twoEdimensionl @xDtA wve (eldF ve rests re identi(ed s density peks in the polr speF he wve (eld n e deomposed over the whole xEt wve (eld s the inoming nd outgoing wve trins pper in the don spe within the θ a 1 • -89 • nd θ a 91 • -179 • intervlsD respetivelyF he originl wve (eld n then e k projeted with the snverse don rnsform pplied seprtely to the di'erent ngles sD η + (x, t) reltively insensitive to wve hrteristisD ut is prone to the smpling sheme nd the numer nd density of wve gugesF @he distne etween guges should e less thn one third of the shortest wvelengthD while the rry should over more thn one third of the longest wvelength @elmr et lFD PHIRAFA IIT B. Bispectra ine the introdution of ispetrl nlysis y rsselmnn et lF @IWTQAD it hs een used in numerous studies to investigte nonlinerities of wve (elds @eFgF preilih nd quzD IWVRY ilgr nd quzD IWVSY rerers et lFD IWWRD IWWSY xorheim et lFD IWWVY rerers et lFD PHHHY homson et lFD PHHTY he fkker et lFD PHISDAF sn ddition to the power spetrumD whih ontins no phse informtionD the ispetrum B f1,f2 detets phseEoupling etween frequeny omponentsD more spei(lly three frequeniesF hese trid intertions re reE

IIWFigure 17 :

 17 Figure 17: Imaginary part of the bispectrum in m 3 (x 10 -8 ) of the incoming wave signal (η + ) over a 1:80 laboratory slope, at x = 70 m, h = 0.18 m. Black solid lines indicate the cuto between infragravity and sea-swell wave frequencies, f IG . Dashed lines indicate the spectral peak (f = 0.44 Hz) and its higher harmonics. After De Bakker et al. (2015a).

  f -f ounts for the sum intertions in the imginry prt of the ispetrumD nd the term -2 ∞ f =0 B f ,f ounts for the di'erene inE tertionsD s eh prtiulr frequeny n prtiipte simultneously in oth di'erene nd sum intertionsF o study intertions inluding infrgrvity frequenies in more detilD the ispetrum n e further divided into zones with di'erent ontriutions of infrgrvity frequeniesD fter whih S nl n e determined for those seprte zones @he fkker et lFD PHISAF IPI