Excitation of edge waves on a beach by incoming swell is considered on the basis of shallow-water model. Subharmonic resonance mechanism of interaction is analyzed by 

multi-scaled expansion asymptotic techniques. The generation of edge waves between wave breakers is found to have a dynamic threshold. It is defined by intensity and frequency of incoming swell, geometry of a shore zone. Nonlinear no stationary wave solutions for the envelope of interacting edge waves are described by generalized 

Sine-Gordon model. An infinite set of exact solutions are received by the Lamb method for the phase synchronism regime of wave’s interaction

Joo Lee, Kwi

Jung Sun, An

Purini, R.

Shugan, I.

English

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2006-10018-xIL NUOVO CIMENTO Vol. 29 C, N. 4 Luglio-Agosto 2006Interaction of edge waves with swell on a beach(∗)Kwi Joo Lee(1), I. Shugan(1)(2), An Jung Sun(1) and R. Purini(3)(1) Department of Naval Architecture & Ocean Engineering, Chosun UniversityGwangju 501-759, Korea(2) Prochorov General Physics Institute of Russian Academy of Sciences - Moscow, Russia(3) ISAC, CNR - Rome, Italy(ricevuto il 9 Maggio 2006; approvato il 22 Maggio 2006)Summary. — Excitation of edge waves on a beach by incoming swell is consideredon the basis of shallow-water model. Subharmonic resonance mechanism of interac-tion is analyzed by multi-scaled expansion asymptotic techniques. The generation ofedge waves between wave breakers is found to have a dynamic threshold. It is definedby intensity and frequency of incoming swell, geometry of a shore zone. Nonlinearno stationary wave solutions for the envelope of interacting edge waves are describedby generalized Sine-Gordon model. An infinite set of exact solutions are received bythe Lamb method for the phase synchronism regime of wave’s interaction.PACS 92.10.Hm – Ocean waves and oscillations.1. – IntroductionEdge waves are of interest in coastal oceanography because of their propagation alongthe beach with amplitude maximal at the shoreline. They are responsible for the for-mation of beach cusps, generation of rip currents and periodic circulation cells in thenear shore region. Edge waves produce on the beach beautiful run-up patterns althoughpropagation is along the straight shoreline [1].The generation of edge waves has been intensively studied, both experimentally andtheoretically, in the last forty years. Greenspan [2] first demonstrated that large-scaleedge waves can be excited by atmospheric forcing due to storms moving along the coast-line. The typical period of this kind of edge wave is related to the spatial extent ofthe storm area and is of the order of several hours. For small-scale edge waves Guzaand Davis [3] proposed the nonlinear interaction mechanism of edge waves with incom-ing swell. Using the shallow-water approximation, they showed that a monochromaticharmonic wave train of frequency 2ω, normally incident and strongly reflected on a(∗) The authors of this paper have agreed to not receive the proofs for correction.c© Societa` Italiana di Fisica 487488 KWI JOO LEE, I. SHUGAN, AN JUNG SUN and R. PURINIbeach, is unstable to subharmonic standing edge-wave perturbations of frequency ω. Ex-periments [4] on a bounded beach indicated that this subharmonic resonance was thestrongest, and a subharmonic standing edge-wave was preferentially excited. Guza andBowen [5] made a systematic examination of the non-linear mechanism of subharmonicresonance. They used Airys shallow-water approximation as the basis of their theory.In addition to the initial instability of edge waves, the incident and reflected waves werefound to leak energy by radiation due to quadratic nonlinearity. In a further develop-ment of the theory, leading-order nonlinear corrections to the linear dispersion relationof traveling Stokes edge waves were calculated in [6]. Minzoni and Whitham [7] studiedthe excitation of standing subharmonic edge waves by a normally incident, strongly re-flected wave train. They formulated the problem in the full water-wave theory withoutmaking the shallow-water approximation and solved it for beach angles φ = π/2N , whereN is an integer. Their work confirms the results from the shallow-water theory in thesmall-beach-angle limit.The large-scale temporal and spatial modulations of subharmonic edge waves excitedby resonant interactions with normally incident, strongly reflected wave trains, werestudied in [8]. Modulation stability of a propagating edge-wave train was re-examinedand confirmed that the instability, predicted in [4], indeed leads to a series of envelopesolitons.Although many results in edge waves dynamics have been already reached, someimportant questions are still open. Firstly, the effect of the beach geometry on edgewaves has not been analytically studied. All the previous analytical work was done onan open beach. But if the beach is bounded by two sidewalls, which is always the casein experiments, this beach geometry will affect the edge-wave dynamics, sometimes evenexclude the excitation of edge waves. This effect was clearly shown in experiments [4].Secondly, the nonlinear evolution of subharmonic edge waves on a wide beach is stillnot clear. Since in this situation the spatial large-scale modulation arises as well as thetemporal one, the evolution equations of these modulations have been derived in [8]. Butwhat these equations imply about the edge-wave evolution is not known.This paper analyzes both of the above two problems. Excitation and spatial variationsof edge waves along a beach due to its resonance interactions with incoming swell arestudied on the basis of shallow-water model. An infinite number of analytical solutionsdescribing slow space and temporal evolution of edge waves are found for the commonsystem of modulation equations for resonance interaction of waves with quadratic non-linearity.2. – Derivation of modulation equationsWe consider a straight and long beach with constant small slope. Let the meanshoreline coincide with y-axis, and let water be in the region x > 0. The bottom isdescribed byz = −sx, x > 0, s = const,(2.1)where z is the vertical axis, directed upward.Since the slopes are assumed to be small due to the small inclination of the beach wecan use one of the forms of shallow-water equations [5]:ζt + [(sx + ζ)Φx]x + [(sx + ζ)Φy]y = 0;(2.2)INTERACTION OF EDGE WAVES WITH SWELL ON A BEACH 489Φt +12(Φ2x + Φ2y) + gζ = 0,(2.3)here ς = ς(x, y, t) is the sea surface level, Φ = Φ(x, y, t) the velocity potential function,g the gravity acceleration.Eliminating ς from the above equations the single non-linear equation for Φ(x, y, t) iswritten as shown below.Φ ≡ −Φtt + sg[(xΦx)x + xΦyy] =(2.4)= 2(ΦxΦxt + ΦyΦyt) + Φt(Φxx + Φyy)++12(Φ2x + Φ2y)(Φxx + Φyy) + Φ2xΦxx + Φ2yΦyy + 2ΦxΦyΦxy .The linearized approximation for this equation admits two types of solutions. Thefirst solution is the normally incident and strongly reflected waves with frequency 2ω:Φ0 = A0J0(β√x)e−2iωt + c.c.,(2.5)where J0(β√x) is the Bessel function of zero order, which is finite at the shorelinex = 0, A0 is the wave potential amplitude, β = 2(4ω2/gs)1/2, c.c. denotes the complexconjugate.The second solution is an edge wave mode of frequency ω, propagating along thebeach. Here only the lowest mode is considered for simplicity. The expression of theedge wave is given byΦe = Aee−kxei(ky±ωt) + c.c.,(2.6)corresponding for the pair of opposite edge waves:Φe1 = Ae1e−kxei(ky−ωt);(2.7)Φe2 = Ae2e−kxei(ky+ωt),here Ae1, Ae2 are the edge waves potential amplitudes, k is the wave number of edgewaves. We will assume all amplitudes to be small of the order ε = a0k0—the typicalsteepness for gentle waves, where a0 is the characteristic free-surface displacement, 2π/k0is the typical surface wavelength, multi-scaled asymptotic expansions in ε can be usedfor unknown functions.To study subharmonic resonance we will consider the velocity potential Φ as the sumof standing wave (2.5) and the pair of opposite edge waves (2.7):Φ = Φ0 + Φe1 + Φe2 =(2.8)= A0J0(β√x)e−2iωt + Ae1e−kxei(ky−iωt) + Ae2e−kxei(ky+iωt) + c.c.After substitution of expression (2.8) into eq. (2.4) resonant terms of the secondorder for each of the interacting harmonics ei(ky−iωt) , ei(ky+iωt) , e−2iωt will appear inthe right side of the equation. The method of slowly varying amplitudes will be used toeliminate these terms and to construct the uniformly valid solution in the second order490 KWI JOO LEE, I. SHUGAN, AN JUNG SUN and R. PURINIin wave steepness ε [9, 10]. So, all complex wave amplitudes are assumed to be slowlyvaried in space and time:A0 = A0(εy, εt) = A0(Y, T ),(2.9)Ae1 = Ae1(εy, εt) = Ae1(Y, T ),Ae2 = Ae2(εy, εt) = Ae2(Y, T ),where Y = εy, T = εt are slow space and time variables.Interaction of waves takes a place in a narrow stripe of width∼ 1/k along the shoreline.That is why we will use modulation equations, averaged over the x direction with theweight of the eigenfunction e−kx for edge waves [5]. After integration over the x directionand collecting resonant terms for all three harmonics one can receive the following systemof modulation equations for wave amplitudes in the second order:(Ae1)T + Cg(Ae1)Y = MA0Ae2;(2.10)(Ae2)T − Cg(Ae2)Y = MA∗0Ae1;(A0)T = −NAe1A∗e2,where Cg = Δω/Δk = (1/2)√gs/k is the group velocity of edge waves, M =4e−2(ω2gs)2,N = 2e43(ω2gs)2are the coefficients of the wave energy exchange, the asterisk (*) denotesthe complex conjugate. These are the final modulation equations for resonant interactionsbetween opposite traveling edge waves of frequency ω and normal incident standing waveof frequency 2ω.3. – Analysis of resultsHomogeneous-in-space waves interaction can be described by the system of eqs. (2.10)with time derivatives in its left side. It has three no independent integrals of motion(Manly-Row) [11]:|Ae1|2 = |Ae2|2 + const;(3.1)|A0|2 + |Ae1|2 = const;|A0|2 + |Ae2|2 = const.Integrals show the directions of energy exchange in this wave system: the incomingswell can give energy simultaneously to both of the resonance edge waves, and inverselythe pair of interacting edge waves can radiate wave energy out of the beach. Intensityof energy exchange between resonant waves is determined by its amplitudes and thecoefficients of energy exchange M and N on the right side of eqs. (3.1). It increaseswith amplitude, the frequency ω of incoming swell and decreasing with a slope of the seabottom. Periodic-in-time solutions can be expressed as elliptical functions [11].Temporal modulations of edge waves can take place on an open beach. But if thebeach is bounded by two sidewalls (wave breakers), this beach geometry will affect theedge wave dynamics, sometimes even excluding the excitation of edge waves. Clearlythe process of the standing edge waves generation needs the along-shore distance at leastequal to the space period of exciting waves. That means that for preventing of edgeINTERACTION OF EDGE WAVES WITH SWELL ON A BEACH 491Fig. 1. – Interaction of edge waves and a swell in a shore zone.waves generation wave breakers have to cut the shore in smaller “pieces”. This effect isclearly seen in experiments [4]. The spatial large-scale modulation arises in such a case aswell as the temporal one. Most intriguing here are the dynamical conditions, necessaryfor excitation of edge waves by incoming swell. A sketch of this process is presented infig. 1.To study edge wave excitation on a constrained beach zone we will consider stationaryspace modulations with a constant normal incident wave A0.From the first two equations of system (2.10) one can easily obtain the harmonicoscillation equation for the edge waves amplitudes Aei:(Aei)Y Y + χ2Aei = 0,(3.2)where i = 1, 2, χ = A0 8e−2k5/2√gs .The general solution of eq. (3.2),Aei = C0i cos(χY + C1i),(3.3)where C0i, C1i are constants of integration, shows that the edge wave amplitude has slowspace oscillations with wave number χ. The length of edge waves increases with thewavelength of the incoming swell and decreases with its amplitude.Standing edge waves can be excited in between wave breakers spaced at a distance ofD, if both amplitudes Aei satisfy the following homogeneous non-propagating boundaryconditions:Aei(Y = 0) = Aei(Y = D) = 0.(3.4)To satisfy (3.4) the period of the edge wave envelope oscillation T = 2π/χ has to benot more than 2D. That means the existence of dynamic space threshold or minimumnecessary distance D for the possibility of edge waves excitation by the incoming swell:D = π/χ = π√gs8e−2k5/2A0.(3.5)492 KWI JOO LEE, I. SHUGAN, AN JUNG SUN and R. PURINILess intensity of the edge wave formed between the wave breakers is considered as afavorable condition. Therefore the distance between waves breakers has to be made notmore than threshold D (eq. (3.5)) to prevent edge waves excitation.4. – Nonlinear waves interactionGeneral dynamics of resonant energy exchange between waves includes the spatiallarge-scale wave modulation as well as the temporal one. Let us rewrite the commonsystem of resonance interactions (2.10) by the following way:(Ae1)T + Cg(Ae1)Y = LA0Ae2;(4.1)(Ae2)T − Cg(Ae2)Y = LA∗0Ae1;(A0)T = −SAe1A∗e2,where L = 4e−2k2, S = (2/3)k2e4.After introducing new moving variables,ξ = CgT + Y, η = CgT − Y,∂∂T= Cg∂∂ξ+ Cg∂∂η;∂∂Y=∂∂ξ− ∂∂ηwe can rewrite the system of eqs. (4.1):∂Ae1∂ξ=L2CgA0Ae2;∂Ae2∂η=L2CgA0A∗e1;∂A0∂ξ+∂A0∂η= − S2CgAe1A∗e2.Renormalization of unknown functions:a0 =L2CgA0; a1e =√LS2CgAe1; a2e =√LS2CgAe2gives the final system of modulation equations:a1eξ = a0a2e;(4.2)a2eη = a∗0a1e;a0ξ + a0η = −a1ea∗2e.The system of eqs. (4.2) has two first integrals of motion:a1ea∗1e + 2a0a∗0 = g(η);(4.3)a2ea∗2e + 2a0a∗0 = f(ξ).INTERACTION OF EDGE WAVES WITH SWELL ON A BEACH 493Phase synchronism of waves is a typical regime for wave generation problems, so letus assume it here, and that means for all wave amplitudes to be real functions of spaceand time. Following substitution of variables and unknown functions,a1e =√g(η) sin(ϕ), a2e =√f(ξ) sin(ψ);ξ =12∫f(ξ)dξ,η=12∫g(η)dη,where g and f are arbitrary functions of its arguments, gives the system of equations fornew ϕ and ψ functions [12]:ϕξ= sin(ψ),(4.4)ψη= sin(ϕ).The received system of equations is equivalent to the generalized Sine-Gordon equa-tion [12-14] and describes a wide range of nonlinear wave’s phenomena.System of eqs. (4.4) has a striking property: sum and difference of unknown functionssatisfied to standard nonlinear Sine-Gordon [11] equation:(ϕ± ψ)ξη= sin(ϕ± ψ) .(4.5)So, we can easily construct a large (infinite) set of solutions for our model by usingdifferent known solutions of Sine-Gordon equation. If S1 and S2 are two of them, theneqs. (4.4) due to (4.5) will be satisfied by the following functions:ϕ =S1 + S22;(4.6)ψ =S1− S22.As the zero function is evidently the solution of eq. (4.5), it is possible to take S2 = 0,S1—one wave soliton solutions of the Sine-Gordon equation [11], and to receive thebrightest physical example here—constantly running wave solitons of the both edge waveenvelopes:tan(ϕ) = tan(ψ) = C exp(λη +ξ /λ),(4.7)where λ and C are free parameters. By taking two different solutions of (4.4) as (ϕ1, ψ1),(ϕ2, ψ2) then the sum and the difference of them will satisfy (4.5):(ϕ1 + ψ1), (ϕ2 + ψ2), (ϕ1 − ψ1), (ϕ2 − ψ2).(4.8)And using (4.7) it is possible to construct a subset of new solutions of the system (4.4)as(ϕ1 + ψ1 + ϕ2 + ψ2)/2; (ϕ1 + ψ1 − ϕ2 − ψ2)/2,(4.9)(ϕ1 + ψ1 − ϕ2 + ψ2)/2; (ϕ1 + ψ1 + ϕ2 − ψ2)/2.494 KWI JOO LEE, I. SHUGAN, AN JUNG SUN and R. PURINIBy repeating this procedure it is possible to construct an infinite number of solutions ofeqs. (4.4) and to describe the different regimes of many waves interaction.5. – ConclusionsResonant interactions between incoming waves and edge waves significantly dependon factors like wave frequency and amplitude of incoming waves, slope of the beach.Intensity of energy exchange between waves increases with amplitude and frequency of theincoming swell. The normal to the shoreline incident wave can give energy simultaneouslyto a pair of opposite resonance edge waves, and inversely a pair of interacting edge wavescan leak wave energy out of a beach. Analysis of spatial modulation enables us to studythe excitation of edge waves in between wave breakers. Less intensity of the edge waveformed between the wave breakers is considered as a favorable condition. The distancebetween wave breakers can be adjusted for less generation of edge waves. As the incidentwave length increases the distance between wave breakers can be increased, and increasingthe amplitude of the incoming wave leads to decreasing of the minimum distance betweenwave breakers. Consideration of fully nonlinear modulation model leads to an infinitenumber of wave solutions and describes various regimes of the many waves interaction.∗ ∗ ∗This study was supported by research funds from Chosun University 2005.REFERENCES[1] Komar P., Beach Processes and Sedimentation, 2nd edition (Prentice-Hall, NJ) 1998.[2] Greenspan H., J. Fluid Mech., 1 (1956) 574.[3] Guza R. and Davis R., J. Geophys. Res., 79 (1974) 1285.[4] Guza R. and Inman D., J. Geophys. Res., 80 (1975) 2997.[5] Guza R. and Bowen A., J. Marine Res., 34 (1976) 269.[6] Whitham G., J. Fluid Mech., 74 (1976) 353.[7] Minzoni A. and Whitham G., J. Fluid Mech., 79 (1977) 273.[8] Akylas T., J. Fluid Mech., 132 (1983) 197.[9] Debnath L., Non-Linear Water Waves (University of Florida) 1994.[10] Mei C., The Applied Dynamics of Ocean Surface Waves (World Scientific) 1989.[11] Ablowitz M. and Segur H., Solitons and the Inverse Scattering Transform (SIAM,Philadelphia) 1981.[12] Voliak K., Lyakhov G. and Shugan I., Phys. Lett. A, 96 (1993) 53.[13] Lyakhov G. and Suyazov N., Phys. Vibrations, 9 (2001) 79.[14] Krasnoslobodtsev A., Lyakhov G. and Suyazov N., Dokl. Phys., 47 (2002) 184.

Interaction of edge waves with swell on a beach

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Interaction of edge waves with swell on a beach

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