Edge Wave Scattering by Coastal Structures on Arbitrary Bathymetry

Journal of Coastal Research 24 6 1536–1544 West Palm Beach, Florida November 2008

Edge Wave Scattering by Coastal Structures onArbitrary BathymetryAlberto Avila, Asuncion Baquerizo, and Miguel A. Losada

Universidad de GranadaGrupo de Puertos y CostasCentro Andaluz de Medio AmbienteAv. del Mediterraneo s/n (CEAMA)18001, Granada, [email protected]

ABSTRACT

AVILA, A.; BAQUERIZO, A., and LOSADA, M.A., 2008. Edge wave scattering by coastal structures on arbitrarybathymetry. Journal of Coastal Research, 24(6), 1536–1544. West Palm Beach (Florida), ISSN 0749-0208.

The formation of rhythmic patterns at the Field Research Facility (U.S. Army Corps of Engineers, Duck, NorthCarolina) under extreme wave climatic conditions is analyzed in terms of the partial standing oscillations formed bythe superposition of the incident edge wave and the wave reflected by the structure. The reflection coefficient at thepier is evaluated with an existing model used for the propagation of an edge wave through a permeable structureperpendicular to the coastline, modified here to deal with an arbitrary beach profile. The mass transport patternsunder the partial standing edge waves are found to resemble the beach cusps and the crescentic bars observed on theshore, north of the pier, after a series of storm events.

ADDITIONAL INDEX WORDS: Infragravity waves, rhythmic forms, beach cusps, crescentic bars, morphodynamics,reflection.

INTRODUCTION

Edge waves are low-frequency water waves trapped to thecoast in the sense that the wave energy propagates parallelto the shoreline and cannot escape from it. In the cross shoredirection, they are in the form of standing oscillations thatreach maximum amplitudes near the shore and decay expo-nentially seaward.

Edge waves can be generated at infragravity frequenciesby wind-generated waves impinging on a beach, forced bywave groups (GALLAGHER, 1971; SCHAFFER, 1990), or by thenonlinear interaction of spectral components (HASSELMANN,1962). They can also be excited by an incident wave trainthrough a nonlinear mechanism leading to synchronous andsubharmonic edge waves, at equal frequencies and at one-half of the frequency of the incoming wave train, as showntheoretically by GUZA and DAVIES (1974) and observed in thelaboratory by GALVIN (1965), HARRIS (1967), and BOWEN andINMAN (1969).

Many studies have noted the existence of rhythmic struc-tured coastal bedforms in the presence of edge waves, accept-ing that these features are the footprint of the hydrodynamicforcing. Some studies assume that they are generated by thecombined action of the incident wave with an alongshore prop-agating edge wave (BOWEN and INMAN, 1971; CONSTANTIN,2001; DYER, 1986; GUZA and BOWEN, 1975, 1981; HUNTLEY

and BOWEN, 1975; INMAN and GUZA, 1982; SEYMOUR and AU-BREY, 1985; VITTORI, DE SWART, and BLONDEAUX, 1999).

DOI: 10.2112/07-0898.1 received 6 June 2007; accepted in revision 30October 2007.

HOLMAN and BOWEN (1982) studied the interaction of any twoedge wave modes and proposed that the associated drift veloc-ity may be responsible for the generation of complex rhythmictopography. Particular cases of this phenomenon are standingand partial standing edge waves.

A classical solution in coastal engineering to prevent theerosion of a beach, or to control the alongshore sedimenttransport, is to place a groin perpendicular to the coastline.Additionally, some beaches may have a pier, also perpendic-ular to the coastline, in order to provide access to deeper wa-ters. Both structures partially reflect the incoming edge waveenergy, which according to HOLMAN and BOWEN (1982) mayaffect the beach morphology.

BAQUERIZO, LOSADA, and LOSADA (2002) proposed a modelfor the propagation of an edge wave through a permeablegroin perpendicular to the coastline in a plane beach with asteep foreshore. BAQUERIZO and LOSADA (2001) extended thesolution for a system of two permeable groins. In this work,the model by BAQUERIZO, LOSADA and LOSADA (2002) is im-proved with the numerical resolution of the edge wave prob-lem for an arbitrary beach profile. The proposed formulationis then used to explore the relation between the partial stand-ing edge waves and the periodic forms observed at the northof the Field Research Facility (U.S. Army Corps of Engineers,Duck, North Carolina) after the passage of a series of stormsin March 2004.

The paper is organized as follows. First, a description ofthe case study including the field site, wave climate condi-tions, and the observed rhythmic features is presented, andthe working hypothesis is formulated. The proposed model is

1537Edge Wave Scattering by Coastal Structures on Arbitrary Bathymetry

Journal of Coastal Research, Vol. 24, No. 6, 2008

Figure 1. Directional wave spectrum at the peak of the strongest storm(March 11, 2004, 0700).

then introduced and applied to the case study. The resultsare used to calculate the mass transport velocity under thepartial standing edge wave, which is compared with therhythmic forms observed at the northern part of the pier.Finally, a discussion about the validity of the working hy-pothesis is given with some conclusions.

FIELD DESCRIPTION AND OBSERVATIONS

The Field Research Facility is situated on a relatively longbeach with almost straight and parallel depth contours andwith an open pier. The presence of edge waves in the site hasbeen reported (BRYAN and BOWEN, 1998; BRYAN, HOWD, andBOWEN, 1998) and related to short-scale rhythmic forms(COCO et al., 2004; VAN ENCKEVORT et al., 2004).

It is known that under extreme wave energy conditions,edge waves and other infragravity oscillations dominate thepower spectra in the inner surf zone and in the swash zone(CONSTANTIN, 2001; EVANS, 1988; LIPPMAN, HOLMAN, andBOWEN, 1997). Therefore, data obtained from the coastalmonitoring program during March and April of 2004, when aseries of storm events were observed, is selected for the anal-ysis (FIELD RESEARCH FACILITY, 2004).

The more severe storm, around March 11, 2004, had a max-imum significant wave height of 4.5 m and was accompaniedby a total tide (astronomical plus meteorological) of about 1.2m. A second moderate storm arrived on March 16, 2004, andlasted for three days, followed by a more severe storm onMarch 28, 2004, with a maximum significant wave heightslightly greater than 3 m. Wave spectra during the threestorms have similar double peak forms with roughly the samepeak frequencies. Figure 1 presents the directional wavespectrum at the peak of the first storm, recorded at a buoylocated at the 8 m depth contour. It shows three narrow-banded peaks at angular frequencies �a � 0.46 s�1 (P1 andP2) and T � 0.89 s�1 (P3).

Figure 2 presents snapshots taken by the Argus systembefore the storm events, at the peak of the strongest storm,and after the final storm. It can be observed that during the

storms, the increase in the mean water level moves the swashlimit to the upper part of the beach face. Beach cusps beginto form during the first storm, but only at the northern sideof the pier. During the second and third events, these fea-tures are reinforced, leaving well-formed cusps with distanc-es between consecutive horns of about 30 m.

Moreover, the bathymetry survey taken on April 7, 2004,after the storms had passed, shows that the initial alongshorebar observed on February 23, 2004 (Figure 3a) was trans-formed into a crescentic bar with distances between crests ofabout 150 m (Figure 3b), and again, only at the northern sideof the pier.

WORKING HYPOTHESIS

A vast amount of literature exists addressing the originand maintenance of rhythmic features in the surf zone.Among the plausible explanations, the presence of partialstanding edge waves has been pointed out by HOLMAN andBOWEN (1982). There is evidence of other mechanisms ca-pable of generating periodic forms, acting as a major force orreinforcing others in a complex interaction process (ORTEGA

SANCHEZ, LOSADA, and BAQUERIZO, 2003).However, in this particular case, some facts suggest that

the scattering of edge waves at the pier could be a significantdriving force in their formation. First, under storm decayingconditions, wave breaking generally corresponds with lowsteepness plunging waves, a condition found by GUZA andBOWEN (1976) to excite edge waves. In fact, under similarstorm conditions, field measurements at the site gave evi-dence of their presence (BRYAN and BOWEN, 1998; BRYAN,HOWD, and BOWEN, 1998). Second, it is found that the spatialscale of the periodic forms has the same order of magnitudeas both the wavelength of the edge waves previously mea-sured at the site and the wavelengths linked to the subhar-monic frequencies of the peaks of the wave spectrum. Third,intriguingly, the forms are present only at the side of the pierthat is affected by the edge wave reflection at the structure.

In this work, it is hypothesized that (1) southward propa-gating edge waves are excited at the synchronous and sub-harmonic frequencies of the dominant periods, (2) these edgewaves are reflected at the pier forming a partial standingwave, and (3) the drift velocity at the top of the boundarylayer is responsible for the generation of the rhythmic formsidentified after the storms. For the presented climatic con-ditions and bathymetric characteristics, the proposed model,briefly described in the following section, is used to test theworking hypothesis. Next, the validity of the working hy-pothesis is examined within the context of the existing debateregarding the origin and maintenance of rhythmic features.

EDGE WAVE SCATTERING BY AN ACROSS-SHOREPOROUS STRUCTURE

An infinitely long beach with straight and parallel depthcontours is considered. The origin of the reference frame islocated at the pier with the x-axis perpendicular to the shore,pointing seaward, and the y-axis opposite to the direction ofthe propagation of the incident edge wave (Figure 4). Thebeach profile is defined by an arbitrary function, h(x), which

1538 Avila, Baquerizo, and Losada


Figure 2. Photographs of the Field Research Facility, showing the mean level elevation during the strongest storm and shore configuration before andafter the three events.



Figure 3. Bathymetries (a) before and (b) after the storm events.

Figure 4. Reference frame and schematic definition of the problem.

provides water depth. In order to express the solution at eachside of the structure in terms of a discrete number of eva-nescent modes, it is assumed that in the x-direction, the do-main is bounded by a vertical impermeable wall located faroffshore at x � xw. As the amplitude of the edge waves decaysexponentially, the magnitude of xw can be moderate.

BAQUERIZO, LOSADA, and LOSADA (2002) considered thatthe width of the structure is much smaller than the along-shore wavelength. This assumption allows it to be treatedmathematically as a thin perforated barrier that divides thedomain into two regions (Figure 4), identified by subscripts1 (y � 0) and 2 (y � 0).

As the edge wave propagates through the coastal structure,its energy is partially reflected, partially transmitted, andpartially dissipated inside the structure. In region 1, the so-lution consists of the superposition of a quasi-standing waveand a series of evanescent modes, while in region 2, the so-lution includes the wave transmitted through the structureand the corresponding series of evanescent modes. The freesurface can be expressed as

N�ik y ik y �i�t0 n� (x, y � 0, t) � A � (x)e R � (x)e e ,�1 0 0 n n[ ]n�0

N�ik y �ik y �i�t0 n� (x, y � 0, t) � A � (x)e � R � (x)e e (1)�2 0 0 n n[ ]n�0

where t is time, � is the angular frequency, and kn are theevanescent wave numbers. The first term of �1 represents theincoming edge wave with amplitude A0; the second term con-tains the reflected edge wave, n � 0, with unknown ampli-tude, R0, and the N evanescent modes with unknown ampli-tudes, Rn (n � 0). �2 contains the transmitted edge wave modewith unknown amplitude, A0 � R0, and a set of N evanescent

modes. The evanescent modes decay rapidly away from thebarrier in the alongshore direction.

The wave numbers, kn, and their corresponding cross shoreamplitudes, �n(x), are calculated numerically with a Runge-Kutta formula for the angular frequency, �, and the bathy-metric profile, h(x), as described in Appendix 1. The coeffi-cients Rn are obtained by a mode matching method that in-cludes the head loss at the structure. The head loss is for-mulated following the model of quadratic head loss andscattering of long waves by MEI, LIU, and IPPEN (1974), asdescribed in Appendix 2.

APPLICATION OF THE MODEL

A representative beach profile extracted from a survey onFebruary 23, 2004, was selected for the analysis (Figure 5a).The solution of the dispersion equation obtained for this pro-file is illustrated in Figure 5b together with URSELL’s (1952)well-known solution calculated for two adjusted beach slopes,one representative of the shore-face bathymetry and the oth-er equal to the mean offshore slope. The differences betweenthe curves obtained with the numerical solution and the sim-plified profiles demonstrate the importance of the shorewardpart of the beach profile on the edge wave behavior, in par-ticular for the longer edge waves.

As the offshore spectra present two different narrow-band-ed peaks (Figure 1), the incident waves are comprised of twoclearly identifiable angular frequencies �a � 0.46 s�1 and �b

� 0.89 s�1, which approach the beach with almost normalincidence. According to INMAN and GUZA (1982), the mostlikely excited edge waves are mode 0 for the subharmonicfrequencies �sh-a � �a/2 � 0.23 s�1 and �sh-b � �b/2 � 0.445s�1. Synchronous edge waves at modes 0 and 1 may also bepresent although with smaller amplitudes (GUZA and INMAN,1975). GALLAGHER’s (1971) mechanism predicts edge wavesforced at �b � �a, which roughly equals �sh-a. Although thesetheories emphasize the importance of the lower modes, BOW-EN and GUZA (1978) and EVANS (1988) pointed out that it isnot clear that only the lower modes are significant.

According to Figure 1, most of the spectrum energy lies inthe interval of the positive angles (see Figure 4 for compassdefinition), meaning that the propagation direction of the



Figure 5. (a) Representative beach profile and (b) dispersion relationcompared with two simplifications.

edge waves is southward. Accordingly, the northern part ofthe pier (region 1) is affected by the edge wave reflection.

Under the considered wave conditions, the characteristicsof plausible edge wave modes 0 to 4 at the synchronous andsubharmonic frequencies are presented in Table 1. Amongthem, the wavelengths of modes 0 and 4 at the subharmonicfrequencies �sh-b and �sh-a, respectively, are in correspondencewith the length scales of the rhythmic features observed afterthe storms.

The proposed model has been applied to analyze the inter-ference of the coastal structure with these edge waves. As-suming a value of the head loss coefficient f � 2, the modelpredicts reflection coefficients R1 � 0.1 (for �sh-a � 0.23 s�1,mode n � 4) and R1 � 0.14 (for �sh-b � 0.445 s�1, mode n �0). Different values of f, as discussed in the following section,slightly modify the amplitude of the partially standing oscil-lations but do not change the general pattern.

For the partial standing oscillations obtained at the north-ern part of the pier, the lagrangian drift velocity (LONGUET-

HIGGINS, 1953), often referred to as the mass transport ve-locity, above the bottom boundary layer was computed as anindication of the trend of the sediment transport pattern. Thecontour plots of their absolute values, normalized by the max-imum values in the domain, are presented in Figures 6 and7 together with the corresponding normalized maps of max-imum amplitudes of the free surface (absolute value of Equa-tion 1).

The oscillation formed in region 1 for mode 0 of the edgewave at �sh-b � 0.445 s�1 shows maximum amplitudes in thealongshore direction at a spacing of L/2 � 22.4 m (Figure 6a),a value that is close to the distance between cusps observedat the Field Research Facility. The contour plot of the netmass transport velocity (Figure 6b) shows maximums withthe same periodicity. Figure 6c illustrates the coupling be-tween the mass transport and the shoreline features in a rec-tified snapshot of the Argus system from April 6, 2004. It isconvenient to stress that these forms appear only at thenorthern part of the pier, where the partial standing oscil-lation occurs.

For mode 4 of the subharmonic frequency �a � 0.46 s�1,relatively high amplitudes are obtained not only at the shore-line but also over the bar with distances between quasi-an-tinodes of approximately L/2 � 129 m (Figure 7a). The netmass transport velocity (Figure 7b) shows an alongshore var-iability with the same periodicity and maximum values overthe shoreward slope of the bar. It is possible that this partialstanding edge wave is related to the crescentic bar measuredin the bathymetry of April 7, 2004 (Figures 7c and d).

DISCUSSION

In spite of the agreement between the results of the modeland the observed features, some aspects related to the valid-ity of the applied formulation need to be addressed: first, itslimitations to estimate the partial standing edge wave and,second, the role played by edge waves in the formation of theperiodic forms. As for the model, its restriction to paralleldepth contours is not considered to affect the results of thecurrent study case. The only significant deviations from thisassumption is the scour hole around the piles, whose size issmall compared to the wavelengths of the analyzed edgewaves. KURKIN, PELINOVSKY, and POLOUKHINA (2006)showed that slow variations in the alongshore direction canboth strengthen and weaken edge waves; however, this maynot affect the qualitative description of the partial standingedge wave.

GALLETTA and VITTORI (2004) studied the nonlinear ef-fects on medium-scale standing edge wave evolution in termsof the nonlinearity parameter, calculated as the ratio be-tween the edge wave amplitude and a representative waterdepth. They showed that for moderate and large values ofthis parameter, nonlinear effects may modify the edge wavedynamics, modulating the amplitudes at time scales muchlonger than the edge wave periods. However, they also foundthat the dispersion relationship remains practically unaffect-ed. The selection of the study case, under extreme conditions,intends to minimize this complexity assuming that the infra-



Figure 6. (a) Maximum amplitudes and (b) mass transport velocity of the subharmonic edge wave at �b � 0.445 s�1, mode 0. (c) Coupling of masstransport with a rectified photo to compare with beach cusps.

Table 1. Wavelengths associated with the peak frequencies.

AngularFrequency (s�1) Period, T (s)

Wavelength of the Edge Wave Mode n, Ln (m)

n � 0 n � 1 n � 2 n � 3 n � 4

�a � 0.46 13.7(synchronous) 42.11 81.46 88.02 97.09 102.9627.3(subharmonic) 134.06 179.41 211.70 237.13 258.06

�b � 0.89 7.1(synchronous) 13.91 33.74 40.13 43.13 44.7114.1(subharmonic) 44.73 84.45 92.05 101.28 107.25

gravity energy is well developed and that its interaction withthe gravity range is limited.

Regarding the value of the friction coefficient, f, used in thehead loss formulation, most studies in the literature are lim-ited to strongly intrusive structures, e.g., OZSOY (1977), butto the authors’ knowledge, very little information exists aboutits order of magnitude for highly porous structures, like pilepiers. Therefore, an arbitrarily extrapolated value f � 2 waschosen. The value of the coefficient may affect the reliabilityof the quantitative output of the model, but as noted previ-ously, it does not modify the qualitative behaviour.

Although the limitations of the model do not affect thequalitative description of the edge waves, this work, as well

as many other explanations of the existence of rhythmic fea-tures, necessitates a deductive reasoning. Therefore, there isnot enough confidence to proceed to the following step of thescientific method, i.e., to proceed in the inductive phase andto formulate a physical law.

Truthfully, the hypothesis that the observed rhythmic fea-tures are caused by these oscillations falls into an open de-bate about the origin and maintenance of topographic fea-tures. Some models, like the one proposed in this work, as-sume that sediment is passively driven by the fluid motion,and, therefore, features are just the footprint of the hydro-dynamic forcing (see BLONDEAUX, 2001, for a review). Thesemodels are met with some scepticism because they neglect



Figure 7. (a) Maximum amplitudes and (b) mass transport velocity of the subharmonic edge wave at �a � 0.23 s�1, mode 4. (c) Comparison with thecrescentic bar of the bathymetry, plan view. (d) Representative beach profile.

the coupled motion of the bed and the fluid. On the otherhand, self-organization models explain the morphology as theresult of the feedback between the underlying topographyand the fluid motion, and they do not need, in principle, aspatial hydrodynamic pattern (COCO, HUNTLEY, andO’HARE, 1999; JOHNSON, 1910; WERNER and FINK, 1993).

There is no doubt about the important role played by thefeedback between the bed and the fluid in the growth andmaintenance of the bed forms. However, it is still unclearwhether this mechanism can solely explain their generation(HOLLAND and HOLMAN, 1996). It seems reasonable to rec-oncile the two types of models by admitting that edge wavesmay initially trigger the system, which then evolves due tothe feedback with the oscillatory motion (HOLLAND and HOL-MAN, 1996; HUNTLEY, 1980; RENIERS, ROELVINK, and VAN

DONGEREN, 2000).

CONCLUSIONSThe existence of certain rhythmic topography observed af-

ter a series of consecutive storms in the northern part of the

Field Research Facility pier in Duck, North Carolina, wasstudied. It is hypothesized that these rhythmic features havetheir origin in the southward propagating edge waves, whichare excited at the synchronous and subharmonic frequenciesof the dominant periods and reflected at the pier.

A model based on BAQUERIZO, LOSADA, and LOSADA

(2002), but improved to deal with an arbitrary beach profile,is used to formulate the interaction of the incident edge wavewith a permeable coastal structure perpendicular to theshore. For edge waves at the subharmonic frequencies of thepeaks of the wind wave spectra, it is found that the wave-lengths and the locations of maximum amplitudes are relatedto the characteristics of the beach cusps and the crescenticbars observed after three studied storm events. Additionally,the periodicity of the magnitude of the mass transport veloc-ity patterns under the partial standing edge waves confirmsthe similarity. The absence of these forms on the other sideof the pier reinforces the hypothesis that edge waves are in-volved in the formation of these morphological features.



ACKNOWLEDGMENTS

The field data used in this work was obtained from the webpage of the Field Research Facility of the Coastal and Hy-draulics Laboratory of the U.S. Army Corps of Engineers. Theauthors appreciate the effort of people and institutions thatmake the sharing of such valuable information possible. Thework done by the first author was funded by the Consejo Na-cional de Ciencia y Tecnologıa (CONACYT, Mexico). The sec-ond and third authors are indebted to the Comision Inter-ministerial de Ciencia y Tecnologıa (CICYT, Spain), contractNo. CTM2005-06583/MAR and the Junta de Andalucıa, con-tracts No. RNM968 and No. RNM1573.

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APPENDIX 1

Dispersion Relation

For a free surface elevation, �, given by

i(kx��t)�(x, y, t) � �(x)e , (A1.1)

the linear shallow water equation reads

d d�2 2gh (� � ghk )� � 0, (A1.2)� �dx dx

where t is time, g is the gravity acceleration, �, k, and � ��(x) are, respectively, the angular frequency, wave number,and the cross shore amplitude profile of the edge wave, andh is water depth.

For the sake of simplicity, in order to avoid the problem-atic boundary condition at a shoreline with zero waterdepth, it is assumed that h � 0 and a null flux is requiredacross this small step (SCHONFELDT, 1995). The problemreduces to solving Equation (A1.2) with the following bound-ary conditions:

d� d�� 0� ; � 0� . (A1.3)x�x x�xs wdx dx

BAQUERIZO, LOSADA, and LOSADA (2002) solved the prob-lem analytically for a simple idealized beach profile with asteep foreshore, a plane slope, and a horizontal shelf. In thiswork, an arbitrary beach profile is considered, and Equations(A1.2) and (A1.3) are solved numerically by a Runge-Kuttaformula. The dispersion relationship relates the values of kand �, for which the problem has a nontrivial solution. For agiven angular wave frequency, there exists a finite numberof real solutions of the dispersion relationship, correspondingto the progressive edge wave modes. The infinite discrete ev-anescent modes required to match the solutions at the piercorrespond to purely imaginary values of k.

APPENDIX 2

Calculation of the Coefficients Rn

It is assumed that the wavelength is larger than the widthof the structure and its effect on head loss can be evaluated

assuming the structure is a thin grooved plate. Solutions of�, Equation (1), for each side of the structure must match atthe barrier (y � 0), which is imposed by the following con-ditions: (1) the alongshore velocity is zero at the solid part ofthe structure, (2) alongshore velocities must match acrossgaps, and (3) surface elevations at both sides of the structurediffer in a momentum loss term.

The momentum loss is formulated through a linear frictionterm, equivalent to the model of quadratic head loss and scat-tering of long waves by MEI, LIU, and IPPEN (1974). Denotingthe solid part of the structure as P and the permeable partor gaps as H, the matching conditions at the coastal struc-ture, y � 0, are

�1 � 0 x ∈ Py�y�0

� �1 2� x ∈ Hy� y�y�0 y�0

(� � � )� � C (x)v x ∈ H (A2.1)1 2 y�0 e

where Ce � ( f/2g)(8/3�)�� is a linear-equivalent friction term,f is a head-loss coefficient that depends on the porous char-acteristics of the structure, and v is velocity.

Conditions of Equation (A2.1) are joined in a mixed bound-ary condition at the structure (ROLDAN, LOSADA, and LO-SADA, 1992; SNEDDON, 1966) according to:

N⎧k A � (x) � k R � (x) � 0�0 0 0 n n n

n�0⎪ at solid plates⎨G(x) � (A2.2)N gk gkn 02 C R � (x) � C A � � 0� e n n e 0 0� �� n�0⎪

at gaps⎩

Equation (A2.2) is used to solve coefficients Rn by minimizingG(x) with the least-squares method. This yields the complexlinear system

G(x)G(x)* dx � 0; n � 0, 1, 2, . . . , N (A2.3)� Rnx

of N 1 results (Rns) and N 1 equations, where G(x)*denotes the complex conjugate of the function G(x). The co-efficients of the propagating and evanescent modes are thenobtained as solutions of an optimization problem that has tobe solved using an iterative method because the friction termdepends on the solution itself.

� RESUMEN �

Se analiza la formacion de patrones morfologicos rıtmicos en el Field Research Facility (U.S. Army Corps of Engineers, Duck, North Carolina) bajo condicionesclimaticas de oleaje severas en terminos de la oscilacion parcialmente estacionaria que resulta de superponer una onda de borde incidente con la reflejada en laestructura. El coeficiente de reflexion se evalua mediante un modelo previo que estudia la propagacion de una onda de borde a traves de una estructura permeableperpendicular a la costa, modificado para trabajar con un perfil de playa arbitrario. Los patrones del transporte de masa obtenidos para la oscilacion parcialmenteestacionaria se parecen a las formas cuspidales y a las barras crescenticas que se observan despues de las tormentas exclusivamente al note del espigon.

Edge Wave Scattering by Coastal Structures on Arbitrary Bathymetry

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