Corresponding author: F. Carbone, School of Mathematical Sciences, University College of Dublin, Dublin, Ireland. (email@example.com)
 Wave impact and runup onto vertical obstacles are among the most important phenomena which must be taken into account in the design of coastal structures. From linear wave theory, we know that the wave amplitude on a vertical wall is twice the incident wave amplitude with weakly nonlinear theories bringing small corrections to this result. In this present study, however, we show that certain simple wave groups may produce much higher runups than previously predicted, with particular incident wave frequencies resulting in runup heights exceeding the initial wave amplitude by a factor of 5, suggesting that the notion of the design wave used in coastal structure design may need to be revisited. The results presented in this study can be considered as a note of caution for practitioners, on one side, and as a challenging novel material for theoreticians who work in the field of extreme wave-coastal structure interaction.
 The robust design of various coastal structures (such as sea-walls and breakwaters) relies on the accurate estimation of the wave loading forces. To this end, engineers have introduced the notion of the so-called design wave. Once the particular characteristics of this design wave are determined, the pressure field inside the bulk of fluid is usually reconstructed (in the engineering practice) using the Sainflou or Goda  semiempirical formulas. However, there is a difficulty in determining the wave height to be used in design works. Sometimes, it is taken as the significant wave height H1/3, but in other cases, it is H1/10 (the average of 10% highest waves) that is substituted into the wave pressure formulas. If we take, for example, an idealized sea state which consists only of a single monochromatic wave component with amplitude a0, its wave height H0 can be trivially computed
Consequently, the design wave will have also the height equal to 2a0.
 In the present study, we show that even such simple monochromatic sea states, subject to the nonlinear dynamics over a constant bottom, can produce much higher amplitudes on a vertical wall. Namely, we show below that some wave frequencies can lead to an extreme runup of the order of ≈5.5a0 on the cliff. The results presented in this study suggest that the notion of the design wave has to be revisited. Moreover, the mechanism elucidated in this work can shed some light onto the freak wave phenomenon in the shallow water regime, where we recall in this context that over 80% of reported past freak wave events have been in shallow waters or coastal areas [Nikolkina and Didenkulova, 2011; O'Brien et al., 2013].
 It is well known that wave propagation on the free surface of an incompressible homogeneous inviscid fluid is described by the Euler equations combined with nonlinear boundary conditions on the free surface [Stoker, 1957]. However, this problem is difficult to solve over large domains, and consequently, simplified models are often used. In particular, in this study, we focus our attention on long wave propagation. A complete description of wave processes, including collisions and reflections, is achieved by employing two-way propagation models of Boussinesq type [Bona and Chen, 1998]. Taking into account the fact that we are interested here in modeling (potentially) high amplitude waves, we adopt the fully nonlinear Serre-Green-Naghdi (SGN) equations [Serre, 1953; Green et al., 1974; Green and Naghdi, 1976; Zheleznyak and Pelinovsky, 1985], which make no restriction on the wave amplitude. Only the weak dispersion assumption is adopted in the mathematical derivations of this model [Wei et al., 1995; Lannes and Bonneton, 2009; Dias and Milewski, 2010].
 We consider a two-dimensional wave tank with a flat impermeable bottom of uniform depth d=const, filled with an incompressible, inviscid fluid (see Figure 1). The Cartesian coordinate system Oxy is chosen such that the y-axis points vertically upward and the horizontal x-axis coincides with the undisturbed water level y=0. The free surface elevation with respect to the still water level is denoted by y=η(x,t), and hence, the total water depth is given by h(x,t)=d+η(x,t). Denoting the depth-averaged horizontal velocity by u(x,t), the SGN system reads [Lannes and Bonneton, 2009; Dias and Milewski, 2010; Clamond and Dutykh, 2012]:
where g is the acceleration due to gravity.
 The SGN system possesses Hamiltonian and Lagrangian structures [Li, 2002; Clamond and Dutykh, 2012] and conservation laws for mass, momentum, potential vorticity, and energy [Li, 2002] (Dutykh et al. arXiv:1104.4456). From a more physical perspective, the SGN model combines strong nonlinear effects with some dispersion that approximates well the full water wave dynamics. This model has been previously validated by extensive comparisons with experimental data for wave propagation and runup [Chazel et al., 2011; Tissier et al., 2011; Carter and Cienfuegos, 2011] (Dutykh et al. arXiv:1104.4456).
 One of the most important questions in water wave theory is the understanding of wave interactions and reflections [Linton and McIver, 2001; Berger and Milewski, 2003; Clamond et al., 2006] and the interaction of solitary waves has also been extensively studied [Zabusky and Kruskal, 1965; Bona et al., 1980; Fenton and Rienecker, 1982; Craig et al., 2006]. By using symmetry arguments, one can show that the head-on collision of two equal solitary waves is equivalent to the solitary wave/wall interaction in the absence of viscous effects.
 The accurate determination of the maximum wave height on a wall is of primary importance for applications. Several analytical predictions for periodic or solitary wave runup Rmax in terms of the dimensionless wave amplitude α=a0/d have been developed: linear theory [Mei, 1989] Rmax/d=2α, third-order theory [Su and Mirie, 1980] Rmax/d=2α+1/2α2+3/4α3, and nonlinear shallow water theory [Mirchina and Pelinovsky, 1984] .
 These results have been confirmed in previous experimental [Maxworthy, 1976], theoretical [Byatt-Smith, 1988], and numerical [Fenton and Rienecker, 1982; Craig et al., 2006] studies. All these theories agree on the fact that the wave height on the wall is 2 times the incident wave amplitude plus higher order corrections. This conclusion provides a theoretical justification for the use of a wave height such as H1/3 in the design wave definition.
2 Numerical Study
 From a practical point of view, however, the reasoning presented above contains at least one serious flaw—in real-world conditions, waves seldom come isolated but rather as groups. In this section, we show numerically how simple wave groups can produce much higher runups than expected from the existing theoretical predictions.
 Our numerical wave periods cover a range between 20 s and 1100 s (i.e., for 6<d<30 m and g=9.81 m/s2), from long swells to tsunami waves. Extreme runups are obtained for wave periods which are in between swell periods and tsunamis periods, corresponding possibly to tsunamis generated by underwater landslides. Moreover, we take as initial conditions waves which are not exact solutions to the equations. Naturally, they deform as they evolve toward the vertical wall. This deformation is reminiscent of the transformation of waves over sloping bathymetries.
2.1 Numerical Scheme and Setup
 Let us consider a flat channel of constant depth d and length L. This channel is bounded on the right by a rigid vertical wall and by a wavemaker on the left (see Figure 1). We use dimensionless variables where lengths are normalized with d, speeds with , and time with . This scaling is equivalent to setting g=1 m/s2, d=1 m in the governing equations (2) and (3).
 In order to solve numerically the SGN equations, we use the high-order finite-volume scheme described in Dutykh et al. (arXiv:1104.4456). This scheme has been successfully validated against analytical solutions and experimental data [Hammack et al., 2004]. For the time integration, we use the classical fourth-order Runge-Kutta scheme [Shampine, 1994; Shampine and Reichelt, 1997]. The computational domain is divided into equal intervals (i.e., control volumes) such that we have N=1000 control volumes per wavelength. The convergence study showed that this grid provides a good trade-off between accuracy and overall computational time. Note that the extreme runup values reported in this study can slightly increase under mesh refinement, which decreases the effect of numerical dissipation. All simulations start with the rest state η(x,t=0)≡0, u(x,t=0)≡0. The wavemaker generates on the left boundary, during a time T, the following periodic free surface disturbance with corresponding fluid velocity:
where the amplitude a0=0.05, ω∈0.01,0.25, is the Heaviside function, and cs is the wave speed (Dutykh et al. arXiv:1104.4456). An important remark should be made on the initialization of the problem: sometimes spurious high-frequency standing waves are generated when numerical simulations of nonlinear progressive waves are initialized using linear waves (in particular, for deep water waves). In such cases, the problem should be initialized by suppressing the spurious generation of standing waves. However, a nonlinear simulation can be initialized with simple linear waves if the runtime is long enough to adjust the wave shape [Dommermuth, 2000]. As can be seen in the bottom panel of Figure 1, the amplitude of the first harmonic remains constant during the propagation while the amplitudes of the higher harmonics tend to increase until a constant value is reached, as illustrated in Dommermuth . For this reason, no adjustment is required in our case.
 We generate only a finite number Nw of waves with period T0=2π/ω, and thus, the wave generation time T is defined as T=NwT0. The monochromatic deviation of the free surface at the left boundary is then propagated toward the right wall under the SGN dynamics. The length L of the computational domain and the final simulation time Tf are chosen adaptively in order to allow all important interactions and to prevent any kind of reflections with the left generating boundary:
with λ being the wavelength corresponding to the frequency ω.
2.2 Numerical Results
 We begin our numerical experiments by considering a single sinusoidal wave interacting with the solid wall. In Figure 2, we show three snapshots of the single-wave evolution at three different times (first three panels on Figures 2a–2c), i.e., before reaching the wall, during the impact, and right after the reflection. The initial sinusoidal wave undergoes steepening during its propagation. The runup on the vertical wall is shown on the bottom panel of Figures 2a. The maximal dimensionless wave elevation Rmax≃0.10245 on the wall reaches roughly twice the incident wave amplitude a0=0.05 (at t≃70). This result is in good agreement with previous numerical studies on the solitary waves interactions [Cooker et al., 1997; Pelinovsky et al., 1999; Chambarel et al., 2009] even if the incident shape is not exactly the same. The maximal relative runup Rmax/a0≃2.34 is achieved for ωmax=0.145. The value of Rmax is slowly decreasing for ω>ωmax.
 When two waves are injected into the domain, the dynamics is similar to the single-wave case. However, with two waves, the nonlinear effects become even more apparent (see the bottom panel of Figure 2b).
 In a certain range of wave periods (ω∈(0.01,0.05)), the so-called dispersive shock waves are formed [Wei et al., 1995; Tissier et al., 2011]. This particular type of solution has been extensively studied theoretically and numerically in El et al. . When the second wave impinges on the first reflected wave, a dispersive shock wave forms and propagates toward the wall. As shown in the bottom panel of Figure 2b, the maximal amplification is achieved when the second wave hits the wall due to nonlinear interactions between two counter-propagating waves. However, we underline that with only two waves, one can achieve a maximal runup on the wall Rmax of almost four incident wave amplitudes a0: Rmax/a0≃3.8, for ω=0.021. Such high runup values are possible due to the energy transfer between the first reflected wave and the second incoming wave. The dependence of the maximal runup Rmax on the incident wave frequency ωand the number Nw of incident waves is shown in Figure 3. One can see from this figure that the optimal energy transfer due to dispersive shocks happens for three incident waves (see the bottom panel of Figure 2c). In this case, the maximal runup is observed around ωmax=0.035, and the amplification is equal to Rmax/a0≃5.43. However, the energy transfer process is saturated for three waves.
 We performed similar computations with four incident waves (also shown on Figure 3), and the maximal runup is not higher than with three waves. Consequently, we focus now only on the optimal three-wave case. The three regimes (hyperbolic, equilibrium, and dispersive) are illustrated on Figure 4, where we show the space-time dynamics of the three-wave system. The left panel shows the hyperbolic regime. On the central panel, strong dispersive shocks can be observed, while on the right panel, the dynamics is smoothed by the dispersion. In the last case, the amplification is mainly produced by the linear superposition of the incident and reflected waves. The reflection and interaction are clearly observed by smooth secondary peaks in the space-time plots (see Figure 4, bottom panels).
 The wave interactions described above strongly depend on the frequency ω of the impinging waves as can be seen in the bottom panels of Figure 4, where we show the wave records on the wall for several values of the frequency ω. As the wave frequency increases, the wavelength shortens and the dispersive effects become gradually more important. Around ωmax, the dispersive effects are balanced with nonlinearities to produce the most pronounced dispersive shock waves. Starting from ω≃0.11, we enter into the dispersive regimes where the waves become regularized.
3 Conclusions and Perspectives
 In the present study, we investigated numerically the interaction between a wave and a vertical wall in the framework of the Serre-Green-Naghdi (SGN) equations. These equations combine strong nonlinear and weak dispersive effects. We explored the whole range of wavelengths from the hyperbolic regime (including shock waves) to smooth dispersive waves (). More importantly, we showed that the wave runup on the wall is strongly dependent on the incident wave frequency, the dependence not being monotonic. In particular, there is a fixed frequency ωmax which provides the maximum long wave runup on the vertical wall. The function Rmax(ω) is monotonically increasing up to ωmax and monotonically decreasing for ω>ωmax (at least within the range of considered numerical parameters). The dynamics can be conventionally divided into three main stages: propagation of the wave group toward the wall along with the front steepening and other nonlinear deformations, first wave runup and its reflection from the wall, and projection of the reflected energy again onto the wall by subsequent incident waves. The maximal observed amplification ≈5.5a0 is achieved with only three incident waves. For example, on a 10 m water depth the critical period is equal approximatively to Tmax≈180 s(ωmax≡2π/Tmax). Such a wave period can be generated by small underwater landslides.
 The results presented in this study shed light onto extreme wave runups on vertical cliffs and similar coastal structures. Moreover, in view of these results, the definition of the design wave has to be revisited. Our suggestion would be to take at least 3H1/3 or even 3H1/10. The present results also shed some new light on the mysterious accumulations of large boulders on cliff tops up to 50 m high on the deep water coasts, especially on the west coast of Ireland [Hansom and Hall, 2009; Williams, 2010; O'Brien et al., 2013]. The emplacement of these megaclasts is usually attributed to extreme storm waves, but there are also those who believe that tsunamis are the most probable explanation of boulder ridges in these areas [Kelletat, 2008; Scheffers et al., 2009, 2010].
 In future investigations, more general wave groups have to be studied to unveil their potential for focusing on the walls. We recall that so far, we considered only simple idealized monochromatic waves. In addition, we are going to investigate the effect of the forces exerted by incident waves on vertical obstacles, which can be different from the purely kinematic amplitude focusing presented in this study. In other words, it is not clear whether the highest wave will produce the highest dynamic pressure spike on the wall. The effect of the wave amplitude is to be investigated as well since all the processes under consideration are highly nonlinear. Some theoretical explanation of these phenomena is also desirable. However, the difficulty is rather high again because of important nonlinearities mentioned hereinabove. We claim that no linear theory is sufficient to provide a satisfactory explanation.
 This work was funded by ERC under the research project ERC-2011-AdG 290562-MULTIWAVE and SFI under grant SFI/12/ERC/E2227. The authors wish to acknowledge the SFI/HEA Irish Centre for High-End Computing (ICHEC) for the provision of computational facilities and support under the project “Numerical simulation of the extreme wave run-up on vertical barriers in coastal areas.” We would like also to thank Alizée DUBOIS who obtained the first preliminary results on this subject during her visit to University College Dublin in May 2012.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.