## 1 Introduction

[2] 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*[1928] or *Goda* [2010] 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 *H*_{1/3}, but in other cases, it is *H*_{1/10} (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 *a*_{0}, its wave height *H*_{0} can be trivially computed

Consequently, the design wave will have also the height equal to 2*a*_{0}.

[3] 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 ≈5.5*a*_{0} 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].

[4] 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].

[5] 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 *O**x**y* 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.

[6] 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).

[7] 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.

[8] 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 *R*_{max} in terms of the dimensionless wave amplitude *α*=*a*_{0}/*d* have been developed: linear theory [*Mei*, 1989] *R*_{max}/*d*=2*α*, third-order theory [*Su and Mirie*, 1980] *R*_{max}/*d*=2*α*+1/2*α*^{2}+3/4*α*^{3}, and nonlinear shallow water theory [*Mirchina and Pelinovsky*, 1984] .

[9] 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 *H*_{1/3} in the design wave definition.