Series expansions of unknown fields n n Z ϕ Φ = å in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions n Z are determined by solving localSturm-Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to n Z cannot be compatible with the physical boundary conditions of Φ , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in (Athanassoulis & Belibassakis 1999 J. Fluid Mech. 389, 275-301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly-convergent representation of the field Φ , valid for any smooth, nonplanar boundaries and any smooth enough Φ . This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.

1. Introduction 2. Formulation of the problem 2.1 Classical formulation of the problem 2.2 The Hamiltonian Coupled-Mode Theory 2.2.1.. Exact vertical series expansion of the wave potential 2.2.2 The Hamiltonian coupled-mode system 2 6. Interaction of solitary waves with bathymetry and vertical walls 6.1 Reflection of solitary waves on a vertical wall 6.2 Shoaling of solitary waves over a plane beach 6.3 Reflection of shoaling solitary waves on a vertical wall at the end of a sloping beach 6.4 Propagation of a solitary wave over a sinusoidal patch 6.5 Transformation of a solitary wave over a 3D bathymetry with banks and trenches 7. Discussion and conclusions Acknowledgments Appendix A: An outline of the derivation of the Hamiltonian Coupled Mode System Appendix B: Calculation of the basic vertical integrals (a) Vertical integrals of the form ( ; ) m J s Z and ( ; ) m J s W (b) Vertical integrals of the form ( ; ) n m J s Z Z for 0 ,1, 2 s = (c) Vertical integrals of the form ( ; ) n m J s W Z for 0 ,1 s = Appendix C. Fast and accurate calculation of local wavenumbers ( , ) n k t x Appendix D: Finite-difference linear system for the substrate problem Eqs. (17a,b) Appendix E. Description of the 3D bathymetry References List of abbreviationsBEM boundary element method CMS coupled-mode system DNM direct numerical methods DtN Dirichlet-to-Neumann (operator) FD finite difference FDM finite difference method FEM finite element method HCMS Hamiltonian coupled-mode system HCMT Hamiltonian Coupled-Mode Theory NLPF nonlinear potential flow NR Newton-Raphson (method) RK Runge-Kutta (method) SGN Serre-Green-Nagdhi (equations) AbstractThis paper deals with the implementation of a new, efficient, non-perturbative, Hamiltonian coupled-mode theory (HCMT) for the fully nonlinear, potential flow (NLPF) model of water waves over arbitrary bathymetry Papoutsellis and Athanassoulis (2017) (arxiv.org/abs/1704.03276). Applications considered herein concern the interaction of solitary waves with bottom topographies and vertical walls both in two-and three-dimensional environments. The essential novelty of HCMT is a new representation of the Dirichlet-to-Neumann operator, which is needed to close the Hamiltonian evolution equations. This new representation emerges from the treatment of the substrate kinematical problem by means of exact semi-separation of variables in the instantaneous, irregular, fluid domain, established recently by Athanassoulis & Papoutsellis (2017) (https://doi.org/10.1098/rspa.2017.0017). The HCMT ensures an efficient dimensional reduction of the exact NLFP, being able to treat an arbitrary bathymetry as simply as the flat-bottom case, without domain transformation. A key point for the efficient implementation of the method is the fast and accurate evaluation of the space-time varying coefficients appearing in some of its equations. In this paper, all varying coefficients are calculated analytically, resulting in a refined version of the theory, characterized by improved accuracy at significantly reduced computationa...

We present a new Hamiltonian formulation for the non-linear evolution of surface gravity waves over a variable impermeable bottom. The derivation is based on Luke’s variational principle and the use of an exact (convergent up to the boundaries) infinite-series representation of the unknown wave potential, in terms of a system of prescribed vertical functions (explicitly dependent on the local depth and the local free-surface elevation) and unknown horizontal modal amplitudes. The key idea of this approach is the introduction of two unconventional modes ensuring a rapid convergence of the modal series. The fully nonlinear water-wave problem is reformulated as two evolution equations, essentially equivalent with the Zakharov-Craig-Sulem formulation. The Dirichlet-to-Neumann operator (DtN) over arbitrary bathymetry is determined by means of a few first modes, the two unconventional ones being most important. While this formulation is exact, its numerical implementation, even for general domains, is not much more involved than that of the various simplified models (Boussinesq, Green-Nagdhi) widely used in engineering applications. The efficiency of this formulation is demonstrated by the excellent agreement of the numerical and experimental results for the case of the classical Beji-Battjes experiment. A more complicated bathymetry is also studied.

Two methods to treat wave breaking in the framework of the Hamiltonian formulation of free-surface potential flow are presented, tested, and validated. The first is an extension of Kennedy et al. (2000)'s eddyviscosity approach originally developed for Boussinesq-type wave models. In this approach, an extra term, constructed to conserve the horizontal momentum for waves propagating over a flat bottom, is added in the dynamic free-surface condition. In the second method, a pressure distribution is introduced at the free surface that dissipates wave energy by analogy to a hydraulic jump (Guignard and Grilli, 2001). The modified Hamiltonian systems are implemented using the Hamiltonian Coupled-Mode Theory, in which the velocity potential is represented by a rapidly convergent vertical series expansion. Wave energy dissipation and conservation of horizontal momentum are verified numerically. Comparisons with experimental measurements are presented for the propagation of a breaking dispersive shock wave following a dam break, and then incident regular waves breaking on a mildly sloping beach and over a submerged bar.Keywords: wave breaking, Hamiltonian formulation of water waves, eddyviscosity, fully nonlinear water waves Highlights• Two wave breaking techniques are tested in a fully nonlinear model. • Breaking is implemented by extending the Hamiltonian Coupled-Mode