In this work, we model extreme waves that occur due to Mach reflection through the intersection of two obliquely incident solitary waves. For a given range of incident angles and amplitudes, the Mach stem wave grows linearly in length and amplitude, reaching up to 4 times the amplitude of the incident waves. A variational approach is used to derive the bidirectional Benney–Luke equations, an asymptotic equivalent of the three-dimensional potential-flow equations modelling water waves. This nonlinear and weakly dispersive model has the advantage of allowing wave propagation in two horizontal directions, which is not the case with the unidirectional Kadomtsev–Petviashvili (KP) equation used in most previous studies. A variational Galerkin finite-element method is applied to solve the system numerically in Firedrake with a second-order Störmer–Verlet temporal integration scheme, in order to obtain stable simulations that conserve the overall mass and energy of the system. Using this approach, we are able to get close to the 4-fold amplitude amplification predicted by Miles.

Offshore structures such as wind turbines, ships and platforms are designed to resist loads and stresses applied by winds, currents and water waves. These three factors can cause damage or destroy these structures when their effect is underestimated. Designers and engineers must take into account the effect of not only each of these phenomena separately, but also their interaction, which can increase their adverse effects. In this work, we focus on the impact of extreme waves created from the propagation of an obliquely incident solitary wave along the side of a ship (a wave–structure interaction) or its impact with another identical obliquely incident wave (a wave–wave interaction). These two cases are mathematically equivalent since reflection at a rigid wall (represented here by the ship's side) is modelled through the boundary condition of no normal flow at the wall, which is equivalent to the intersection of two identical waves travelling in opposite directions, in which case a virtual wall is formed. The study of extreme, freak or rogue waves resulting from reflection at a wall or interaction of waves has spawned different theories in the last 50 years, some of which are subsequently reviewed.

Top: top view of a channel containing an incident solitary wave
propagating in the

The objective of the present work is to apply a theory first introduced by

This theory holds in the case of small-but-finite wave amplitude,
shallow-but-finite water depth, and weak nonlinearity, that is,

The purpose of the present work is to derive and apply a stable numerical
scheme able to estimate the solution over a long distance of propagation, in
order to model high-amplitude waves and to confirm the transition from
regular to Mach reflection happening for

The remainder of this paper is organized as follows: the modified
Benney–Luke-type model is derived from the variational principle for an
inviscid and incompressible fluid

Our water-wave model is derived using a variational approach that ensures
conservation of mass, momentum and energy. In a basic sea state with extreme
waves, these conservation properties are essential given the different length
scales involved. Starting from Luke's variational principle for an inviscid
fluid with a free surface

Three-dimensional water-wave domain with rest depth

Water-wave equations are often adequately described by the potential-flow
approximation. In the absence of vorticity, the fluid velocity

To derive the Benney–Luke model, the velocity potential

The Kadomtsev–Petviashvili equation for small-amplitude solitons can be
derived from the Benney–Luke variational principle Eq. (

O-type and (3142)-type solitons as represented by ^{©}IOP Publishing. Reproduced with permission. All rights reserved.

Schematic plan showing the link between the scaling of the three systems of equations involved in the derivation of the exact solution and critical condition for which Miles' and Kodama's predictions hold in the Benney–Luke approximation.

In Sect.

As a first step in the computational solution, the Benney–Luke model needs to be discretized in space and time, on a meshed domain. This section explains the methods used to discretize the domain and the equations.

A continuous Galerkin finite-element method is used to discretize the
solutions in space. The variables

the domain in which the equations are solved, and the kind of mesh to use (e.g. quadrilaterals, the spatial dimension);

the order and type of polynomials used;

the type of expansion for the unknowns (e.g. continuous Galerkin, Lagrange polynomials);

the function space of the unknowns and test functions; and

the weak formulations discretized in time.

After incorporating the FEM expansions, the space-discrete form of the
variational principle Eq. (

Definition of the domains in the two cases described in the text:

In this section, the domain is specified and discretized in order to evaluate

The interaction of two solitary waves can be modelled using either two
obliquely intersecting channels, with incident solitons propagating along
each channel (see scheme a in Fig.

The domain is described by the length of the wall

Soliton surface deviations obtained for an initial amplitude

Domain discretization using quadrilaterals in Gmsh. In order to reduce computational requirements, mesh refinement is restricted to only the region adjacent to the wall.

The behaviour of the incident and stem waves in the cases of an oblique
incident soliton (Eq.

In order to evaluate

The numerical amplification of the stem wave is compared with the predictions
of modified Miles' theory applied to our Benney–Luke model
Eqs. (

Comparison between the expected amplification (solid line) from
Miles (Eq.

Numerical results and predictions for the reflected and stem waves in
the case of regular reflection, i.e.

Miles' theory also predicts different directions of propagation of the stem
and reflected waves in the cases of regular and Mach reflections. While in
the first case, characterized by

Numerical results and predictions for the reflected and stem waves in
the case of Mach reflection, i.e.

Figure

Prediction of the minimal distance needed by the stem wave to reach
at least twice its initial amplitude in a sea state with characteristic wave
height

Figure

The present model Eq. (

There are some limits to the current model. As already concluded in previous
studies, the wave needs to propagate over a long distance (relative to its
wavelength) in order to reach its maximal amplitude. Consequently, the
numerical domain needs to be large and the mesh fine enough to estimate the
wave crests accurately. This numerical requirement increases the
computational time. A compromise between the accuracy of the simulations and
the running time is therefore needed. This constraint is important because
near the transition from Mach to regular reflection a slight change in the
incident wave amplitude modifies dramatically the interaction parameter and
consequently the predictions of the stem and reflected waves. Therefore, a
careful analysis of the numerical results must be made. For the same reason,
simulations for

One may wonder how likely it is that solitary waves would undergo reflection
in an open ocean. Interaction of obliquely incident waves on the sides of a
ship leads to an increasing wave amplitude, sometimes reaching the deck. This
phenomenon is called “green water” and has been studied experimentally and
numerically by the Maritime Research Institute Netherlands (MARIN) to limit
the damage caused by waves on ships

The present model can also be used to predict the impact of extreme (i.e.
freak or rogue) waves on structures. Indeed, when the stem wave reaches more
than twice the amplitude of the incident wave, it can be viewed as a freak
wave since it has similar properties in terms of nonlinearity, dispersivity
and high amplitude. Table

Finally, the present work can also be used as a starting point for the
modelling of the interaction of three obliquely incident line solitons, which
should lead to a 9-fold-amplified resulting wave that can also be generated
in wave tanks.

O. Bokhove suggested this calculation to Y. Kodama, personal communication, who performed the calculation using the KP equation at the “Rogue waves” international workshop held at the Max Planck Institute in 2011, Dresden, Germany.

The implementation of our discretization of the Benney–Luke equations is an
example in Firedrake,

The Störmer–Verlet scheme Eq. (

The authors declare that they have no conflict of interest.

The research was funded by the Marie Curie Fellowship, as part of the
European Industry Doctorate (EID) SurfsUp project. The Firedrake
implementation of our discretization of the Benney–Luke equations is an example in Firedrake,