Water wave problem with inclined walls (Q2085610)

The authors study the propagation of two-dimensional free-surface incompressible, homogeneous potential flows with a bathymetry consisting of two inclined walls with a 45-degrees-slope. First, they propose a description of the free surface as the image of the surface level at rest through the flow associated with an artificial divergence-free and irrotational velocity. The evolution equations for free-surface potential flows can then be described in terms of the physical and artificial harmonic velocity potentials. The role played by the artificial velocity is however implicit, as the question of the existence or determination of such an artificial velocity field for free surfaces at a given time is not pursued by the authors. Instead the authors turn to approximate explicit expressions, using the weakly nonlinear assumption of small deformations. By neglecting cubic and higher-order nonlinearities, they arrive at explicit (approximate) expressions for the trace of the physical and artificial velocity potentials at the surface level at rest. Using a decomposition of harmonic functions into normal modes and explicit expressions provided at the linear level thanks to the special geometry of the bathymetry, these expressions can be translated into evolution equations for the spectral coefficients. In other words, these expressions are closed evolution equations involving spectral multipliers, and as such are natural counterparts of ``Whitham-Boussinesq'' models for flat topographies. The study is completed with numerical experiments, computing wave amplitudes and run-up for near-monochromatic and near-bichromatic waves.