# Nonlinear water waves over strongly varying bottom topography

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A fully nonlinear time-stepping model for water wave motion over strongly varying topography

in three dimensions is presented. The modl is fully dispersive, fully nonlinear and, and also very rapid. The kinematic and dynamic boundary

condition at the free surface are used to derive the prognostic equations. Conservation of mass yields two integral equations for the normal velocity at the free surface and the wave potential at the sea floor. These are inverted analytically be means of Fourier transform. Various levels of nonlinearity of the equations are derived. A highly efficient computational scheme is obtained by the FFT-part of the formulation. Computations exemplify how a very long tsunami with leading depression running into very shallow water develop very short waves, that in the beginning are linear, developing then into a train of solitary waves of

large amplitude. Numerical examples on the formation of very strong ocean surface waves - rogue waves - are given.