Extended Boussinesq equations for rapidly varying topography

We developed a new Boussinesq-type model which extends the equations of Madsen and Sørensen [1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly varying bathymetry. Coastal Engineering 18, 183–204.] by including both bottom curvature and squared bottom slope terms. Numerical experiments were conducted for wave reflection from the Booij's [1983. A note on the accuracy of the mild-slope equation. Coastal Engineering 7, 191–203] planar slope with different wave frequencies using several types of Boussinesq equations. Madsen and Sørensen's model results are accurate in the whole slopes in shallow waters, but inaccurate in intermediate water depths. Nwogu's [1993. Alternative form of Boussinesq equation for nearshore wave propagation. Journal of Waterway, Port, Coastal and Ocean Engineering 119, 618–638] model results are accurate up to 1:1 (V:H) slope, but significantly inaccurate for steep slopes. The present model results are accurate up to the slope of 1:1, but somewhat inaccurate for very steep slopes. Further, numerical experiments were conducted for wave reflections from a ripple patch and also a Gaussian-shaped trench. For the two cases, the results of Nwogu's model and the present model are accurate, because these models include the bottom curvature term which is important for the cases. However, Madsen and Sørensen's model results are inaccurate, because this model neglects the bottom curvature term.