  In the present study, we propose a modified version of the Nonlinear Shallow
Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the
case when the bottom undergoes some significant variations in space and time.
The model is derived from a variational principle by choosing an appropriate
shallow water ansatz and imposing some constraints. Our derivation procedure
does not explicitly involve any small parameter and is straightforward. The
novel system is a non-dispersive non-hydrostatic extension of the classical
Saint-Venant equations. A key feature of the new model is that, like the
classical NSWE, it is hyperbolic and thus similar numerical methods can be
used. We also propose a finite volume discretisation of the obtained hyperbolic
system. Several test-cases are presented to highlight the added value of the
new model. Some implications to tsunami wave modelling are also discussed.
