Run-Up of Linear Long Waves on Uniform Slope in Lagrangian Description

The case of linear, two-dimensional long waves on a uniform slope is considered. It is assumed that the fluid is nonviscous and incompressible. In the present paper the description of the long wave proposed by Wilde (Wilde, Chybicki 2004) is based on the fundamental assumption that the vertical material lines of fluid remain vertical during the entire motion. The equations of motion are derived with the help of a variational formulation of the problem. The Lagrangian is the difference between the kinetic and potential energy of the fluid. In the paper a correction followed from dispersion to the results obtained by Shuto is presented.