Climb of a bore on a beach. Part 1. Uniform beach slope

The shoreward travel of a bore into water at rest on a beach of uniform slope is studied to elucidate why, in a class of problems—mainly gas-dynamical ones involving non-uniform shock propagation—similarity solutions seem to act like magnets attracting other solutions. For the shallow-water problem, the real magnet is shown to be the shore singularity of the governing differential equations. The shore singularity of the solution is shown to be a directional singularity of the water acceleration, for a fairly wide range of conditions, and a rather detailed asymptotic approximation for the bore development near shore is deduced.