A note on trapped Gerstner waves

Abstract

The progressive Gerstner wave is studied theoretically in a rotating inviscid ocean when the bottom has a constant slope. Exact solutions are found in Lagrangian coordinates without applying the hydrostatic assumption for the pressure. For waves propagating with the coast to the right in the Northern Hemisphere, trapped Gerstner wave solutions are found for slope angles β with respect to the horizontal in the range 0 < β ≤ π/2. In particular, the Stokes edge wave transforms into the deep‐water coastal Kelvin wave in the vertical wall limit β = π/2. It is shown that standing waves obtained by superposition of oppositely traveling Gerstner wave trains in the nonrotating case are not exact solutions of the Lagrangian equations. This is discussed in relation to observed beach features and runup patterns.