Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution

Abstract

Two-dimensional periodic waves beneath an elastic sheet resting on the surface of an infinitely deep fluid are investigated using a high-order series-expansion technique. The solution is found to have certain features in common with capillary-gravity waves; specifically, there is a countably infinite set of values of the flexural rigidity of the sheet at which the series solution fails, and these values are conjectured to be bifurcation points of the solution. Limiting waves of maximum height are found at each value of the flexural rigidity investigated. These are characterized by a cusp singularity in the elastic bending moment at the wave crest, and infinite fluid pressure there. For at least one value of the flexural rigidity, the series solution shows that the wave of maximum height also travels with infinite speed.