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Surface waves of large amplitude beneath an elastic sheet. I. High-order series solution. (English) Zbl 0607.76015

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 countable 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 serious solution shows that the wave of maximum height also travels with infinite speed.


MSC:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76T99Two-phase and multiphase flows
76M99Basic methods in fluid mechanics
35Q99PDE of mathematical physics and other areas