Two-dimensional flexural–gravity waves of finite amplitude in deep water

Steady periodic and solitary waves propagating in a 2D fluid bounded above by a flexible sheet—which may be viewed as modelling an ice sheet—are considered in deep water. The non-linear elastic model is based on the special Cosserat theory of hyperelastic shells proposed by Toland (2008, Steady periodic hydroelastic waves. Arch. Ration. Mech. Anal., 189, 325–362) for this problem. Numerical solutions are computed via conformal mapping and a pseudo-spectral method. New solitary waves are found by using a continuation method to follow the branch of elevation waves. The results extend Guyenne and Părău's findings (2012, Computations of fully non-linear hydroelastic solitary wave on deep water. J. Fluid Mech., 713, 307–329). It is shown that, for periodic waves, far along the branches the profiles become overhanging and ultimately approach configurations with a trapped bubble at their troughs.