The water wave - ice floe interaction and associated integral equation problems.

The water wave - ice floe interaction is introduced by reviewing the work done to date on the problem. Several mathematical models, incorporating hitherto unexplored and possibly significant mechanisms of the interaction, are then constructed and investigated. In the first place, the effect of a plane wave incident at any angle upon a semi-infinite elastic sheet of constant thickness is considered, using linearised shallow water theory. The solution for the velocity potential under the ice is discussed for various values of the physical parameters, and in the most interesting case, numerical calculations are made to determine the relevance of such factors as ice thickness and angle of incidence. Secondly, a semi-infinite sheet of variable thickness is examined and the particular case treated when this thickness has a sinusoidal form. Ranges of incident wavelengths corresponding to a progressive wave solution under the ice are calculated. Also, an ice thickness having a rectangular wave form is considered with similar results. Attention is then turned to the problem of the existence of a progressive wave in an infinite array of rigidly held, equally spaced floes. Two different approaches are employed to reduce the resulting potential problem to weakly singular integral equations, which in turn are solved by a perturbation method, and, in the general case, by a numerical technique. It is found that complex wave groups can be constructed satisfying the problem, but that simple progressive waves do not exist. In an attempt to make analytic inroads on the above mentioned integral equations, some aspects of singular integro-differential equations are investigated, and methods developed by which these may be solved. The closely associated generalised Riemann-Hilbert problem is also discussed and two integro-differential equations arising in aerodynamic theory are solved as examples of the techniques proposed.