On the solution of the problem of scattering of surface-water waves by the edge of an ice cover

The mixed boundary-value problem arising in the study of scattering of two-dimensional time-harmonic surface-water waves by a discontinuity on the surface boundary conditions, separating the clean surface and an ice-covered surface, is solved completely in the case of an in­ nite depth of water. The main problem is reduced to that of solving a singular integral equation, of the Carleman type, over a semi-­ nite range and the explicit solution of the original problem is determined. Neat and computable expressions are derived for the two most important quantities, known as the re®ection and transmission coe ̄ cients, occurring in such scattering problems and tables of numerical values of these quantities are presented for speci­ c choices of a parameter modelling the ice cover. The absolute values of the re®ection and transmission coe ̄ cients are presented graphically. The present method of solution of the boundary-value problem produces simple expressions for the principal unknowns of the problem at hand and thus provides an easily understandable alternative to the rather complicated Wiener{Hopf method used previously.