Water wave scattering by a circular cylinder submerged in water with an ice-cover

When a train of surface water waves is normally incident on a horizontal circular cylinder fully submerged in deep water with a free surface, it experiences no reflection, although there is a change of phase of the transmission coefficient. It is shown here that the same phenomenon also holds good when the deep water has an ice-cover instead of free surface. The problem is reduced to solved an integral equation of either a first kind or a second kind. Both the integral equations produce systems of infinite linear equations which are solving by truncation. It is observed that while the absolute value of the transmission coefficient is almost unity, its phase varies, which is depicted graphically against the wave number in a number of figures. When the ice-cover is replaced by a free surface, the corresponding figures for the phase angle can be identified with results found in the literature.