# Mass Loading Model of Ice

One particular problem that illustrates the progression of solution techniques for floating plate problems is that of the scattering of incoming ocean waves by shore-fast sea ice, modelled by a semi-infinite sheet of ice. The first attempt at a solution was by Weitz and Keller 1950 who solved a simplified problem by using the Wiener-Hopf technique to obtain an extremely simple expression for the modulus of the reflection coefficient, [math]\displaystyle{ R }[/math]. This formula is given by

where [math]\displaystyle{ \alpha_0 }[/math] and [math]\displaystyle{ k_0 }[/math] are the wavenumbers of the propagating waves in the water and in the ice respectively (both are real), and was pointed out by both Shapiro and Simpson 1953 and Keller and Weitz 1953. (Note that although Wadhams 1986 showed that there is an error in the paper by Shapiro and Simpson 1953, their formula for [math]\displaystyle{ R }[/math] is still valid as that mistake only pertained to an energy conservation theorem and an expression for the modulus of the transmission coefficient.) Their solution was obtained by ignoring the elastic properties of the ice, and simply modelling the ice as a collection of independent point masses. This model is referred to as the "mass loading model", which, although somewhat inaccurate for ice sheets (for example it predicts that wavelengths should shorten on moving from water into ice, Squire et al. 1995), has proved reasonably successful for wave propagation through frazil or pancake ice (Wadhams and Holt 1991). It effectively reduces the order of the thin plate boundary condition from five to one, eliminating the need to apply additional conditions at the ice-edge.