JOURNAL OF GEOPHYSICALRESEARCH,VOL. 98, NO. C7, PAGES 12,537-12,542, JULY 15, 1993 Finite-Floe Wave Reflection and Transmission Coefficients From Semi-Infinite Model MICHAEL MEYLAN AND VERNON A. SQUIRE Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand A model to describe the reflection and transmission of ocean waves by a single ice floe is developed from the semi-infinite modelof Fox and Squire(1990, 1991). This is done by considering the coefficients for the transition from ice to water in the semi-infinite case in terms of those from water to ice. Finite-floe reflection and transmission coefficients,/• and T, respectively, are then found as the solution of a set of four simple simultaneousequations. The properties of/• and T are investigated, and examples of their absolutevaluesare given for severalgeometries. compares well with the predictions of a precise model in the case of deep water. These results suggestthat the analytical model described has applicationsto defining the sea state within marginal ice zones, given the floe size and ice thicknessdistributions and the incoming sea wave spectrum. INTRODUCTION Fox and Squire [1990,1991] have reported a precise linear model to determine the reflection and transmission coeffi- cients for oceanwaves propagating into a sheetof shore fast seaice floatingon water of any finite depth, assuming that the semi-infinite region-cw < x < 0, 0 • y • H is openwa- ter and the semi-infinite region0 _< x • cw, 0 • y < H is ice- covered. The model is precise in the sense that the complete set of velocity potentials is included;further, the lineariza- tion of the hydrodynamics has been shown to be valid for physically reasonable -wave heights and periods [Fox,1992]. It compares favorably with the few data that exist [Squire, 1984], especially when some velocity-dependent damping is introduced into the constitutive relation for the ice [Squire and Fox, 1992] using,for example,the method of Robinson andPalmer [1989].The model has recently been generalized to include obliquely incidentocean waves [Fox and Squire, 1993]. In this paper we are concerned,not with a semi-infinite ice cover,but with an ice floe of limited spatial extent such as might be foundwithin a marginal ice zone(MIZ). The ice is thereforeassumed to occupy a region0 _< x _< 1 of the sea surface. Ocean waves incident on the ice floe from the open searegion -c• < x < 0 will be partially reflected,with a proportion of the incoming wave energy being transmit- ted into the wa. ters beyond the floe. In a MIZ this energy will then likely meet another floe and again be partially re- flected and transmitted. With each new floe encountered the energy will decrease, leading to spatial attenuation of waves penetrating the pack ice. This has been observed on many occasions; see for exampleLiu et al. [1991], Robin [1963], Squire and Moore [1980], Wadhams [1975, 1978], and Wadhams et al. [1986, 1988]. Attempts to model the phe- nomenon either have used singleice floes in a scattering model [Wadhams, 1986]or haveassumed that the packice forms a continuum of some kind andhave then superimposed some dissipative mechanism [Liu et al., 1991; Squire,1993; Wadhams, 1973; Weber, 1987]. While the second methodis Copyright 1993 by the American Geophysical Union. Paper number 93JC00940. 0148-0227 / 93/ 93J C-00940$ 05.00 mathematically more elegant, it is really the single icefloes themselves whichcause the wavedecay and it is therefore more physically meaningful to represent the MIZ asa large number ofdiscrete ice floes and cakes satisfying some prede- fined floe size and floe thickness distribution. Unfortunately, the determination of the reflection and transmission coeffi- cients fora single elastic (orviscoelastic) floating body isnot simple, and the values used in the literature are based on an approximation firstproposed by Hendrickson et al. [1962] and Hendrickson [1966],and later modified by Wadhams [1986]. The approximation, which is essentially identical to that used in the semi-infinite case prior to Fox and Squire [1990], uses incomplete velocity potentials which cannot be made to satisfy all the boundary conditions precisely. Fox and Squire [1990] show that this leads to significant sys- tematic errors in the semi-infinite case, particularly at short periods, and we would expect similar deficiencies to occur in the finite-floe case. It is therefore of value to investigate the application of the precise Fox and Squire modelto an ice floe of finite length as opposed to the semi-infinite sheet of shore fast ice for which it was intended. The procedure described in this paper rests onthesimple observation that theremustbe a relationship between the reflection and transmission coefficients forwaves entering an icesheet andthose for waves leaving the same icesheet. This is a well-known result of optics for electromagnetic waves travelling between two media of different refractive index. In the present context, unlike the optics case, an incoming ocean wavewill generate additionalpotentials on both sides of the ice edge whichare necessary to precisely solve the system of equations but which decay rapidly away from the edge. These potentials are of two types: in the open water they are the sum:of an infinite number of evanescentmodal eigenfunctions; Within the ice they are made up of acomplex conjugate pair of damped travelling wave modes together with a similar infinite sum of evanescentmodes. A short distance fromthe edge, determined by the incom- ingwavelength and the icethickness, only the propagating waves will remain; anice-coupled wave propagating into the icewhich suffers noattenuation in a perfectly elastic model, and an open waterwave reflected back into the open sea. Thetransmitted ice-coupled wave will reach thefar edge of the floe, where it will be partiallytransmitted out into the open sea andpartially reflected back toward the front edge to again suffer some reflection. This process will, in prin- 12,537