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WAVE PROPAGATION ACROSS A CRACK IN AN ICE SHEET

This article presents a mathematical model and a method of solution for the wave propagation across a crack in an ice sheet. Wiener-Hopf (W-H) technique is used to derive analytical solutions of the problem, which leads to simple description of the conditions at the crack. The conditions at the crack are represented by a 4-by-4 matrix and a vector in the W-H solutions. Thus it is possible to study the effects of the crack on the wave propagation using the matrix and the vector. As a beginning of mathematical modelling of the crack conditions, the connection at the crack is mimicked by a rotational and a vertical springs. These springs transmit the vertical and gradient differences from one ice sheet to the other. The wave propagation is studied under varying spring constants. INTRODUCTION This paper presents a W-H solution of wave propagation across a crack where two semi-infinite floating elastic plates are joined by various transition conditions (see Fig. 1). There has been a steady development of analytical solutions of floating elastic plates and water waves. A simple case of water-plate interaction problem is solved in (Balmforth and Craster, 1999) and (Evans and Davies, 1968) using the W-H technique. (There are too many papers on the W-H to list them all here.) A similar technique is used in (Chung and Fox, 2002) to solve a plate-plate interaction with an open gap and a rigid joint. In a slightly different situation, (Chung and Linton, 2003) gives analytical solutions of plate-water-plate problem using the residue calculus technique. A complementary problem, water-plate-water with free edge conditions is solved using the W-H in (Tkacheva, 2002). Wave scattering by a long crack is studied in (Evans and Porter, 2003) using a Green’s function for a floating elastic plate. The family of problems concerning floating elastic plates, which can be solved analytically, is growing. However, the conditions at the edges of the plates have been usually assumed to be free. In this paper we exploit the fact that the W-H technique can, with some modifications shown in (Chung and Fox, 2002), incorporate more complex transition conditions than the simple open gap or rigid joint.