Scattering of flexural waves by multiple narrow cracks in ice sheets floating on water

An explicit solution is derived for the reflection and transmission of flexural-gravity waves propagating on a uniform elastic ice sheet floating on water which are obliquely-incident upon any number, N, of narrow parallel cracks of arbitrary separation. The solution is expressed in terms of a system of 2N linear equations for the jumps in the displacements and gradients across each of the cracks. A number of interesting features of the problem are addressed including the scattering by periodically-spaced arrays of cracks, the existence of localised edge wave solutions which travel along each of the cracks and examples of non-uniqueness, or trapped waves, in the case of four cracks. The problem of wave reflection by a semi-infinite periodic array of cracks is also formulated exactly in terms of a convergent infinite system of equations and relies on certain properties of the so-called Bloch problem for wave propagation through infinite periodic array of cracks.