Interaction between water waves and elastic plates: using the residue calculus technique

This paper reports a new method of computing the coefficients of the modal expansion of the velocity potential for the so-called linear hydroelasticity problem. Two examples of the hydroelasticity problems are studied, semi-infinite plate and finite-gap cases. The finite-gap depicts the situation when the ocean surface is covered with a thin elastic plate and the plate has a gap of a constant width. The mathematical model in this paper, a thin elastic plate coupled with an inviscid, incompressible fluid, is often used to describe sea-ice sheets and floating runways. The model can describe the dynamics of these objects for the wavelengths and wave-magnitudes typically found in the ocean. This article reports mainly the technical side of the method of solution and does not dwell on the geophysical and offshore engineering aspects of the problem. Boundary-value problems of this kind, concerning a body of fluid with discontinuous boundary conditions, are usually numerically solved using the so-called mode-matching technique \cite{fox2,Lawrie99}. The mode-matching technique uses the fact that the velocity potential of the fluid can be expressed as a modal expansion over the roots of the dispersion equation for the boundary conditions, e.g., elastic plates or open water. The modal expansion for the regions are then matched at the discontinuity. A system of equations for the coefficients of the modes is obtained as the result. The method of solution described in this paper, the Residue Calculus Technique (RCT) uses a complex function, which is constructed in such a way that the coefficients of the modes correspond to the residues at the function's poles. The closed form solution for the wave propagation across an infinite crack in an ice sheet has been reported in \cite{Williams02} (using Green's function for an elastic plate) and in \cite{Chung00} (using the Wiener-Hopf technique based \cite{eavns}). The method here closely follows the method described in \cite{Linton01} (rigid plate), particularly the introduction of the finite-length correction terms. The formula is based on the solution for the semi-infinite plate problem. The interaction between the ocean waves and the semi-infinite plate is solved in \cite{Linton02} using the RCT. A difficulty arises when the plate is elastic, because there are two modes with complex wavenumbers (neither the real nor the imaginary part is zero). This is in contrast to the case of a rigid plate, where the vertical modes are $\cos\pi n\left( z+H\right) /H$, $n=0,1,2,...$ under the rigid plate. In order for the coefficients to be determine completely, the boundary conditions to be applied at the plate ends called the edge conditions, are required. In \cite{Linton02} the edge conditions were accommodated by introducong two arbitrary constants in the associated complex function. To the authors' knowledge, this method of incorporating the edge conditions to the RCT has not been reported before