## Abstract

In this chapter we consider the two-dimensional interaction of an incident wave with a flexible floating dock or very large floating platform (VLFP) with finite draft. The water depth is finite. The case of a rigid dock is a classical problem. For instance Mei and Black (J. Fluid Mech. 38: 499–511, 1969) have solved the rigid problem, by means of a variational approach. They considered a fixed bottom and fixed free surface obstacle, so they also covered the case of small draft. After splitting the problem in a symmetric and an antisymmetric one, the method consists of matching of eigenfunction expansions of the velocity potential and its normal derivative at the boundaries of two regions. In principle, their method can be extended to the flexible platform case. Recently we derived a simpler method for both the moving rigid and the flexible dock (Hermans in J. Eng. Math. 45: 39–53, 2003). However we considered objects with zero draft only. In this chapter we present our approach for the case of finite, but small, draft. The draft is small compared to the length of the platform to be sure that we may use as a model, for the elastic plate, the thin plate theory, while the water pressure at the plate is applied at finite depth. The method is based on a direct application of Green’s theorem, combined with an appropriate choice of expansion functions for the potential in the fluid region outside the platform and the deflection of the plate. The integral equation obtained by the Green’s theorem is transformed into an integral-differential equation by making use of the equation for the elastic plate deflection. One must be careful in choosing the appropriate Green’s function. It is crucial to use a formulation of the Green’s function consisting of an integral expression only. In Sect. 9.4 we derive such a Green’s function for the two-dimensional case. One may derive an expression as can be found in the article of Wehausen and Laitone (Encyclopedia of Physics, vol. 9, Springer, pp. 446–814, 1960) after application of Cauchy’s residue lemma. In the three-dimensional case one also may derive such an expression. The advantage of this version of the source function is that one may work out the integration with respect to the space coordinate first and apply the residue lemma afterwards. In the case of a zero draft platform this approach resulted in a dispersion relation in the plate region and an algebraic set of equations for the coefficients of the deflection only. Here we derive a coupled algebraic set of equations for the expansion coefficients of the potential in the fluid region and the deflection.

## Keywords

Dispersion Relation Velocity Potential Flexural Rigidity Fluid Region Rigid Dock## References

- 4.A.J. Hermans, A boundary element method for the interaction of free-surface waves with a very large floating flexible platform. J. Fluids Struct.
**14**, 943–956 (2000) CrossRefGoogle Scholar - 5.A.J. Hermans, Interaction of free-surface waves with a floating dock. J. Eng. Math.
**45**, 39–53 (2003) MathSciNetMATHCrossRefGoogle Scholar - 6.A.J. Hermans, The ray method for the deflection of a floating flexible platform in short waves. J. Fluids Struct.
**17**, 593–602 (2003) CrossRefGoogle Scholar - 8.C. Hyuck, C.M. Linton, Interaction between water waves and elastic plates: Using the residue calculus technique, in
*Proceedings of the 18th IWWWFB*, ed. by A.H. Clément, P. Ferrant (Ecole Centrale de Nantes, Nantes, 2003) Google Scholar - 10.C.C. Mei, J.L. Black, Scattering of surface waves by rectangular obstacles in waters of finite depth. J. Fluid Mech.
**38**, 499–511 (1969) MATHCrossRefGoogle Scholar - 11.M.H. Meylan, V.A. Squire, The response of a thick flexible raft to ocean waves. Int. J. Offshore Polar Eng.
**5**, 198–203 (1995) Google Scholar - 16.M. Roseau,
*Asymptotic Wave Theory*(North-Holland, Amsterdam, 1976) MATHGoogle Scholar - 19.J.V. Wehausen, E.V. Laitone, Surface waves, in
*Encyclopedia of Physics*, vol. 9 (Springer, Berlin, 1960), pp. 446–814. Also http://www.coe.berkeley.edu/SurfaceWaves/ Google Scholar