we write the velocity potential φ, pressure p, and vertical displacement of the structure to in the following normalized form:
where a is the amplitude of incident wave, ρ the fluid density, and g the gravitational acceleration.
Suffix I represents quantities related to the incident wave, S the scattering component, and j the radiation component of ｊ-th mode of motion with complex amplitude Xｊ. In the definition of mode indices, not only rigid-body motions, but also a set of 'general' modes to be used for representing elastic deformations are included. Note that pI = 0 on z = 0 and that the disturbance due to existence of the structure can be represented by the pressure distribution on the free surface, pj(j = S,1,2,…)on z = 0, because the draft is substantially zero.
In the analysis to follow, the length dimensions are nondimensionalized in terms of L/2, and thus the structure exists in the region of |x|＜ 1 and |y|＜ b = B/L on ｚ= 0.
The dynamic and kinematic boundary conditions on the free surface are written as
where G(x, y, z) denotes the Green function satisfying the homogeneous equation with pｊ = 0 in (5) and is written for the finite-depth case in the form
