Green function with dissipation and side wall effect in wave tanks

The Green function with viscous dissipation associated with a pulsating source in a wave tank is considered within the theory of visco-potential flow presented in Chen & Dias (2010). The so-called Tank Green function (TGF) with the side-wall effect in wave tanks is formulated as a formal sum of open-sea Green functions representing the infinite images between two parallel side walls of the source in the tank. The introduction of fluid viscosity gives rise to a decay factor in the TGF which is not singular any more at the wavenumbers associated with the transversal resonances. Furthermore, the new integrals representing the far-field waves obtained by analyzing the truncated infinite series of the wave component of the open-sea Green function are expressed as the analytical formulations involving the complementary error function. 1. Tank Green function with dissipation A Cartesian coordinate system is defined by placing the (x, y)-plane coincided with the undisturbed free surface and the z-axis oriented positively upward. The x-axis is coincident with the center plane of the tank whose width is denoted by b. Under the assumption of fairly perfect fluid (Chen & Dias, 2010), the TGF G(P,Q) representing the velocity potential at a field point P (x, y, z) in the wave tank due to a pulsating source of unit strength located at the point Q(x′, y′, z′), satisfies the following set of equations: ∇2G(P,Q) = δ(P −Q) P ⊂ D (1a) Gz − k̄G − i4αGzz = 0 z = 0 (1b) Gz = 0 z = −h (1c) Gy = 0 y = ±b/2 (1d) where δ(·) is the Dirac function and the parameters (k̄, α) are defined as k ̄ = ω2/g and α = µω/(ρg) with ω the frequency of pulsating source, g the acceleration of the gravity, µ the fluid viscosity and ρ the fluid density. The solution of (1) can be obtained by considering an infinite number of images of the source between two parallel side-walls, that is: G(P,Q) = n=−