New formulation of the finite depth free surface Green function

Abstract

For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [Communs. Pure Appl. Math. 3:45–101, 1950] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain $(0, \infty )$. When the source point together with a field point is on the free surface, John’s integral and its gradient are not convergent since the integration $\int ^\infty _\kappa$ of the corresponding integrands does not tend to zero in a uniform manner as $\kappa$ tends to $\infty$. Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice.

Appendix A: Derivation of John’s Green function

For the completion of the analysis, we follow John [11] to show the derivation of the Green function

$$\begin{aligned} G=\frac{1}{r}+\frac{1}{r_0}+G_1. \end{aligned}$$

The use of the Hankel transformation (18) produces

$$\begin{aligned} {{\mathcal {H}}}\left( \frac{1}{r}+\frac{1}{r_0}\right)= & {} \frac{1}{k}\left( \mathrm{{e}}^{-k|z-\zeta |} +\mathrm{{e}}^{-k|z+\zeta +2h|}\right) . \end{aligned}$$

Then applying the Hankel transformation to the Laplace equation (1) and employing the bottom boundary condition (3), we have

$$\begin{aligned} {\mathcal {H}}(G_1 )=\frac{1}{k} A_0(k) \cosh k(z+h) \end{aligned}$$

(A.1)

for a function $A_0$. Therefore applying again the Hankel transformation to the free surface condition (2), we have, for z close to 0,

$$\begin{aligned} 0\approx & {} (\partial _z-\nu ){\mathcal {H}}(G_1) +\partial _z{{\mathcal {H}}}\left( \frac{1}{r}+\frac{1}{r_0}\right) -\nu {{\mathcal {H}}}\left( \frac{1}{r}+\frac{1}{r_0}\right) \nonumber \\= & {} \frac{1}{k} A_0(k) [k\sinh k(z+h)-\nu \cosh k(z+h)]- \frac{k+\nu }{k}\left( \mathrm{{e}}^{-k(z-\zeta )} +\mathrm{{e}}^{-k(z+\zeta +2h)}\right) . \end{aligned}$$

(A.2)

This implies that

$$\begin{aligned}&A_0(k)=\frac{2(\nu +k)\mathrm{{e}}^{-kh}\cosh k(\zeta +h)}{k\sinh kh-\nu \cosh kh }. \end{aligned}$$

Therefore the desired Green function is obtained by rewriting (A.1) as

$$\begin{aligned}&G_1=H^{-1}\left( \frac{1}{k} A_0(k) \cosh k(z+h)\right) , \end{aligned}$$

or the desired Green function

$$\begin{aligned} G= & {} \frac{1}{r}+\frac{1}{r_0}+\int _L\frac{2(\nu +k)\mathrm{{e}}^{-kh}\cosh k(\zeta +h)\cosh k(z+h)}{k\sinh kh-\nu \cosh kh }J_0(kR)\mathrm{{d}}k. \end{aligned}$$

(A.3)

The integral pass L passing beneath the pole is determined by the asymptotic behaviour (4).