  For a pulsating free surface source in a three-dimensional finite depth fluid
domain, the Green function of the source presented by John [F. John, On the
motion of floating bodies II. Simple harmonic motions, Communs. Pure Appl.
Math. 3 (1950) 45-101] 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.
