The fundamental solution of the ship-motion problem, known as the ship-motion Green function associated with a pulsating and translating source, is formulated and analyzed based on the weakly damped free-surface flows. The inclusion of viscous effects modifies the dispersion relation, resulting in the existence of three wavenumbers instead of two, as observed in inviscid fluids. The Green function with viscous effects is examined in relation to these three complex wavenumbers, and novel formulations of Havelock's type are developed. Special attention is paid to the analysis of several numerical issues and the treatment to resolve them. The wavenumber integral function involved is analyzed using contour integration in the complex plane, leading to a classical formulation that includes the exponential integral function and a constant function representing the contributions of residues. Additionally, new series and Taylor expansions are developed to facilitate the integration of the ship-motion Green function on ship hull and the free surface. Furthermore, derivatives and antiderivatives of the wavenumber integral function have been obtained for all three formulations. Numerical examples are included to illustrate the soundness of the formulations and their efficiency in numerical evaluations. At the critical frequency, in particular, the Green function with viscous effects is shown to be finite and its magnitude decreases along the horizontal distance from the source. Finally, benefits of introducing viscous effects in the fundamental solution are examined and emphasized in the discussion and conclusions.

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