A new numerical evaluation scheme is developed of the Green function which is essential in the boundary-value problem of the motions of a ship with forward speed in waves. The single integral expression derived by Bessho' is used and its integration is performed along the path of steepest descent on which the integrand does not oscillate. Using this numerical steepest-descent method, we can evaluate the Green function with less computing time as well as enough accuracy, even if both of the field and source points are close to the free surface. With this efficient scheme incorporated in the panel method, radiation and diffraction problems are solved for a submerged spheroid. Convergence and accuracy of the solution are numerically checked by increasing the number of discretized panels on the body. In the radiation problem, contributions of the steady perturbation potential to the body boundary condition and to the unsteady pressure are consistently taken into account. These effects on the added-mass and damping coefficients are investigated by comparing the obtained results with another numerical results taking only the unform-flow contribution into consideration. In the diffraction problem, computations are performed for the wave exciting forces and the diffraction wave pattern on the free surface. These computed results are compared with corresponding experiments measured by the authors, and found to be in favorable agreement.