rized. The combination may be especially important for deep transoms.

Although, according to Lechter (6), the method of collocation point shifting and analytic differentiation may break down at low Froude numbers, several enhancements might be made to the Rankine singularity method that was used to obtain numerical approximations to free-surface potential flows. They are related to the more general problem of calculating the linearized free-surface potential flow near arbitrarily shaped ships advancing into calm water. The first would be to eliminate the finite differencing at the hull waterline. This would eliminate all finite differences except those at the hull-transom intersection. Second, the relationship between free-surface paneling and collocation point shifting should be clarified. When a second small longitudinal shift downstream is used, damping is introduced into the wave pattern. Further, a more or less rectangular grid of panels on the free surface damps the waves in the Kelvin wave pattern more than the arrangement used in this paper, in which the computational free-surface domain is swept back at a 45-degree angle. The effect is most noticeable in the diverging waves seen in contour plots. Finally, it might be useful to consider whether something similar to the DtN exact boundary condition of Keller and Givoli (14) could be used for the outer boundaries of the computational domain to handle the boundary conditions there more accurately and to reduce the size of the computational domain.

ACKNOWLEDGMENT

This work was supported in part by the Applied Hydromechanics Research program of the Applied Research Division of the Office of Naval Research.

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