Singular velocity potentials are derived which satisfy the linear free-surface boundary condition and appropriate initial condition or radiation condition, but which are singular in the same form as a spheroidal harmonic. Thus these singularities can be utilized to satisfy boundary conditions on submerged spheroids without integrating conventional point sources or dipoles over the body surface or axis, in an analogous manner to the use of spheroidal harmonics in an unbounded fluid. In addition to the general unsteady case, equations are derived for the special cases of steady forward motion and of sinusoidal time-dependence, either at zero or constant forward velocity of the spheroid. Wave-free singularities are derived for the case of zero forward speed, and applications to problems in wave resistance and in seakeeping are discussed briefly.

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