We consider a floating or submerged body in deep water translating parallel to the undisturbed free surface with a steady velocity \(U\) while undergoing small oscillations at frequency \(\omega\). It is known that for a single source, the solution becomes singular at the resonant frequency given by \(\tau\equiv U\omega/g= {1\over 4}\), where \(g\) is the gravitational acceleration. We show that for a general body, a finite solution exists as \(\tau\to {1\over 4}\) if and only if a certain geometric condition (which depends only on the frequency \(\omega\) but not on \(U\)) is satisfied. For a submerged body, a necessary and sufficient condition is that the body must have a non-zero volume. As an illustration, we provide an analytic (closed-form) solution for the case of a submerged circular cylinder oscillating near \(\tau={1\over 4}\).