On the Wave Field Due to a Moving Body Performing Oscillations in the Vicinity of the Critical Frequency

A two-dimensional submerged body translating under a free surface with steady velocity U while performing small oscillations with frequency ω is considered. It has been known for a long time that for a single source the solution becomes unbounded at the critical frequency, which is given by τ=Uω/g=1/4 where g is the acceleration of gravity. It was therefore believed that also the motion due to an oscillating body was unbounded at this frequency. It has, however, in the last few years been shown that this motion is bounded for τ=1/4. In this paper previous results are discussed, and the strong variation of the forces with respect to ω close to τ=1/4 is examined. Recently, a mathematical argument was given that the motion at the critical frequency is bounded for bodies with nonzero cross-section area. It is proved that also the motion generated by a thin foil with zero cross-section area is bounded at τ=1/4.