Kelvin–Froude wake patterns of a traveling pressure disturbance

Abstract

According to Kelvin, a point pressure source uniformly traveling over the surface of deep calm water leaves behind universal wake pattern confined within 39° sector and consisting of the so-called transverse and diverging wavefronts. Actual ship wakes differ in their appearance from both each other and Kelvin’s prediction. The difference can be attributed to a deviation from the point source limit and for given shape of the disturbance quantified by the Froude number F. We show that within linear theory effect of arbitrary disturbance on the wake pattern can be mimicked by an effective pressure distribution. Further, resulting wake patterns are qualitatively different depending on whether water-piercing is present or not (“sharp” vs “smooth” disturbances). For smooth pressure sources, we generalize Kelvin’s stationary phase argument to encompass finite size effects and classify resulting wake patterns. Specifically, we show that there exist two characteristic Froude numbers, F1 and F2>F1, such as the wake is only present if F≳F1. For F1≲F≲F2, the wake consists of the transverse wavefronts confined within a sector of an angle that may be smaller than Kelvin’s. An additional 39° wake made of both the transverse and diverging wavefronts is found for F≳F2. If the pressure source has sharp boundary, the wake is always present and features additional interference effects. Specifically, for a constant pressure line segment source mimicking slender ship the wake pattern can be understood as due to two opposing effect wakes resembling (but not identical to) Kelvin’s and originating at segment’s ends.