Exponential asymptotics and capillary waves

Abstract

Recently developed techniques in exponential asymptotics beyond all orders are employed on the problem of potential flows with a free surface and small surface tension, in the absence of gravity. Exponentially small capillary waves are found to be generated on the free surface where the equipotentials from singularities in the flow (for example, stagnation points and corners) meet it. The amplitude of these waves is determined, and the implications are considered for many quite general flows. The asymptotic results are compared to numerical simulations of the full problem for flow over a polygonal plough and for flow round a right-angled corner, and they show remarkably good agreement, even for quite large values of the surface tension parameter.