Exponential asymptotics for steady capillary ripples on steep gravity waves #

Josh Shelton

11:10 Tuesday in 2Q49.

Part of the IMA Lighthill-Thwaites prize session.

Abstract #

A steadily travelling water wave, under the action of both gravity and surface tension, is one of the most fundamental problems in fluid dynamics. Physically these waves exist in the regime of small surface tension. However, mathematically this is a singular limit for which vital effects in the solution profile appear beyond-all-orders of an asymptotic expansion for small surface tension. These correspond to parasitic capillary ripples, which in nature are observed to form on travelling surface waves.

In this talk, I will demonstrate how the inclusion of these beyond-all-order terms (which are exponentially small) corrects a confusing narrative in potential flow theory extending back to historical works by Longuet-Higgins, Chen & Saffman, and Vanden-Broeck. The derivation of these exponentially-small components requires the use of modern techniques in exponential asymptotics, in which the Stokes phenomenon occurs across Stokes lines in the analytic continuation of the leading-order asymptotic solution (a gravity wave).