2013/04/23 by Philippe H. Trinh, S. Jonathan Chapman · 17 citations
Earth and Planetary Sciences · Engineering · Mathematics · Physics and Astronomy · #Asymptotic analysis #Divergence (linguistics) #Exponential function #Flow (mathematics) #Fluid dynamics and aerodynamics studies #Fourier series #Nonlinear system #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Series (stratigraphy) #Surface (topology) #math-ph #math.MP #msc:30E10 #msc:41A60 #msc:76B07 #msc:76B15 #msc:76B20 #msc:76B45 #physics.flu-dyn
paper · pdf · doi:10.1017/jfm.2013.110
published in Journal of Fluid Mechanics 724, 367-391 (Cambridge University Press)
arxiv created 2013/04/23 · openalex publication_date 2013/04/29 · arxiv updated 2015/10/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Abstract When traditional linearized theory is used to study gravity–capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity–capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity–capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem.