New gravity–capillary waves at low speeds. Part 1. Linear geometries
New gravity–capillary waves at low speeds. Part 1. Linear geometries
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新的低速重力-毛细波第 1 部分：线性几何。
DOI：
10.1017/jfm.2013.110
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发表时间：
2013
影响因子：
3.7
通讯作者：
S. Jonathan Chapman
中科院分区：
文献类型：
--
作者：
Philippe H. Trinh;S. Jonathan Chapman
摘要当用传统的线性化理论来研究重力-毛细波时，通常假定物体的几何形状在一维或几维上是很小的。为了保持非线性性质的障碍，渐近展开在低弗劳德数或低邦德数的限制，可以得出，但在这里，解决方案总是预测无波表面在每一个订单。这是因为波实际上是指数级的小，因此超越了正则渐近的所有阶;它们的形成是渐近级数发散和相关斯托克斯现象的结果。通过应用指数渐近的技巧，我们发现了一类新的重力-毛细波的存在性，通常的线性解形式，但一个特殊的情况。在本文中，我们提出的初始理论推导出这些波通过研究重力毛细流动的线性化步骤。这将使用两种方法来完成：在第一种方法中，我们使用傅立叶变换的标准方法导出表面波;在第二种方法中，我们使用指数渐近法导出相同的结果。最终，这两种方法给出了相同的结果，但在概念上，它们为研究低弗劳德数，低邦德数问题提供了不同的见解。
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.