Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation
Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation
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DOI：
10.1111/sapm.12128
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发表时间：
2015-07
影响因子：
2.7
通讯作者：
S. Dyachenko;P. Lushnikov;A. D. F. Mathematics;U. I. Urbana-Champaign;Usa;-. D. O. Mathematics;U. Arizona;Statistics;U. N. Mexico;-. M. I. O. Physics;Russia.
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文献类型：
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作者：
S. Dyachenko;P. Lushnikov;A. D. F. Mathematics;U. I. Urbana-Champaign;Usa;-. D. O. Mathematics;U. Arizona;Statistics;U. N. Mexico;-. M. I. O. Physics;Russia.
分析了具有自由表面和无限深的理想不可压缩流体二维势流的Stokes波的复解析结构。Stokes波是一种完全非线性的周期性重力波，以恒定速度传播.为了高精度地找到Stokes波，并研究了在波峰处2π/3弧度角时Stokes波接近极限形式的情况，进行了四重（32位）和变精度（200位以上）的模拟。使用保角映射将Stokes波的自由流体表面映射到真实的线，流体域映射到下复半平面。斯托克斯波完全由上复半平面中的复奇点表征。这些奇异性通过复平面中斯托克斯波的有理（Padé）插值来解决。Padé近似对复极点密度的收敛性随数值精度的提高和逼近极点数目的增加而增加，表明Stokes波的奇点只有由分支割连接的分支点.收敛密度是跨越分支切口的跳跃。斯托克斯波的每个水平空间周期λ存在一个平方根分支点，其位于距真实的线的距离vc处。标度波高H/λ从线性极限H/λ=0增加到临界值Hmax/λ，标志着从几乎线性波的极限到强非线性极限Stokes波（也称为最大高度Stokes波）的过渡。这里，H是物理变量中从波峰到波谷的波高。当奇点到达流体表面时，极限斯托克斯波出现。给出了不同高度Stokes波的Padé近似表。这些表允许恢复斯托克斯波的相对精度至少为10−26。表中的极点数量从近线性斯托克斯波的几个增加到大约一百个极点，高度非线性斯托克斯波，vc/λ为10−7。
Complex analytical structure of Stokes wave for two‐dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave prop agating with the constant velocity. Simulations with the quadruple (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with 2π/3 radians angle on the crest. A conformal map is used that maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half‐plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half‐plane. These singularities are addressed by rational (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé approximation to the density of complex poles with the increase in the numerical precision and subsequent increase in the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one square‐root branch point per horizontal spatial period λ of Stokes wave located at the distance vc from the real line. The increase in the scaled wave height H/λ from the linear limit H/λ=0 to the critical value Hmax/λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called the Stokes wave of the greatest height). Here, H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least 10−26. The number of poles in tables increases from a few for near‐linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with vc/λ∼10−7.