Structure and location of branch point singularities for Stokes waves on deep water
Structure and location of branch point singularities for Stokes waves on deep water
复制标题
深水斯托克斯波分支点奇点的结构和位置
DOI：
--
复制
发表时间：
2016
影响因子：
3.7
通讯作者：
P. Lushnikov
中科院分区：
文献类型：
--
作者：
P. Lushnikov
斯托克斯波是一种在无粘性流体中以恒定速度传播的有限振幅周期性重力波。使用斯托克斯波的自由流体表面到实轴的共形映射以及将流体域映射到下复半平面上来分析斯托克斯波的复杂解析结构。斯托克斯波的每个空间周期有一个平方根分支点，位于上复半平面，距实轴距离 $v_{c}$。斯托克斯波高度的增加导致 $v_{c}$ 接近于零，并且限制斯托克斯波形成在 $v_{c}=0$ 处。极限斯托克斯波具有 $2/3$ 幂律奇点，在波峰上形成 $2{\rm\pi}/3$ 弧度角，这与适用于任意小但非零 $v_{c}$ 的平方根奇点有质的不同，使得零 $v_{c}$ 的极限非常重要。通过将平方根的分支切割交叉到黎曼曲面的第二层和随后的更高层，以在每层距实轴 $\pm v_{c}$ 的距离处找到耦合平方根奇点，可以解决该限制。片的数量是无限的，斯托克斯波在所有这些片中的解析延拓与每片内奇点处的半整数幂级数展开一起被发现。据推测，前导阶的非限制斯托克斯波由无限数量的嵌套平方根奇点组成，这也意味着在第三层和更高层中存在远离实轴和虚轴的附加平方根奇点。当 $v_{c}$ 消失时，这些嵌套的平方根形成极限斯托克斯波的 $2/3$ 幂律奇点。
The Stokes wave is a finite-amplitude periodic gravity wave propagating with constant velocity in an inviscid fluid. The complex analytical structure of the Stokes wave is analysed using a conformal mapping of a free fluid surface of the Stokes wave onto the real axis with the fluid domain mapped onto the lower complex half-plane. There is one square root branch point per spatial period of the Stokes wave located in the upper complex half-plane at a distance $v_{c}$ from the real axis. The increase of Stokes wave height results in $v_{c}$ approaching zero with the limiting Stokes wave formation at $v_{c}=0$ . The limiting Stokes wave has a $2/3$ power-law singularity forming a $2{\rm\pi}/3$ radians angle on the crest which is qualitatively different from the square root singularity valid for arbitrary small but non-zero $v_{c}$ , making the limit of zero $v_{c}$ highly non-trivial. That limit is addressed by crossing a branch cut of a square root into the second and subsequently higher sheets of the Riemann surface to find coupled square root singularities at distances $\pm v_{c}$ from the real axis at each sheet. The number of sheets is infinite and the analytical continuation of the Stokes wave into all of these sheets is found together with the series expansion in half-integer powers at singular points within each sheet. It is conjectured that a non-limiting Stokes wave at the leading order consists of an infinite number of nested square root singularities which also implies the existence in the third and higher sheets of additional square root singularities away from the real and imaginary axes. These nested square roots form a $2/3$ power-law singularity of the limiting Stokes wave as $v_{c}$ vanishes.