开尔文船波

Integrand of Kelvin Wake Integral

Kelvin Ship Wake Integrand contour Maple plot

当船只以速度V驶过深水湖面，波形的幅度在相对于船只为静止的极坐标（${\displaystyle \rho ,\phi }$中在船只的速度矢量方向，${\displaystyle \phi =0}$），由下列公式表示

${\displaystyle K(\phi ,\rho )=\int _{-\pi /2}^{\pi /2}cos(\rho {\frac {cos(\theta +\phi )}{cos^{2}\theta }}d\theta }$

其中${\displaystyle \rho =gr/V^{2}}$

${\displaystyle {\frac {1}{\rho }}={\frac {V^{2}}{gr}}}$是福祿數的平方${\displaystyle Fr^{2}}$

${\displaystyle g}$为重力常数${\displaystyle l}$为船的长度。

上列K函数是下列多鞍点积分的正数部分：

${\displaystyle K(\phi ,\rho )=Re(\int _{-\infty }^{\infty }\exp(i*\rho *f(\theta ,\rho )d\theta )}$ 其中，多鞍点积分的核函数为

${\displaystyle f(\theta ,\phi )=-{\frac {cos(\theta +\phi )}{cos^{2}\theta }}}$

此核函数是一个多鞍点函数，振荡剧烈如图

求其极点，

${\displaystyle {\frac {df(\theta ,\phi )}{d\theta }}={\frac {sin(\theta +\phi )}{cos(\theta )^{2}}}-{\frac {2*cos(\theta +\phi )*sin(\theta )}{cos(\theta )^{3}}}=0}$

解之，得

${\displaystyle \theta _{1}=arctan({\frac {(1/4)*(1+{\sqrt {(1-8*tan(\phi )^{2}))}}}{tan(\phi )}})=-arctan({\frac {(1/4)*(-1+{\sqrt {(}}1-8*tan(\phi )^{2}))}{tan(\phi )}})}$

由此

${\displaystyle \phi _{1}=19.47}$度,

${\displaystyle \phi _{2}=-19.47}$度

这就是凯尔文船波的V型波包线的夹角，最早由凯尔文男爵发现，而且角度与船速无关.至于波纹本身则与船速矢量的夹角为

${\displaystyle \theta =\pi -19.47=35.3}$°

Kelvin Wake (Maple density plot)

开尔文船波波形

开尔文船波积分${\displaystyle K(\phi ,\rho )}$必须通过数值积分计算。开尔文男爵根据被积分函数在积分区间内剧烈震荡的特点，提出了驻相法（Method of Stationary Phase）。

原理：当被积分函数剧烈震荡时，除了在极点外，震荡的被积分函数正负相抵消，因此可以将此被积分函数在极点的值作为整个积分的近似，驻相法乃是拉普拉斯方法的推广。

被积分函数 ${\displaystyle f(\theta ,\phi )=-{\frac {cos(\theta +\phi )}{cos^{2}\theta }}}$ 的两个极点是：

${\displaystyle \theta _{p}=arctan({\frac {(1/4)*(1+{\sqrt {(1-8*tan(\phi )^{2}))}}}{tan(\phi )}})}$

${\displaystyle \theta _{m}=-arctan({\frac {(1/4)*(-1+{\sqrt {(}}1-8*tan(\phi )^{2}))}{tan(\phi )}})}$

令

${\displaystyle f_{m}=f(\theta _{m},\phi )={\frac {sin((1/2)*\phi -(1/2)*arcsin(3*sin(\phi )))}{sin((1/2)*\phi +(1/2)*arcsin(3*sin(\phi )))}}}$

${\displaystyle f_{p}=f(\theta _{p},\phi )={\frac {cos((1/2)*\phi +(1/2)*arcsin(3*sin(\phi )))}{cos(-(1/2)*\phi +(1/2)*arcsin(3*sin(\phi )))}}}$

${\displaystyle fbar:=1/2*(f_{p}+f_{m})}$

${\displaystyle D2F={\frac {d^{2}F(\theta ,\phi )}{d\theta ^{2}}}}$

${\displaystyle D2F_{p}=D2F(\theta _{p},\phi )}$

${\displaystyle D2F_{m}=D2F(\theta _{m},\phi )}$

${\displaystyle \Delta$ :=(3/4*(f_{m}-f_{p}))^{(}2/3)}

${\displaystyle u={\sqrt {\frac {\Delta ^{1/2}}{2}}}*({\frac {1}{\sqrt {D2F_{p}}}}+{\frac {1}{\sqrt {-D2F_{m}}}})}$

${\displaystyle v={\sqrt {\frac {2}{\Delta ^{1/2}}}}*({\frac {1}{\sqrt {D2F_{p}}}}-{\frac {1}{\sqrt {-D2F_{m}}}})}$

${\displaystyle K(\phi ,\rho )\approx 2*\pi *(u*cos(\rho *fbar)*AiryAi(-\rho ^{(}2/3)*\Delta )/\rho ^{(}1/3)+v*sin(\rho *fbar)*AiryAi(1,-\rho ^{(}2/3)*\Delta )/\rho ^{(}2/3))}$

开尔文船波的波峰，由下列两个参数方程式描述

${\displaystyle x:=X*sin(\beta )*(1-(1/2)*sin(\beta )^{2})}$

${\displaystyle y:=X*sin(\beta )^{2}*cos(\beta )/(2*M)}$

• §36.13 Kelvin’s Ship-Wave Pattern

• Frank J. Oliver, NIST Handbook of Mathematical Functions, 2010, Cambridge University Press
• Jame Lighthill Waves in Fluids, Cambridge University Press 1979