开尔文船波

${\displaystyle Fr={\frac {V}{\sqrt {gl}}}}$

多鞍点函数积分

Integrand of Kelvin Wake Integral

Kelvin Ship Wake Integrand contour Maple plot

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

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

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

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

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

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

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

${\displaystyle \phi _{1}=19.47}$ 度,

${\displaystyle \phi _{2}=-19.47}$

${\displaystyle \theta =\pi -19.47=35.3}$ °[1]

开尔文驻相法

Kelvin Wake (Maple density plot)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

腳註

1. James LightHill, p274
2. ^ James Lighthill p275
3. ^ Frank Oliver, p790-791
4. ^ Shu, Jian-Jun. Transient Marangoni waves due to impulsive motion of a submerged body. International Applied Mechanics. 2004, 40 (6): 709–714. doi:10.1023/B:INAM.0000041400.70961.1b.
5. ^ Shu, Jian-Jun. Transient free-surface waves due to impulsive motion of a submerged source. Underwater Technology. 2006, 26 (4): 133–137. doi:10.3723/175605406782725023.
6. ^ Frank Oliver, p790-795
7. ^ James LightHill,p277

参考文献

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