开尔文船波

我不是女神ヾ 2022-07-13 01:27 142阅读 0赞

# \[[编辑][Link 1]\] #

维基百科，自由的百科全书

[![400px-Fjordn_surface_wave_boat.jpg][]][400px-Fjordn_surface_wave_boat.jpg 1]

开尔文船波

[![Wake.avon.gorge.2boats.arp.750pix.jpg][]][Wake.avon.gorge.2boats.arp.750pix.jpg 1]

**开尔文船波**（Kelvin Wake, Kelvin ship wave）。鸭子或船只在深水湖面经过时，在后面会激起一道V形的波。[开尔文男爵][Link 2]最先对船波进行数学研究，因此称为开尔文船波。船波动形状和[福禄数][Link 3]\{\\displaystyle Fr\}![\{\\displaystyle Fr\}][displaystyle Fr]有密切关系。

\{\\displaystyle Fr=\{\\frac \{V\}\{\\sqrt \{gl\}\}\}\}![\{\\displaystyle Fr=\{\\frac \{V\}\{\\sqrt \{gl\}\}\}\}][displaystyle Fr_frac _V_sqrt _gl]

其中g为重力常数，V是船速，l是船的长度。

令船的长度\{\\displaystyle l=k\*\{\\frac \{V^\{2\}\}\{g\}\}\}![\{\\displaystyle l=k\*\{\\frac \{V^\{2\}\}\{g\}\}\}][displaystyle l_k_frac _V_2_g] 则\{\\displaystyle Fr=\{\\frac \{1\}\{\\sqrt \{k\}\}\}\}![\{\\displaystyle Fr=\{\\frac \{1\}\{\\sqrt \{k\}\}\}\}][displaystyle Fr_frac _1_sqrt _k].

对于长度大而速度低的轮船，Fr数小，开尔文船波主要是长波，其波前与速度矢量的夹角比较小。

而小快艇，长度小，速度高，Fr 数大，开尔文船波则以短波长的水波为主，而波前则与速度数量成较大的夹角。[\[1\]][1]

开尔文船波动研究，对于船舶的设计有重要意义，因为船舶的马力，有一部分消耗在激起船波。利用Fr数与速度成正比，与长度的平方根成反比的规律，可以利用小的模型，缩小船长\{\\displaystyle M^\{2\}\}![M^\{2\}][M_2]倍，同时缩小速度M倍，可以在实验室中模拟海上巨舟。[\[2\]][2]

## 目录 ##

\[隐藏\]

*  [1多鞍点函数积分][1 1]
 *  [2开尔文驻相法][2 1]
 *  [3外部链接][3]
 *  [4脚注][4]
 *  [5参考文献][5]

## 多鞍点函数积分\[[编辑][Link 4]\] ##

[![220px-Integrand_of_Kelvin_Wake_Integral.png][]][220px-Integrand_of_Kelvin_Wake_Integral.png 1]

Integrand of Kelvin Wake Integral

[![220px-Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png][]][220px-Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png 1]

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 \}![\{\\displaystyle \\rho ,\\phi \}][displaystyle _rho _phi]中在船只的速度矢量方向，\{\\displaystyle \\phi =0\}![\\phi=0][phi_0]），由下列公式表示[\[3\]][3 1]

\{\\displaystyle K(\\phi ,\\rho )=\\int \_\{-\\pi /2\}^\{\\pi /2\}cos(\\rho \{\\frac \{cos(\\theta +\\phi )\}\{cos^\{2\}\\theta \}\}d\\theta \}![\{\\displaystyle K(\\phi ,\\rho )=\\int \_\{-\\pi /2\}^\{\\pi /2\}cos(\\rho \{\\frac \{cos(\\theta +\\phi )\}\{cos^\{2\}\\theta \}\}d\\theta \}][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 \\rho =gr/V^\{2\}\}][displaystyle _rho _gr_V_2]

\{\\displaystyle \{\\frac \{1\}\{\\rho \}\}=\{\\frac \{V^\{2\}\}\{gr\}\}\}![\{\\displaystyle \{\\frac \{1\}\{\\rho \}\}=\{\\frac \{V^\{2\}\}\{gr\}\}\}][displaystyle _frac _1_rho _frac _V_2_gr]是[福禄数][Link 3]的平方\{\\displaystyle Fr^\{2\}\}![\{\\displaystyle Fr^\{2\}\}][displaystyle Fr_2]

\{\\displaystyle g\}![g][]为重力常数\{\\displaystyle l\}![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 )\}![\{\\displaystyle K(\\phi ,\\rho )=Re(\\int \_\{-\\infty \}^\{\\infty \}\\exp(i\*\\rho \*f(\\theta ,\\rho )d\\theta )\}][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 f(\\theta ,\\phi )=-\{\\frac \{cos(\\theta +\\phi )\}\{cos^\{2\}\\theta \}\}\}][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\}![\{\\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 _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 )\}\})\}![\{\\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 _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\}![\{\\displaystyle \\phi \_\{1\}=19.47\}][displaystyle _phi _1_19.47]度,

\{\\displaystyle \\phi \_\{2\}=-19.47\}![\{\\displaystyle \\phi \_\{2\}=-19.47\}][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.[\[4\]][4 1][\[5\]][5 1]至于波纹本身则与船速矢量的夹角为

\{\\displaystyle \\theta =\\pi -19.47=35.3\}![\{\\displaystyle \\theta =\\pi -19.47=35.3\}][displaystyle _theta _pi -19.47_35.3]°[\[1\]][1]

## 开尔文驻相法\[[编辑][Link 5]\] ##

[![800px-Maple_density_plot_of_Kelvin_Wake.png][]][800px-Maple_density_plot_of_Kelvin_Wake.png 1]

Kelvin Wake (Maple density plot)

[![600px-Kelvin_Ship_wave_plot.png][]][600px-Kelvin_Ship_wave_plot.png 1]

开尔文船波波形

开尔文船波积分\{\\displaystyle K(\\phi ,\\rho )\}![\{\\displaystyle K(\\phi ,\\rho )\}][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][Link 6]的推广。[\[6\]][6]

被积分函数 \{\\displaystyle f(\\theta ,\\phi )=-\{\\frac \{cos(\\theta +\\phi )\}\{cos^\{2\}\\theta \}\}\}![\{\\displaystyle f(\\theta ,\\phi )=-\{\\frac \{cos(\\theta +\\phi )\}\{cos^\{2\}\\theta \}\}\}][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 \_\{p\}=arctan(\{\\frac \{(1/4)\*(1+\{\\sqrt \{(1-8\*tan(\\phi )^\{2\}))\}\}\}\{tan(\\phi )\}\})\}][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 \\theta \_\{m\}=-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\_\{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_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 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 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 fbar:=1/2\*(f\_\{p\}+f\_\{m\})\}][displaystyle fbar_1_2_f_p_f_m]

\{\\displaystyle D2F=\{\\frac \{d^\{2\}F(\\theta ,\\phi )\}\{d\\theta ^\{2\}\}\}\}![\{\\displaystyle D2F=\{\\frac \{d^\{2\}F(\\theta ,\\phi )\}\{d\\theta ^\{2\}\}\}\}][displaystyle D2F_frac _d_2_F_theta _phi _d_theta _2]

\{\\displaystyle D2F\_\{p\}=D2F(\\theta \_\{p\},\\phi )\}![\{\\displaystyle D2F\_\{p\}=D2F(\\theta \_\{p\},\\phi )\}][displaystyle D2F_p_D2F_theta _p_phi]

\{\\displaystyle D2F\_\{m\}=D2F(\\theta \_\{m\},\\phi )\}![\{\\displaystyle D2F\_\{m\}=D2F(\\theta \_\{m\},\\phi )\}][displaystyle D2F_m_D2F_theta _m_phi]

\{\\displaystyle \\Delta :=(3/4\*(f\_\{m\}-f\_\{p\}))^\{(\}2/3)\}![\{\\displaystyle \\Delta :=(3/4\*(f\_\{m\}-f\_\{p\}))^\{(\}2/3)\}][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 u=\{\\sqrt \{\\frac \{\\Delta ^\{1/2\}\}\{2\}\}\}\*(\{\\frac \{1\}\{\\sqrt \{D2F\_\{p\}\}\}\}+\{\\frac \{1\}\{\\sqrt \{-D2F\_\{m\}\}\}\})\}][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 v=\{\\sqrt \{\\frac \{2\}\{\\Delta ^\{1/2\}\}\}\}\*(\{\\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 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 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[\[7\]][7]

\{\\displaystyle x:=X\*sin(\\beta )\*(1-(1/2)\*sin(\\beta )^\{2\})\}![\{\\displaystyle x:=X\*sin(\\beta )\*(1-(1/2)\*sin(\\beta )^\{2\})\}][displaystyle x_X_sin_beta _1-_1_2_sin_beta _2]

\{\\displaystyle y:=X\*sin(\\beta )^\{2\}\*cos(\\beta )/(2\*M)\}![\{\\displaystyle y:=X\*sin(\\beta )^\{2\}\*cos(\\beta )/(2\*M)\}][displaystyle y_X_sin_beta _2_cos_beta _2_M]

## 外部链接\[[编辑][Link 7]\] ##

*  [§36.13 Kelvin’s Ship-Wave Pattern][36.13 Kelvin_s Ship-Wave Pattern]

## 脚注\[[编辑][Link 8]\] ##

1.  ^ [**1.0**][1.0] [**1.1**][1.1] James LightHill, p274
2.  **[^][Link 9]** James Lighthill p275
3.  **[^][Link 10]** Frank Oliver, p790-791
4.  **[^][Link 11]** 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][doi_10.1023_B_INAM.0000041400.70961.1b].
5.  **[^][Link 12]** 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][doi_10.3723_175605406782725023].
6.  **[^][Link 13]** Frank Oliver, p790-795
7.  **[^][Link 14]** James LightHill,p277

## 参考文献\[[编辑][Link 15]\] ##

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

[分类][Link 16]：

*  [特殊函数][Link 17]
 *  [流体力学][Link 18]

[Link 1]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=0&summary=/*%20%E9%A6%96%E6%AE%B5%20*/%20
[400px-Fjordn_surface_wave_boat.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Fjordn_surface_wave_boat.jpg/400px-Fjordn_surface_wave_boat.jpg
[400px-Fjordn_surface_wave_boat.jpg 1]: https://zh.wikipedia.org/wiki/File:Fjordn_surface_wave_boat.jpg
[Wake.avon.gorge.2boats.arp.750pix.jpg]: https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Wake.avon.gorge.2boats.arp.750pix.jpg/400px-Wake.avon.gorge.2boats.arp.750pix.jpg
[Wake.avon.gorge.2boats.arp.750pix.jpg 1]: https://zh.wikipedia.org/wiki/File:Wake.avon.gorge.2boats.arp.750pix.jpg
[Link 2]: https://zh.wikipedia.org/wiki/%E9%96%8B%E7%88%BE%E6%96%87%E7%94%B7%E7%88%B5
[Link 3]: https://zh.wikipedia.org/wiki/%E7%A6%8F%E7%A5%BF%E6%95%B8
[displaystyle Fr]: https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee418eba0739c49e861e127d14d63c54d1dab50
[displaystyle Fr_frac _V_sqrt _gl]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f59e8764b5a9ecb9a9afe68715167a0479a679e2
[displaystyle l_k_frac _V_2_g]: https://wikimedia.org/api/rest_v1/media/math/render/svg/a47725252ad9604bdf24003e848d4f0dfd503d48
[displaystyle Fr_frac _1_sqrt _k]: https://wikimedia.org/api/rest_v1/media/math/render/svg/379ff428a3125dde764722e353b7ed4d52beb6ae
[1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-L274-1
[M_2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9c3e3aa1aec0e135b5d342181d9ee7f9814f87
[2]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-L275-2
[1 1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#.E5.A4.9A.E9.9E.8D.E7.82.B9.E5.87.BD.E6.95.B0.E7.A7.AF.E5.88.86
[2 1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#.E5.BC.80.E5.B0.94.E6.96.87.E9.A9.BB.E7.9B.B8.E6.B3.95
[3]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#.E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5
[4]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#.E8.85.B3.E8.A8.BB
[5]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#.E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE
[Link 4]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=1
[220px-Integrand_of_Kelvin_Wake_Integral.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Integrand_of_Kelvin_Wake_Integral.png/220px-Integrand_of_Kelvin_Wake_Integral.png
[220px-Integrand_of_Kelvin_Wake_Integral.png 1]: https://zh.wikipedia.org/wiki/File:Integrand_of_Kelvin_Wake_Integral.png
[220px-Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png/220px-Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png
[220px-Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png 1]: https://zh.wikipedia.org/wiki/File:Kelvin_Ship_Wake_Integrand_contour_Maple_plot.png
[displaystyle _rho _phi]: https://wikimedia.org/api/rest_v1/media/math/render/svg/a8256f2207e3dc22e57156a5a8a60c7d8899ac55
[phi_0]: https://wikimedia.org/api/rest_v1/media/math/render/svg/2c96cdab103e6c884877c86d6e5db6e471a167d5
[3 1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-F790-1-3
[displaystyle K_phi _rho _int _-_pi _2_pi _2_cos_rho _frac _cos_theta _phi _cos_2_theta _d_theta]: https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0d743290fc0d0a4af905444e263ed5aeb2f5dd
[displaystyle _rho _gr_V_2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d16fef975b514e273d4d4ea20b08ad0595e7f
[displaystyle _frac _1_rho _frac _V_2_gr]: https://wikimedia.org/api/rest_v1/media/math/render/svg/85d0e782f76774a445baaa6627a45f03aa9d327f
[displaystyle Fr_2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f19ca5183cda0fa6d57963e8c57e86886c09cc32
[g]: https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77
[l]: https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac
[displaystyle K_phi _rho _Re_int _-_infty _infty _exp_i_rho _f_theta _rho _d_theta]: https://wikimedia.org/api/rest_v1/media/math/render/svg/1e65e7ee08127ca4cc78d80fd3cc5b43cfec41e3
[displaystyle f_theta _phi _-_frac _cos_theta _phi _cos_2_theta]: https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6643337e5b43be921b1b36117d28e9abc35e2a
[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]: https://wikimedia.org/api/rest_v1/media/math/render/svg/c62d8b5ffee8420c73782d81b1b5c75c18fe94b9
[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]: https://wikimedia.org/api/rest_v1/media/math/render/svg/731a35d12901bd78b37d7079669654b6219f6294
[displaystyle _phi _1_19.47]: https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb66195650b1257d62125ce3774ac82d55e77b1
[displaystyle _phi _2_-19.47]: https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6795a626ab063a2b73b8ddfe203063b26a8355
[4 1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-4
[5 1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-5
[displaystyle _theta _pi -19.47_35.3]: https://wikimedia.org/api/rest_v1/media/math/render/svg/526ab6c347486197029facb18536ef553cbfa481
[Link 5]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=2
[800px-Maple_density_plot_of_Kelvin_Wake.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/0/06/Maple_density_plot_of_Kelvin_Wake.png/800px-Maple_density_plot_of_Kelvin_Wake.png
[800px-Maple_density_plot_of_Kelvin_Wake.png 1]: https://zh.wikipedia.org/wiki/File:Maple_density_plot_of_Kelvin_Wake.png
[600px-Kelvin_Ship_wave_plot.png]: https://upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Kelvin_Ship_wave_plot.png/600px-Kelvin_Ship_wave_plot.png
[600px-Kelvin_Ship_wave_plot.png 1]: https://zh.wikipedia.org/wiki/File:Kelvin_Ship_wave_plot.png
[displaystyle K_phi _rho]: https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa60a6f038adb24952b1fffd0f790a1f1ea9683
[Link 6]: https://zh.wikipedia.org/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E6%96%B9%E6%B3%95
[6]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-F790-5-6
[displaystyle _theta _p_arctan_frac _1_4_1_sqrt _1-8_tan_phi _2_tan_phi]: https://wikimedia.org/api/rest_v1/media/math/render/svg/d053f4da80b65ca44c855d3efd33487138800290
[displaystyle _theta _m_-arctan_frac _1_4_-1_sqrt _1-8_tan_phi _2_tan_phi]: https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c444d9a9ca3785c5e772414968b94f6031e356
[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]: https://wikimedia.org/api/rest_v1/media/math/render/svg/d2bf2fa3e819cc4a596850a0d5aeebd6dbf1525f
[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]: https://wikimedia.org/api/rest_v1/media/math/render/svg/c8835739b8e0d6e557feea2ac0e1758e9c2d8240
[displaystyle fbar_1_2_f_p_f_m]: https://wikimedia.org/api/rest_v1/media/math/render/svg/12942f0ef302d5e90ccc4019f9a59c7b2386a688
[displaystyle D2F_frac _d_2_F_theta _phi _d_theta _2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/457ee941608a34dbdee675f755d953d3b0d24db6
[displaystyle D2F_p_D2F_theta _p_phi]: https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb0f0ed999179542892b929392342cdb249a306
[displaystyle D2F_m_D2F_theta _m_phi]: https://wikimedia.org/api/rest_v1/media/math/render/svg/3e02b20e6c29d50ecea82debb0696719d35615a6
[displaystyle _Delta _3_4_f_m_-f_p_2_3]: https://wikimedia.org/api/rest_v1/media/math/render/svg/c59de0e223dc9f0a8b813d36206e99fc70453da1
[displaystyle u_sqrt _frac _Delta _1_2_2_frac _1_sqrt _D2F_p_frac _1_sqrt _-D2F_m]: https://wikimedia.org/api/rest_v1/media/math/render/svg/a82635ace43fa52c4bfcc428797899c2342554ab
[displaystyle v_sqrt _frac _2_Delta _1_2_frac _1_sqrt _D2F_p_-_frac _1_sqrt _-D2F_m]: https://wikimedia.org/api/rest_v1/media/math/render/svg/f54193e14f3493bd4ce74b89ec1b84e314cf8000
[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]: https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9cd66b269d9b8973353e149c802ce2b00ed425
[7]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_note-L277-7
[displaystyle x_X_sin_beta _1-_1_2_sin_beta _2]: https://wikimedia.org/api/rest_v1/media/math/render/svg/bea3025b36926924c56c1c0eb010a9772baa7530
[displaystyle y_X_sin_beta _2_cos_beta _2_M]: https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f32f87f8f75a4c8ec82e65893f2713844c9c5f
[Link 7]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=3
[36.13 Kelvin_s Ship-Wave Pattern]: http://dlmf.nist.gov/36.13
[Link 8]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=4
[1.0]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-L274_1-0
[1.1]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-L274_1-1
[Link 9]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-L275_2-0
[Link 10]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-F790-1_3-0
[Link 11]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-4
[doi_10.1023_B_INAM.0000041400.70961.1b]: https://dx.doi.org/10.1023%2FB%3AINAM.0000041400.70961.1b
[Link 12]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-5
[doi_10.3723_175605406782725023]: https://dx.doi.org/10.3723%2F175605406782725023
[Link 13]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-F790-5_6-0
[Link 14]: https://zh.wikipedia.org/wiki/%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2#cite_ref-L277_7-0
[Link 15]: https://zh.wikipedia.org/w/index.php?title=%E5%BC%80%E5%B0%94%E6%96%87%E8%88%B9%E6%B3%A2&action=edit&section=5
[Link 16]: https://zh.wikipedia.org/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB
[Link 17]: https://zh.wikipedia.org/wiki/Category:%E7%89%B9%E6%AE%8A%E5%87%BD%E6%95%B0
[Link 18]: https://zh.wikipedia.org/wiki/Category:%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6