尾波 （英语 ：wake ）是固体 在划过流体 （特别是液体 ）表面时在尾部产生的V形传播的波 ，例如水鸟 或船舶 匀速游过水体 时在水面激起的后方波纹。因为由英国 的开尔文男爵 ——物理学家 威廉·汤姆森（William Thomson，1824～1907）最先对船波进行数学研究，因此也称为开尔文船波 （Kelvin wake或Kelvin ship wave ）。
船只尾波的鸟瞰图 从前方观测的船只尾波
船形物体的尾波形状和福禄数
F
r
{\displaystyle Fr}
F
r
=
V
g
l
{\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
l
=
k
⋅
V
2
g
{\displaystyle l=k\cdot {\frac {V^{2}}{g}}}
F
r
=
1
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，而波前则与速度矢量成较大的夹角。[ 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
M
2
{\displaystyle M^{2}}
[ 2]
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（
ρ
,
ϕ
{\displaystyle \rho ,\phi }
ϕ
=
0
{\displaystyle \phi =0}
[ 3]
K
(
ϕ
,
ρ
)
=
∫
−
π
/
2
π
/
2
cos
ρ
cos
(
θ
+
ϕ
)
cos
2
θ
d
θ
{\displaystyle K(\phi ,\rho )=\int _{-\pi /2}^{\pi /2}\cos \rho {\frac {\cos(\theta +\phi )}{\cos ^{2}\theta }}d\theta }
其中
ρ
=
g
r
/
V
2
{\displaystyle \rho =gr/V^{2}}
1
ρ
=
V
2
g
r
{\displaystyle {\frac {1}{\rho }}={\frac {V^{2}}{gr}}}
福禄数 的平方
F
r
2
{\displaystyle Fr^{2}}
g
{\displaystyle g}
l
{\displaystyle l}
上列K函数是下列多鞍点积分的正数部分：
K
(
ϕ
,
ρ
)
=
ℜ
(
∫
−
∞
∞
exp
(
i
ρ
f
(
θ
,
ρ
)
d
θ
)
{\displaystyle K(\phi ,\rho )=\Re (\int _{-\infty }^{\infty }\exp(i\rho f(\theta ,\rho )d\theta )}
f
(
θ
,
ϕ
)
=
−
cos
(
θ
+
ϕ
)
cos
2
θ
{\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，
d
f
(
θ
,
ϕ
)
d
θ
=
sin
(
θ
+
ϕ
)
cos
(
θ
)
2
−
2
cos
(
θ
+
ϕ
)
sin
(
θ
)
cos
(
θ
)
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
θ
1
=
arctan
(
(
1
/
4
)
(
1
+
(
1
−
8
tan
(
ϕ
)
2
)
)
tan
(
ϕ
)
)
=
−
arctan
(
(
1
/
4
)
(
−
1
+
(
1
−
8
tan
(
ϕ
)
2
)
)
tan
(
ϕ
)
)
{\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
ϕ
1
=
19.47
{\displaystyle \phi _{1}=19.47}
ϕ
2
=
−
19.47
{\displaystyle \phi _{2}=-19.47}
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] [ 5]
θ
=
π
−
19.47
=
35.3
{\displaystyle \theta =\pi -19.47=35.3}
[ 1]
Kelvin Wake (Maple density plot) 开尔文船波波形 开尔文船波积分
K
(
ϕ
,
ρ
)
{\displaystyle K(\phi ,\rho )}
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。[ 6]
被积分函数
f
(
θ
,
ϕ
)
=
−
c
o
s
(
θ
+
ϕ
)
c
o
s
2
θ
{\displaystyle f(\theta ,\phi )=-{\frac {cos(\theta +\phi )}{cos^{2}\theta }}}
θ
p
=
a
r
c
t
a
n
(
(
1
/
4
)
∗
(
1
+
(
1
−
8
∗
t
a
n
(
ϕ
)
2
)
)
t
a
n
(
ϕ
)
)
{\displaystyle \theta _{p}=arctan({\frac {(1/4)*(1+{\sqrt {(1-8*tan(\phi )^{2}))}}}{tan(\phi )}})}
θ
m
=
−
a
r
c
t
a
n
(
(
1
/
4
)
∗
(
−
1
+
(
1
−
8
∗
t
a
n
(
ϕ
)
2
)
)
t
a
n
(
ϕ
)
)
{\displaystyle \theta _{m}=-arctan({\frac {(1/4)*(-1+{\sqrt {(}}1-8*tan(\phi )^{2}))}{tan(\phi )}})}
令
f
m
=
f
(
θ
m
,
ϕ
)
=
s
i
n
(
(
1
/
2
)
∗
ϕ
−
(
1
/
2
)
∗
a
r
c
s
i
n
(
3
∗
s
i
n
(
ϕ
)
)
)
s
i
n
(
(
1
/
2
)
∗
ϕ
+
(
1
/
2
)
∗
a
r
c
s
i
n
(
3
∗
s
i
n
(
ϕ
)
)
)
{\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 )))}}}
f
p
=
f
(
θ
p
,
ϕ
)
=
c
o
s
(
(
1
/
2
)
∗
ϕ
+
(
1
/
2
)
∗
a
r
c
s
i
n
(
3
∗
s
i
n
(
ϕ
)
)
)
c
o
s
(
−
(
1
/
2
)
∗
ϕ
+
(
1
/
2
)
∗
a
r
c
s
i
n
(
3
∗
s
i
n
(
ϕ
)
)
)
{\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 )))}}}
f
b
a
r
:=
1
/
2
∗
(
f
p
+
f
m
)
{\displaystyle fbar:=1/2*(f_{p}+f_{m})}
D
2
F
=
d
2
F
(
θ
,
ϕ
)
d
θ
2
{\displaystyle D2F={\frac {d^{2}F(\theta ,\phi )}{d\theta ^{2}}}}
D
2
F
p
=
D
2
F
(
θ
p
,
ϕ
)
{\displaystyle D2F_{p}=D2F(\theta _{p},\phi )}
D
2
F
m
=
D
2
F
(
θ
m
,
ϕ
)
{\displaystyle D2F_{m}=D2F(\theta _{m},\phi )}
Δ
:=
(
3
/
4
∗
(
f
m
−
f
p
)
)
(
2
/
3
)
{\displaystyle \Delta :=(3/4*(f_{m}-f_{p}))^{(}2/3)}
u
=
Δ
1
/
2
2
∗
(
1
D
2
F
p
+
1
−
D
2
F
m
)
{\displaystyle u={\sqrt {\frac {\Delta ^{1/2}}{2}}}*({\frac {1}{\sqrt {D2F_{p}}}}+{\frac {1}{\sqrt {-D2F_{m}}}})}
v
=
2
Δ
1
/
2
∗
(
1
D
2
F
p
−
1
−
D
2
F
m
)
{\displaystyle v={\sqrt {\frac {2}{\Delta ^{1/2}}}}*({\frac {1}{\sqrt {D2F_{p}}}}-{\frac {1}{\sqrt {-D2F_{m}}}})}
K
(
ϕ
,
ρ
)
≈
2
∗
π
∗
(
u
∗
c
o
s
(
ρ
∗
f
b
a
r
)
∗
A
i
r
y
A
i
(
−
ρ
(
2
/
3
)
∗
Δ
)
/
ρ
(
1
/
3
)
+
v
∗
s
i
n
(
ρ
∗
f
b
a
r
)
∗
A
i
r
y
A
i
(
1
,
−
ρ
(
2
/
3
)
∗
Δ
)
/
ρ
(
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))}
[ 7]
x
:=
X
∗
s
i
n
(
β
)
∗
(
1
−
(
1
/
2
)
∗
s
i
n
(
β
)
2
)
{\displaystyle x:=X*sin(\beta )*(1-(1/2)*sin(\beta )^{2})}
y
:=
X
∗
s
i
n
(
β
)
2
∗
c
o
s
(
β
)
/
(
2
∗
M
)
{\displaystyle y:=X*sin(\beta )^{2}*cos(\beta )/(2*M)}
Frank J. Oliver, NIST Handbook of Mathematical Functions, 2010, Cambridge University Press
Jame Lighthill Waves in Fluids, Cambridge University Press 1979 
