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)}
