aus Wikisource, der freien Quellensammlung
14
oder
φ
=
−
ω
n
+
1
ϱ
sin
θ
∂
χ
n
∂
θ
,
{\displaystyle \varphi =-{\frac {\omega }{n+1}}\varrho \sin \theta {\frac {\partial \chi _{n}}{\partial \theta }},\,}
φ
¯
=
−
ω
R
n
+
1
sin
θ
∂
Y
n
∂
θ
{\displaystyle {\bar {\varphi }}=-{\frac {\omega R}{n+1}}\sin \theta {\frac {\partial Y_{n}}{\partial \theta }}\,}
n
{\displaystyle n\,}
Ist die inducirende Potentialfunktion
χ
n
=
(
R
ϱ
)
n
+
1
Y
n
,
{\displaystyle \chi _{n}=\left({\frac {R}{\varrho }}\right)^{n+1}Y_{n},\,}
Ψ
i
=
−
4
π
R
2
(
2
n
+
1
)
n
ω
k
(
ϱ
R
)
n
Y
n
′
Ψ
a
=
−
4
π
R
2
(
2
n
+
1
)
n
ω
k
(
R
ϱ
)
n
+
1
Y
n
′
Ω
i
=
4
π
R
(
n
+
1
)
(
2
n
+
1
)
n
ω
k
(
ϱ
R
)
n
Y
n
′
Ω
a
=
−
4
π
R
2
n
+
1
ω
k
(
R
ϱ
)
n
+
1
Y
n
′
ψ
=
−
ω
k
R
n
Y
n
′
u
=
−
1
n
ω
k
∂
Y
n
′
∂
ω
x
v
=
−
1
n
ω
k
∂
Y
n
′
∂
ω
y
w
=
−
1
n
ω
k
∂
Y
n
′
∂
ω
z
φ
¯
=
ω
n
R
sin
θ
∂
Y
n
∂
θ
.
{\displaystyle {\begin{aligned}{\mathit {\Psi _{i}}}&=-{\frac {4\pi R^{2}}{(2n+1)n}}{\frac {\omega }{k}}\left({\frac {\varrho }{R}}\right)^{n}Y_{n}'\\{\mathit {\Psi _{a}}}&=-{\frac {4\pi R^{2}}{(2n+1)n}}{\frac {\omega }{k}}\left({\frac {R}{\varrho }}\right)^{n+1}Y_{n}'\\{\mathit {\Omega _{i}}}&={\frac {4\pi R(n+1)}{(2n+1)n}}{\frac {\omega }{k}}\left({\frac {\varrho }{R}}\right)^{n}Y_{n}'\\{\mathit {\Omega _{a}}}&=-{\frac {4\pi R}{2n+1}}{\frac {\omega }{k}}\left({\frac {R}{\varrho }}\right)^{n+1}Y_{n}'\\\psi &=-{\frac {\omega }{k}}{\frac {R}{n}}Y_{n}'\\u&=-{\frac {1}{n}}{\frac {\omega }{k}}{\frac {\partial Y_{n}'}{\partial \omega _{x}}}\\v&=-{\frac {1}{n}}{\frac {\omega }{k}}{\frac {\partial Y_{n}'}{\partial \omega _{y}}}\\w&=-{\frac {1}{n}}{\frac {\omega }{k}}{\frac {\partial Y_{n}'}{\partial \omega _{z}}}\\{\bar {\varphi }}&={\frac {\omega }{n}}R\sin \theta {\frac {\partial Y_{n}}{\partial \theta }}.\end{aligned}}}
ψ
,
u
,
v
,
w
,
φ
{\displaystyle \psi ,u,v,w,\varphi \,}
n
{\displaystyle n\,}
−
n
−
1
{\displaystyle -n-1\,}
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