Environmental Engineering Reference
In-Depth Information
Remark 1.
Consider the Laplace equation in a spherical coordinate system
Δ
u
(
r
,
θ
,
ϕ
)=
0
.
Assuming that
u
=
R
(
r
)
Y
(
θ
,
ϕ
)
, by separation of variables, we have an equation of
spherical functions.
sin
2
Y
∂ϕ
1
sin
∂
∂θ
θ
∂
Y
∂θ
1
sin
2
∂
+
2
+
n
(
n
+
1
)
Y
=
0
.
θ
θ
Assume that
Y
; by another separation of variables, we arrive
at the associated Legendre equation regarding
(
θ
,
ϕ
)=
Θ
(
θ
)
Φ
(
ϕ
)
. Its solution is
P
n
(
Θ
(
θ
)
cos
θ
)
.The
Φ
(
ϕ
)
-equation
d
2
Φ
m
2
2
+
Φ
=
0
d
ϕ
e
im
ϕ
. Therefore, the equation of spherical functions has the
has the solution
Φ
(
ϕ
)=
solution
P
n
(
e
im
ϕ
. It is called the
spherical harmonic function
and is indepen-
dent of radius
r
. Its real and imaginary parts are
P
n
(
cos
θ
)
cos
θ
)
cos
m
ϕ
,
n
=
0
,
1
,
2
, ··· ,
m
=
0
,
1
,
2
, ··· ,
m
≤
n
,
P
n
(
cos
θ
)
sin
m
ϕ
,
n
=
0
,
1
,
2
, ··· ,
m
=
0
,
1
,
2
, ··· ,
m
≤
n
.
They are called the
spherical functions of order n
.
Remark 2.
Special functions can also be introduced by an expansion of generating
functions. The generating function for a Bessel function is
t
−
1
t
x
2
+
∞
∑
n
=
−
∞
t
n
e
=
J
n
(
x
)
.
ie
iθ
and
x
Let
t
=
=
kr
, thus we obtain
2
+
∞
e
i
kr
cos θ
=
n
=
1
i
n
J
n
(
kr
)
cos
n
θ
,
J
0
(
kr
)+
(A.32)
in which every Bessel function represents an amplitude factor of cylindrical waves
(Section 2.5.2 and Section 2.8).
Consider the solution of the one-dimensional wave equation in the form
e
−
iω
t
u
(
x
,
t
)=
v
(
x
)
.
d
2
v
d
x
2
+
k
2
v
The
v
(
x
)
satisfies
=
0, where
k
=
ω
/
a
is called the
phase constant
.Its
solution can be expressed by
v
0
e
i
kx
v
0
e
i
kr
cos θ
,
v
(
x
)=
=
(A.33)
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