Environmental Engineering Reference
In-Depth Information
Tabl e 7 . 1
Laplace Operator
Δ
2
∂
x
2
+
∂
2
∂
y
2
∂
Two-dimensional Cartisian coordinate system
2
2
∂θ
∂
1
r
∂
∂
r
2
∂
1
Polar coordinate system
r
2
+
r
+
∂
2
2
2
2
∂
x
2
+
∂
y
2
+
∂
Three-dimensional Cartisian coordinate system
∂
∂
∂
z
2
r
∂
∂
2
∂θ
2
1
r
∂
∂
r
2
∂
1
2
+
∂
+
Cylindrical coordinate system
r
r
∂
z
2
r
2
sin
r
2
∂
1
∂
∂
r
1
r
2
sin θ
∂
∂θ
∂
∂θ
Spherical coordinate system
+
θ
∂
r
2
∂ϕ
1
r
2
sin
2
∂
+
2
θ
7.4 Fundamental Solution and the Harmonic Function
In this section we discuss fundamental solutions of homogeneous potential equa-
tions and analyze features of harmonic functions.
7.4.1 Fundamental Solution
A two-dimensional harmonic equation in a polar coordinate system reads
2
u
2
u
∂θ
∂
1
r
∂
u
r
2
∂
1
r
2
+
r
+
2
=
0
.
∂
∂
It reduces to a Euler equation when
u
is axis-symmetric (independent of
θ
)
d
2
u
d
r
2
+
1
r
d
u
d
r
=
0
.
Its general solution is thus
c
1
ln
1
u
=
r
+
c
2
,
2
2
.
where
c
1
and
c
2
are constants, and
r
=
(
x
−
x
0
)
+(
y
−
y
0
)
refers to the solution of harmonic equations
that has continuous derivatives of second order in
A
harmonic function
in a domain
Ω
. The particular solution ln
r
is a harmonic function in a two-dimensional plane with a single discontinuity at
the point
Ω
. It is called the
fundamental solution of two-dimensional harmonic
equations
. It can be shown that
u
(
0
,
0
)
ln
r
satisfies the two-dimensional harmonic
=
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