Environmental Engineering Reference
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equation except at the point
(
0
,
0
)
,
2
u
2
u
∂
x
2
+
∂
y
2
=
0
.
∂
∂
Similarly, a spherically-symmetric three-dimensional harmonic equation reads
r
2
d
u
d
r
1
r
2
d
d
r
=
0
.
Its general solution is
c
1
1
=
r
+
,
u
c
2
2
2
2
.
where
c
1
and
c
2
are constants, and
r
=
(
x
−
x
0
)
+(
y
−
y
0
)
+(
z
−
z
0
)
The particular solution
r
is thus a harmonic function in all three-dimensional
space except at the point
. It is called the
fundamental solution of three-
dimensional harmonic equations
. It can also be shown that
u
(
0
,
0
,
0
)
1
r
=
satisfies the three-
dimensional harmonic equation
2
u
2
u
2
u
∂
x
2
+
∂
y
2
+
∂
z
2
=
0
.
∂
∂
∂
Fundamental solutions play a very important role in examining harmonic func-
tions and in seeking solutions of PDS of potential equations. Consider the electric
field generated by an infinite electric wire passing perpendicularly through the point
M
0
(
x
0
,
y
0
,
0
)
on the
Oxy
-plane and with a uniform electric-charge density 2
πε
.Here
ε
is the dielectric constant. Since the field is uniform at all planes perpendicular to
the wire, it is a typical model of plane fields. The
x
-and
y
-components of electric
field intensity at
M
(
x
,
y
,
0
)
are
+
∞
2
−
2
x
−
x
0
x
−
x
0
2
r
3
x
−
x
0
r
2
E
1
=
3
/
2
d
z
=
cos
α
d
α
=
,
2
(
r
2
+
z
2
)
−
∞
+
∞
y
−
y
0
y
−
y
0
E
2
=
2
√
r
2
z
2
d
z
=
,
r
2
+
−
∞
2
2
. Thus the electric field intensity can
where
z
=
r
tan
α
,
r
=
(
x
−
x
0
)
+(
y
−
y
0
)
be written as
ln
1
r
E
=
−
∇
.
The fundamental solution ln
r
thus represents the potential of the electric field gen-
erated by the wire.
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