Environmental Engineering Reference
In-Depth Information
2. Equations (5.78) and (5.79) both show that
u
2
(
tends to
the characteristic cone surface. Therefore the singularity of the source point prop-
agates along the characteristic surface, where the solution is consequently discon-
tinuous. This is contrary to the physical reality. However,
u
1
(
x
,
y
,
t
)
→
∞
as
M
(
x
,
y
,
t
)
x
,
y
,
t
)
does not have
2
2
A
2
2
,
this drawback. For a point
(
x
,
y
)
such that
(
x
−
x
0
)
+(
y
−
y
0
)
=
(
t
−
t
0
)
A
2
1
(
t
−
t
0
)
e
−
u
1
(
x
,
y
,
t
)=
,
4
a
2
4
π
a
2
(
t
−
t
0
)
which still has a finite value. The
u
1
(
x
,
y
,
t
)
→
∞
occurs only when
t
→
t
0
.The
2
2
t
→
t
0
implies that
(
x
−
x
0
)
+(
y
−
y
0
)
→
0sothat
(
x
,
y
)
is in the neighborhood
of the source point
(
x
0
,
y
0
)
. The singularity is thus confined in the source point
(
x
0
,
y
0
)
without propagation.
5.7 Methods for Solving Axially Symmetric and
Spherically-Symmetric Cauchy Problems
Axially symmetric and spherically-symmetric problems are special cases of two-
dimensional and three-dimensional Cauchy problems. The solution structure the-
orem is valid for them. In this section we use the Hankel transformation and the
spherical Bessel transformation of order zero to solve them.
5.7.1 The Hankel Transformation for Two-Dimensional Axially
Symmetric Problems
An Integral Formula of Bessel Function of Order Zero
The expansion expression of the generating function for the Bessel function is (Ap-
pendix A)
+
∞
∑
e
2
(
t
−
t
−
1
)
t
n
=
J
n
(
x
)
.
n
=
−
∞
By the formula for Laurent series coefficients, we have
π
e
2
(
z
−
z
−
1
)
1
1
2
e
i
x
cos ϕ
d
J
0
(
x
)=
d
z
=
ϕ
,
2
π
i
z
π
|
z
|
=
1
−
π
(
ϕ
+
2
)
. It is transformed to, by a variable transformation
e
i
where
z
=
ϕ
=
θ
+
π
,
2π
1
2
e
−
i
x
cos θ
d
J
0
(
x
)=
θ
,
(5.80)
π
0
which is the integral formula of Bessel function of order zero.
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