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
,
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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