POTENTIAL PROBLEMS 51
r is the distance from the source point to the point under con-
sideration.
H
{
Q
]
is a Hankel function of
the
second kind of zero order,
i.e.
Η^(λτ) = 3
0
(λτ)-\Υ
0
(λτ) (2.97)
where J
0
is a Bessel function of the first kind and Y
0
is one of the
second kind. The Hankel function of order one is
H[
2)
(kr)
= J, (λή - i Y, (λή (2.98)
For three dimensions, the fundamental solution is,
u* = e~
Ur
4nr
and (2.99)
du*
1 /l .„ .
or 4nr \r
Substituting (2.93) into (2.92) we obtain,
u,+
l·
q*dr =
qu*dr (2.100)
If the point
Ί"
is taken to be on the smooth Γ boundary, we will have
for two and three dimensions the following result:
iu,+ uq*dr = qu*dr (2.101)
In general we can write a constant c, instead of \ for a non-smooth
boundary. Hence,
0^ +
=
l·"
uq*dr = qu*dr (2.102)
From now on the usual boundary element matrix technique can be
applied.
Let us now consider how the solution behaves at infinity by
assuming that we have two boundaries, one
a
closed boundary and the
other the one at infinity (Figure 2.12). Formula (2.102) can now be
written as
CM+ I (uq*-qu*)dr= | (qu*-q*u)dr (2.103)