POTENTIAL PROBLEMS 41
Formula (2.60) can now be written as,
N
".·= Σ °J
G
ij
N
(2.62)
<lk= L
a
j
H
kj
Equations (2.62) gives a system of 2N equations which can now be
reduced to a N x N system if the necessary boundary conditions are
applied,
Ui
= w, at N
i
points on Γ
χ
(2.63)
<lk
—
<lk
at
N
2
points on Γ
2
The final system can then be written as,
Ασ = F (2.64)
where the unknowns in the σ vector are the source intensities.
It is important to point out that an 'equilibrium' type requirement
should be introduced in two-dimensional problems. The reason is
that the potentials given by (2.62) are only relative potentials in two-
dimensional problems. This is because of the special logarithmic
character of the solution, i.e.
u*
= - (2π)~
ι
ln(l/r), which implies that
w* does not go to zero at infinity.
As the choice of datum is arbitrary we can write
",·= Σ efiij + C (2.65)
j=i
where C is an unknown constant. The system has now (iV + 1)
unknowns but we can solve it by adding the following 'equilibrium'
condition:
/»
σάΓ = 0 (2.66)
Jr
or in discrete form,
Σ
*^
= °
(
2
·
67
)
where /, is the length of each element. This equation represents the
requirement that the sum of all the sources applied is zero and will