3.1 The Helmholtz Integral Operators
The subroutines compute the discrete form of the integral
operators Lk, Mk, Mkt and Nk that arise in the application
of collocation to integral equation formulations of the Helmholtz equation.
Expressions for the discrete integral operators are derived by approximating
the boundaries by the most simple panels for each of the three cases
- straight line panels for the general two-dimensional case, flat
triangular panels for the general three-dimensional case and
conical panels for the axisymmetric three-dimensional case - and approximating
the boundary functions by a constant on each panel.
For each particular case of boundary division, the discrete form of the
operators is computed using the subroutines H2LC (two-dimensional),
H3LC (three-dimensional) and H3ALC (axisymmetric three-dimensional).
3.1.1 The Helmholtz Operators
The Helmholtz integral operators are defined as follows:
|
{ Lk z}G(p ) º |
ó
õ
G
|
Gk(p,q) z(q) dSq , |
| (3.1) |
|
{ Mk z}G(p) º |
ó
õ
G
|
|
¶Gk
¶nq
|
(p,q) z(q) dSq , |
| (3.2) |
|
{ Mkt z}G(p; up) º |
¶
¶up
|
|
ó
õ
G
|
Gk(p,q) z(q) dSq , |
| (3.3) |
|
{ Nk z}G(p; up) º |
¶
¶up
|
|
ó
õ
G
|
|
¶Gk
¶nq
|
(p,q) z(q) dSq , |
| (3.4) |
where G is a boundary or partial boundary,
nq, up
are unit vectors with nq the unique normal to G at q
and z(q)
is a function defined for q Î G. Gk(p,q)
is the free-space Green's
function for the Helmholtz equation.
3.1.2 Green's functions
Let the Green's functions be denoted by Gk
and they are defined as follows:
|
Gk(p,q) = |
i
4
|
H0(1)(kr) (k Î C \{ 0 }) in two dimensions, |
| (3.5) |
|
Gk(p,q) = |
1
4 p
|
|
eikr
r
|
(k Î C) in three dimensions, |
| (3.6) |
where r = |r|, r = p-q, C is the set
of complex numbers and i is the unit imaginary
number. The function H0(1) is the spherical Hankel function of
the first kind of order zero. The Green's functions (3.5) and (3.6) also
satisfy the Sommerfeld radiation condition for |k| > 0.
For the special case when k = 0 the Helmholtz equation (1.5)
is the Laplace equation. In this particular case the chosen Green's
functions will be
|
G0(p,q) = - |
1
2 p
|
logr in two dimensions, |
| (3.7) |
|
G0(p,q) = |
1
4 p
|
|
1
r
|
in three dimensions. |
| (3.8) |
Note that limk ® 0 Gk(p,q) = G0(p,q)
for the three-dimensional case but not for the two dimensional case and
that G0(p,q) for the two-dimensional case does
not satisfy condition (1.15).
3.1.3 Properties of the Operators
In general for a given function z(p) (p Î S),
{Lk z}G(p) and {Nk z}G(p;up)
are continuous across the boundary G
(for any given unit vector up in the definition of the
latter function).
The { Mk z}G(p) and { Mkt z}G(p)
are discontinuous at G and continuous on the remainder of the
domain.
The operators Mk and Mkt
have the following continuity properties at points in the neighbourhood
of G:
|
{ Mk z}G(p+enp)+ |
1
2
|
z(p) = { Mk z}G(p) = { Mk z}G(p-enp)- |
1
2
|
z(p) , |
|
|
{ Mkt z}G(p+enp; np) - |
1
2
|
z(p) = { Mkt z}G(p; np) = { Mkt z}G(p-enp; np) + |
1
2
|
z(p) , |
|
where p Î G and np is the unit normal to the
G at p. The continuity properties are slightly different
if G is not smooth at p.
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