2 Integral Equation Formulations of the Helm-holtz Equation
In this section some of the integral equation formulations of the
interior Helmholtz equation are given.
The Helmholtz integral operators
Lk, Mk and Mkt are defined as follows:
{ Lk m}P(p) º
ó õ P
Gk(p,q) m(q) dSq (p Î D ÈS ),
{ Mk m}P(p) º
ó õ P
¶Gk
¶nq
(p,q) m(q) dSq (p Î D ÈS ),
{ Mkt m}P(p) º
¶
¶np
ó õ P
Gk(p,q) m(q) dSq (p Î S)
where nq and np are
unit outward normals to the boundary S at
q and p,
P Í S ,
and m(q) is a bounded function defined for
q Î P. Gk(p,q) is the free-space Green's
function for the Helmholtz equation:
Gk(p,q) =
eikr
4 pr
inthreedimensions,
Gk(p,q) = -
i
4
H0(1) (kr) intwodimensions,
where r=p - q, r=|r| and H0(1) is the spherical
Hankel function of the first kind of order zero.
Some restrictions on the regularity of the boundary and boundary functions
are necessary to reformulate the Helmholtz equation as an integral equation
[23].
In this paper it will be assumed that such conditions are satisfied.
An indirect formulation of the interior Helmholtz problem
is derived through writing j as
a single-layer potential. The following equations result:
j(p) = { Lk s}S (p) (p Î D ÈS),
(5)
¶j(p)
¶np
= { Mkt s}S (p)+ c(p) s(p) (p Î S),
(6)
where s is a source density function defined on S, np is the
unit outward normal to the boundary at
p. The function c(p) is defined so that
in two-dimensional problems 2 pc(p) (p Î S)
is the angle subtended by the interior at p and in
three-dimensional problems
4 pc(p) (p Î S)
is the solid angle subtended by the interior at p.
Note that if
S is smooth at p then c(p) = [1/2].
The direct formulation is obtained through
the application of Green's second theorem to the Helmholtz equation
and can be presented as follows: