2  Integral Equation Formulation

In this section the direct and indirect boundary and shell integral equation formulations of the interior Helmholtz equation are given. The boundary and shell integral equations may be regarded as hybrids of their respective standard boundary integral equation formulations as given, for example in [1] and shell integral equation formulations given in [5,6].

2.1  Notation

Let the function v(p) for p Î S be defined as follows
v(p) =
lim
e® 0+ 
 j
np
 (p + enp )  (p Î S),
where np is the unit outward normal to S at p. Each shell is assumed to have two sides or surfaces, let G+ be the upper surface and let G- be the lower surface. The potential j and its derivatives are generally discontinuous at the shell, however they take limiting values on G+ and G-. Let the functions j+ (p), j- (p), v+ (p) and v- (p)   (p Î G) be defined as follows:
j+(p) =
lim
e® 0+ 
 j(p + enp ) ,

j-(p) =
lim
e® 0+ 
 j(p - enp ) ,

v+(p) =
lim
e® 0+ 
 j
np
 (p + enp ) ,

v-(p) =
lim
e® 0+ 
 j
np
 (p - enp ) .
The geometrical function c(p)  (p Î S ÈG) is defined to be the angle subtended by the interior region at p for points on S and the angle subtended the region at G+ for points on G, each angle then divided by 2p.
It is helpful to introduce the functions d(p), n(p), F(p) and V(p) for p Î G which are defined as follows:
d(p) = j+(p) - j-(p)  (p Î G),

n(p) = v+(p) + v-(p)  (p Î G),

F(p) = c(p) j+(p) + (1 - c(p)) j-(p)   (p Î G),

V (p) = c(p) v+(p) - (1 - c(p)) v-(p)   (p Î G).

2.2  Boundary and Shell Conditions

The boundary condition has the form
a(p) j(p) + b(p)v(p) = g(p)  (p Î S),
where a(p), b(p) and g(p) are functions of p on S. The shell conditions are assumed to have the following general form
a(p) d(p) + b(p)n(p) = f(p)  (p Î G),

A(p) F(p) + B(p) V(p) = F(p)  (p Î G),
where a(p), b(p), f(p), A(p), B(p) and F(p) are functions of p on G.

2.3  Integral Operator Notation

The Laplace integral operators Lk, Mk, Mkt, and Nk are defined as follows:
{ Lk m}P(p) º
ó
õ
P 
  Gk (p,q)  m(q)  dSq   (p Î E ÈS ÈG) ,

{ Mk m}P(p) º
ó
õ
P 
   Gk
nq
 (p,q)  m(q) dSq    (p Î E ÈS ÈG) ,

{ Mkt m}P(p) º
np
  
ó
õ
P 
  Gk(p,q)  m(q)  dSq    (p Î S ÈG) ,

{ Nk m}P(p) º
np
ó
õ
P 
   Gk
nq
 (p,q)  m(q) dSq    (p Î S ÈG) ,
where P Ì S ÈG, nq and np are unit outward normal to P when P Ì S or the unit normal to P+ when P Ì G at q, p 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) = i
4
 H0(1)(kr)    in two dimensions,
where r=p - q and r=|r|. The function H0(1) is the spherical Hankel function of the first kind of order zero.

2.4  Direct Integral Equation Formulation

The equations that make up the boundary and shell integral equation formulation of the Helmholtz equation are given in this subsection. For points on the boundary the following equation holds
{ Mk j}S (p) + c(p) j(p) = { Lk v } S (p)+ { Mk d}G (p) - { Lk n}G (p)    (p Î S).
(2)
This equation relates j(p) and v(p) for points p on the boundary S. For points on the shell, we have the following equations:
F(p) = - {Mk j}S(p) + {Lk v }S(p)+ {Mk d}G(p) - {Lk n}G(p)   (p Î G) ,
(3)

V (p) = - {Nk j}S(p) + {Mkt v }S(p)+ {Nk d}G(p) - {Mkt n}G(p)   (p Î G) .
(4)
The value of j(p) for points in the domain are related to the solutions on S and G through the following equation:
j(p) = - { Mk j}S (p) + { Lk v }S (p) + { Mk d}G(p) -{ Lk n}G (p)  (p Î D) .
(5)

2.5  Indirect Integral Equation Formulation

The equations that make up the indirect boundary and shell integral equation formulation of the Helmholtz equation are given in this subsection. For points on the boundary the following equations hold:
j(p) = { Lk s} S (p)+ { Mk d}G (p) - { Lk n}G (p)    (p Î S),
(6)

v (p) = {Mkt s}S(p) + c(p) s(p)+ {Nk d}G(p) - {Mkt n}G(p)   (p Î S),
(7)
where s is generally known as a source density function. These equations relate j(p) and v(p) for points p on the boundary S. For points on the shell, we have the following equations:
F(p) = {Lk s}S(p) + {Mk d}G(p) - {Lk n}G(p)  (p Î G) ,
(8)

V (p) = {Mkt s}S(p) + {Nk d}G(p) - {Mkt n}G(p)   (p Î G) .
(9)
The value of j(p) for points in the exterior domain are related to the solutions on S and G through the following equation
j(p) = { Lk s}S (p) +{ Mk d}G(p) -{ Lk n}G (p)  (p Î D) .
(10)

2.6  Formulation of the Eigenvalue Problem

The eigenvalue problem is that of finding the values of k for which the Helmholtz equation (1) with a homogeneous boundary condition of the form
a(p) j(p) + b(p)v(p) = 0  (p Î S),

a(p) d(p) + b(p)n(p) = 0  (p Î G),

A(p) F(p) + B(p) V(p) = 0  (p Î G),
has non-trivial solutions.