3  Integral Equation Reformulation of the Acoustic Field

The problem of reformulating the Helmholtz equation exterior to a radiating or scattering body as an integral equation that forms a reliable basis for solution by a numerical method has interested researchers for several decades. For the background to this, see references [4], [8], [13], [14]. The similar problem, but exterior to shells, has also received some attention [10], [11], [15]. In this section the direct boundary integral equation of Burton and Miller [9] and the integral equation reformulation for the problem exterior to shells are unified to give the formulation of the full system illustrated in figure 1. A similar technique has been applied to find the boundary and shell integral equation for Laplace problems in reference [16].

3.1  Notation

The Helmholtz 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 is a surface (not necessarily closed), nq, np are unit 'outward' normal to P 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) = eikr
4 pr
      in three dimensions,
(14)
where r=p - q and r=|r|.

3.2  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
a{ Mk j}S (p) - 1
2
 aj(p)+ b{ Nk j}S (p) = a{ Lk v } S (p)+ b{ Mkt v } S (p)+ 1
2
 bv(p)

- a{ Mk d}G (p) - b{ Nk d}G(p) + a{ Lk n}G (p) + b{Mkt n}G(p)    (p Î S)
where a and b are complex numbers and S is smooth at p. 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 d}G(p) - {Lk n}G(p) + {Mk j}S(p) - {Lk v }S(p)  (p Î G) ,

V (p) = {Nk d}G(p) - {Mkt n}G(p) + {Nk j}S(p) - {Mkt v }S(p)  (p Î G) ,
where G is smooth at p. 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) = { Mk j}S (p) -{ Lk v }S (p) + { Mk d}G(p) -{ Lk n}G (p)  (p Î E) .
The velocity at the upper surface of the shell is equal and opposite to that at the lower surface, that is n(p) º 0  (p Î G). Hence the equations above can be simplified as follows:
a{ Mk j}S (p) - 1
2
 aj(p)+ b{ Nk j}S (p) = a{ Lk v } S (p)+ b{ Mkt v } S (p)+ 1
2
 bv(p)

- a{ Mk d}G (p) - b{ Nk d}G(p)    (p Î S)   ,
(15)

F(p) = {Mk d}G(p) + {Mk j}S(p) - {Lk v }S(p)  (p Î G) ,
(16)

V (p) = {Nk d}G(p) + {Nk j}S(p) - {Mkt v }S(p)  (p Î G) ,
(17)

j(p) = { Mk j}S (p) -{ Lk v }S (p) + { Mk d}G (p)   (p Î E) .
(18)