3  Integral Equation Reformulation of the Acoustic Field

The problem of reformulating the exterior Helmholtz equation 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], [5], [11-13]. In this paper the integral equation of Burton and Miller [14] is selected.
Before stating this integral equation, it is helpful to introduce the Helmholtz integral operators Lk, Mk, Mkt, and Nk:
{ Lk m}P(p) º
ó
õ
P 
  Gk(p,q)  m(q)  dSq   (p Î E ÈS),

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

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

{ Nk m}P(p) º
np
ó
õ
P 
   Gk
nq
 (p,q)  m(q) dSq    (p Î S),
where P is a surface (not necessarily closed), nq, np are unit 'outward' normal to the surface P at q, p and m(q) is a density 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,
(12)
where r=p - q, r=|r| and H0(1) is the spherical Hankel function of the first kind of order zero.
The Burton and Miller integral formulation of the Helmholtz equation is as follows:
a{ Mk j}S (p) - ac(p) j(p)+ b{ Nk j}S (p) =                                             

a{ Lk j
n
 } S (p)+ b{ Mkt j
n
 } S (p)+ bc(p) j(p)
np
    (p Î S )
(13)
where a and b are complex numbers. The function c(p) represents the angle subtended by the exterior at p divided by 2 p. Equation (13) relates j(p) and [(j)/(n)](p) for points p on the boundary S. Numerical methods based on this integral equation have the potential for being reliable at all wavenumbers k. The value of j(p) for points p Î E are related to j and [(j)/(n)] on S through the following equation:
j(p) = { Mk j}S (p) -{ Lk j
n
}S (p)     (p Î E)
(14)
Numerical methods based on these equations are applied to exterior acoustic problems in references [5,13,15], to the direct solution of coupled acoustic-structure problems in references [2-4].