3  Rayleigh Integral Formulation

A derivation of the Rayleigh integral formulation is given in Pierce [6] pp214-215. In brief, it relates the velocity potential j(p) at a point p in the exterior E, on the plate G, or on the baffle to the normal velocity v on the plate G. In the standard integral operator notation used in integral equation methods (see Burton [7] for example) the Rayleigh integral is as follows,
j(p) = -2 { Lk v }G (p) .
(10)
In equation (10) the operator Lk is defined by
{ Lk m}G(p) º ó
õ

G 
  Gk (p,q)  m(q)  dSq   (p Î E ÈG) ,
(11)
where Gk(p,q) is a free-space Greens function for the Helmholtz equation. In this paper the Green's function is defined as follows
Gk(p,q) = 1
4 p
 eikr
r
   (k Î \sf C) ,
(12)
where r=|r|, r=p-q, C is the set of complex numbers and i is the unit imaginary number. The Green's function (12) also satisfies the Sommerfeld radiation condition.