3  Rayleigh Integral Formulation

A derivation of the Rayleigh integral formulation is given in Pierce [9]. In brief, it relates the velocity potential j(p) at a point p in the exterior E, on the plate P, or on the baffle to the normal velocity v on the plate P.

3.1  Formulation

In the standard integral operator notation used in integral equation methods (see Kirkup [6] for example) the Rayleigh integral is as follows,
j(p) = -2 { Lk v }P (p)    (p Î PÈE).
(11)
In equation (11) the operator Lk is defined by
{ Lk m}P(p) º ó
õ

G 
  Gk (p,q)  m(q)  dSq   (p Î GÈE) ,
(12)
where Gk(p,q) is a free-space Greens function for the Helmholtz equation and G represents the whole or part of the plate. In this document the Green's function is defined as follows
Gk(p,q) = 1
4 p
 eikr
r
   (k Î \sf R+) ,
(13)
where r=|r|, r=p-q, \sf R+ is the set of positive real numbers and i is the unit imaginary number. The Green's function (13) also satisfies the Sommerfeld radiation condition (6), ensuring that all scattered and radiated waves are outgoing in the farfield.

3.2  Discussion

Using the Rayleigh integral (11), the velocity potential at any point in the exterior field can be computed from the velocity v of the panel by a straightforward integration. However, the velocity of the panel is only known at the outset in cases when the Neumann condition (a(p)=0 and b(p)=1 in (5)) is given. In the case of other boundary conditions, the velocity v of the panel must be determined first. For example if j is known on the panel then v needs to be obtained on the panel first by solving the integral equation (11). Once v is known on the panel, the velocity potential (or sound pressure) can be determined throughout the field by (11).
In this work, the approach to solving (11) with general boundary conditions (5) involves applying the technique of collocation. (In the case of the Neumann boundary condition, the technique reduces to product integration.) It can be observed that one potential difficulty could arise in the numerical solution of equation (11); in the Dirichlet problem the equation (11) is a Fredholm integral equation of the first kind. It is well-known that first kind equations generally do not lend themselves to computational solution (see [1], for example). However, since the integrand in (11) is singular, the equation can be solved by typical integral equation methods such as collocation. The method may be less accurate than you would otherwise expect in this case, but the solution is still possible.