3 Integral Equation Formulations of the Eigenvalue Problems
In this section the homogeneous boundary condition is
enforced on the integral equation formulations in order to obtain
the integral equations for the Helmholtz eigenvalue problem.
The ensuing indirect integral equation reformulation of the Helmholtz
eigenvalue problem with the general boundary condition (2) is
a(p) { Lk m}S (p)+ b(p)
æ è
{ Mkt m}S (p)+ c(p) m(p)
ö ø
= 0 (p Î S)
which arises through the substitution of (5) and (6) into (2)
where s has been replaced by m in (5)-(6).
The indirect formulations for the Dirichlet and Neumann eigenproblems
are as follows:
Dirichlet
{ Lk m}S (p) = 0 (p Î S) ,
Neumann
{ Mkt m}S (p)+ c(p) m(p) = 0 (p Î S) .
The direct formulation for the Dirichlet and Neumann eigenproblems are
as follows:
Dirichlet