6.1 Formulations of the Eigenvalue
Problem
The integral equation formulations of the interior Helmholtz equation
were given in Section 4.1. The distinction here is that the
boundary condition (6.2) is homogeneous and the
characteristic wavenumbers k* and the eigenfunctions
f* are sought, rather than the solution of a
boundary-value problem.
6.1.1 Indirect Formulation
An indirect formulation of the interior Helmholtz problem
is derived by writing f as
a single-layer potential, as first shown in Section
4.1. The following equations result:
|
f(p) = { Lk s}S (p) (p Î D ÈS), |
| (6.5) |
|
v(p) = { Mkt s}S (p; np)+ |
1
2
|
s(p) (p Î S), |
| (6.6) |
where s is a source density function defined on S,
(termed s0 in Section 4.1).
Also v(p) = [(¶f(p))/( ¶np)]
and np is the unit outward normal to the boundary at
p.
The indirect formulations for the Dirichlet and Neumann eigenproblems
are as follows:
|
{ Lk m}S (p) = 0 (p Î S) ,for the Dirichlet boundary condition and |
|
|
{ (Mkt+ |
1
2
|
I) m}S (p;np) (p Î S) for the Neumann boundary condition. |
|
For the Helmholtz eigenvalue problem with the general
boundary condition (6.2) the integral equation formulation
is as follows:
|
a(p) { Lk m}S (p)+ b(p) { (Mkt + |
1
2
|
I) m}S (p) = 0 (p Î S) |
| (6.7) |
which arises through the substitution of (6.5) and (6.6) into (6.2)
where s has been replaced by m in (6.5)-(6.6).
The general solution strategy is to find the solutions k*, m*
of the relevant integral equation defined on S. Each of these
may then be substituted into equation (6.5) to obtain
the solution at the domain points.
6.1.2 Direct Formulation
As in Section 4.1, the direct formulation is obtained through
the application of Green's second theorem to the Helmholtz equation
and can be presented as follows:
|
{ Mk f}S (p) + f(p) = { Lk v }S (p) (p Î D) , |
| (6.8) |
|
{ Mk f}S (p) + |
1
2
|
f(p) = { Lk v }S (p) (p Î S) . |
| (6.9) |
The direct formulation for the Dirichlet and Neumann eigenproblems are
as follows:
|
{ Lk m}S (p) = 0 (p Î S) for the Dirichlet boundary condition, |
|
|
{ (Mk + |
1
2
|
I) m}S (p) = 0 (p Î S) for the Neumann boundary condition, |
|
where f has been replaced by m in (6.8)-(6.9).
For the more general Robin condition (6.2) the eigenproblem
cannot be written so concisely; it is the solution of (6.9)
subject to the boundary condition (6.2).
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