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 D S),
(6.5)
v(p) = { Mkt s}S (p; np)+ 1
2
s(p)      (p 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 S) ,for the Dirichlet boundary condition and
      { (Mkt+ 1
2
I) m}S (p;np)     (p 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 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which arises through the substitution of (6.5) and (6.6) into (6.2) where s has been replaced by m in (6.5)-(6.6).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 D) ,
(6.8)
{ Mk f}S (p) + 1
2
f(p) = { Lk v }S (p)     (p S) .
(6.9)

The direct formulation for the Dirichlet and Neumann eigenproblems are as follows:

     { Lk m}S (p) = 0      (p S)    for the Dirichlet boundary condition,
       { (Mk + 1
2
I) m}S (p) = 0     (p 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where f has been replaced by m in (6.8)-(6.9).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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