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# Definition of the billiard problem

Given the isotropic quadratic dispersion relation corresponding to (5.2), we can choose energy units such that . If the dispersion relation is not isotropic, it can be made so by a re-parametrization of . However, note that will not play any further role. The numerical methods described in this thesis are really about finding the eigenwavenumbers . Therefore the methods are entirely applicable to any other Helmholtz eigenproblem regardless of the dispersion relation, or indeed the existence of an energy' (for instance is physically irrelevant in acoustic problems). The only requirement is that the wavenumber be constant (and isotropic) in the interior.

The billiard has -dimensional volume' and dimensional surface area' , giving a typical length scale . Our eigenproblem can be written

 (5.5) (5.6) (5.7)

where the eigenvalue is . The functionals and return a scalar, and are quadratic in the wavefunction .

The BCs have been incorporated as (5.6) rather than the linear condition (5.3) because satisfaction of the BCs by a wavefunction is then revealed by a single number . This number measures the 2-norm of some error function, and is therefore a non-negative quantity. The error function (e.g. (5.3)) gives the amount by which the desired BCs fail to be obeyed. Heller[91] named tension', and I shall follow suit. The definition of is

 (5.8)

where is a weighting function over the boundary. Arbitrary weighting schemes are possible, but I will stick to which treats all locations on equally. The case of Dirichlet BCs is given by , put more simply, measures the boundary integral of the squared value of the wavefunction. Generally my work will be restricted to this case from now on--however the same techniques would apply to other BCs.

Without a further condition, (5.5) and (5.6) admit the useless solution, for all . Therefore the quadratic functional

 (5.9)

which measures the normalization (norm) in the domain is required. Unit norm is fixed by (5.7).

The solution is now completely determined, when reaches one of the eigenwavenumbers . For other , no solution exists. Therefore in order to be able to define a `best' solution for any given guess at , one of the conditions needs to be relaxed. The condition (5.6) will be replaced by the minimization

 (5.10)

A sweep in will now show the -dependent tension , indicating how well the best at each can match the BCs. When approaches an eigenwavenumber , the tension will drop to zero, indicating all the conditions (5.5), (5.6) and (5.7) are satisfied simultaneously. Such a sweep is shown in Fig. 5.2.

Next: Representation by a Helmholtz Up: Chapter 5: Improved sweep Previous: Brief history of the
Alex Barnett 2001-10-03