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
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 named `tension', and I shall follow suit.
The definition of is
Without a further condition, (5.5) and (5.6) admit the
for all .
Therefore the quadratic functional
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
at , one of the conditions needs to be relaxed.
The condition (5.6) will be replaced by the minimization