  We formulate a systematic elegant perturbative scheme for determining the
eigenvalues of the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0
in two dimensions when the normal derivative of {\psi} vanishes on an irregular
closed curve. Unique feature of this method, unlike other perturbation schemes,
is that it does not require a separate formalism to treat degeneracies.
Degenerate states are handled equally elegantly as the non-degenerate ones. A
real parameter, extracted from the parameters defining the irregular boundary,
serves as a perturbation parameter in this scheme as opposed to earlier schemes
where the perturbation parameter is an artificial one. The efficacy of the
proposed scheme is gauged by calculating the eigenvalues for elliptical and
supercircular boundaries and comparing with the results obtained numerically.
We also present a simple and interesting semi-empirical formula, determining
the eigenspectrum of the 2D Helmholtz equation with the Dirichlet or the
Neumann condition for a supercircular boundary. A comparison of the
eigenspectrum for several low-lying modes obtained by employing the formula
with the corresponding numerical estimates shows good agreement for a wide
range of the supercircular exponent.
