Abstract:
We consider formulations for the Helmholtz operator for periodic media containing
high contrast inclusions in the limit when the wavelength outside the inclusions tends to infinity.
Applications are to problems of electromagnetism. The main focus is on the analysis of the effect of
noncommuting limits, an effect which indicates that linear boundary value problems of electromagnetism
give formally different results for the long wavelength limits in cases where highly conducting
inclusions have refractive indices of different orders of magnitude. Specifically, the effective moduli
of the homogenized material will depend on the path used to approach the origin in the coordinate
space {wave number, (normalized refractive index of the inclusions)-¹}. This mathematical observation
gives a physical subtlety which is studied in this paper. The dispersion relation for the lowest
frequency (or acoustic mode) is investigated, as are the conditions for existence of an acoustic mode.
Cases of both nondispersive and dispersive inclusions are considered.
