Noncommuting limits in electromagnetic scattering: Asymptotic analysis foran array of highly conducting inclusions

We consider formulations for the Helmholtz operator for periodic media cont aining high contrast inclusions in the limit when the wavelength outside th e inclusions tends to infinity. Applications are to problems of electromagn etism. The main focus is on the analysis of the effect of noncommuting limi ts, an effect which indicates that linear boundary value problems of electr omagnetism give formally different results for the long wavelength limits i n cases where highly conducting inclusions have refractive indices of diffe rent orders of magnitude. Specifically, the effective moduli of the homogen ized material will depend on the path used to approach the origin in the co ordinate space {wave number, ( normalized refractive index of the inclusion s)(-1)}. This mathematical observation gives a physical subtlety which is s tudied 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 co nsidered.