Spatial Bounds on the Effective Complex Permittivity for Time-Harmonic Waves in Random Media

Abstract. We consider wave propagation in random cell materials when the wavelength is finite, so that scattering effects must be taken into account. An effective dielectric coefficient is introduced, which in general is a spatially dependent function, yet reduces, under the infinite wavelength assumptions, to the constant effective parameter in the quasistatic limit. We present an upper bound on the effective permittivity and a bound on its spatial variations that depends on the maximum volume of the inhomogeneities and the contrast of the medium. Numerical experiments illustrate the rigorous results.