Multiple Scattering by Multiple Spheres: A New Proof of the Lloyd-Berry Formula for the Effective Wavenumber

Abstract. We provide the first classical derivation of the Lloyd–Berry formula for the effective wavenumber of an acoustic medium filled with a sparse random array of identical small scatterers. Our approach clarifies the assumptions under which the Lloyd–Berry formula is valid. More precisely, we derive an expression for the effective wavenumber which assumes the validity of Lax’s quasicrystalline approximation but makes no further assumptions about scatterer size, and then we show that the Lloyd–Berry formula is obtained in the limit as the scatterer size tends to zero.