A Study of the E ective Properties of Complex Scatterers Using Multiresolution Decomposition

A systematic study of the across scale coupling phenomenology in scattering problems is addressed using the theory of multiresolution decomposition and orthogonal wavelets. By projecting an integral equation formulation of the scattering problem onto a set of subspaces that constitutes a multiresolution decomposition of L 2 (R), one can derive two coupled formulations. The rst governs the macroscale response, and the second governs the microscale response. By substituting the formal solution of the latter in the former, a new self consistent formulation that governs the macroscale response component is obtained. This formulation is written on a macroscale grid, where the e ects of the microscale heterogeneity is expressed via an across scale coupling operator. This operator can also be interpreted as representing the e ective properties of the microstructure. We study the properties of this operator versus the characteristics of the Green function and the microstructure for various structural acoustics problems, using general asymptotic considerations. Speci c examples of scattering from a thin, linearly elastic, uid loaded plate with surface mass density or bending sti ness variations are provided. We show that mass microscale variation has virtually no e ect on the macroscale response, and sti ness microscale variations can signi cantly e ect the macroscale response. Classes of sti ness variations that have an identical macroscale response are derived using a one dimensional local constitutive relation developed to govern the macroscale eld components. The results are supported by numerical examples. Acknowledgment This research was supported in part by THE ISRAEL SCIENCE FOUNDATION administered by THE ISRAEL ACADEMY OF SCIENCE AND HUMANITIES