A wavelet approach to effective modal analysis of complex lamination

Sturm-Liouville equations and eigenfunctions/eigenvalues problems are of the most fundamental tools in mathematical physics with applications pertaining to statics (electromagnetics or mechanics), theory of wave propagation, sound and vibration, quantum mechanics, and more. The eigenvalues and the associated eigenfunctions (or modes) convey indispensable information about the physics of the problem and constitute a solid basis for a plethora of analytical and computational techniques or modal analysis. The article investigates the relations between the small scale structure of a system heterogeneity and the eigenvalues and modes of the corresponding Sturm-Liouville problem. In particular, multiresolution theory and wavelets are used to derive a new boundary value formulation governing the large scale features of the modes. Numerical examples are provided, and the computation of the large scale component of a two dimensional Green function in a finely laminated waveguide structure is demonstrated and discussed.