Phononic crystal with rigid scatterers: Infinite density versus infinite elasticity

Abstract

Models involving materials with extreme parameters, while usually unreachable in practice, play important role in different areas of physics, for example, black body and ideal gas in thermodynamics, perfect conductor, and ideal diamagnetic in electrodynamics. In acoustics, the concept of rigid (or hard) scatterer is widely used to study propagation of sound in heterogeneous media with high-acoustic contrast between the constituents. Here, we report the results obtained for band structure calculations of phononic crystals with rigid scatterers. A scatterer with infinite acoustic impedance is modeled by approaching either the mass density (ρ) or the elastic modulus (λ) to infinity. It is shown, using the plane-wave expansion method, that in both cases the dispersion equation contains singular matrices with elements related to the form factor of the crystal lattice. However, this singularity does not affect calculations of the band structure in the case λ → □. Unlike this, in the limiting case of infinite density, the dispersion equation becomes meaningless. We explain the mathematical reason of this drastic difference and propose a regularization numerical procedure. Our general results are illustrated by a particular case of elastic superlattice when the dispersion relation is available analytically.