Discipline: Physics
Subcategory: Physics (not Nanoscience)
Session: 1
Room: Captial
Dmitrii Shymkiv - University of North Texas
Co-Author(s): Arkadii Krokhin, University of North Texas, TX
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.Plane wave expansion method is a well-known technique on how to calculate the bandstructure of the crystal. Since it doesn’t involve the use of boundary conditions, it can be used for different geometries of the scatterer without additional complexity. However, due to the same reason one has to use the same wave equation for both scatterer and the background. It leads to difficulties and unphysical results [1, 2] if solid-fluid system is considered. In this work, we explain the mathematical reason of these issues as well. In crystals with air background and solid scatterers the approximation of rigid scatterer is valid due to high acoustical contrast. In order to make calculations easier we would like to find what material properties of the scatterer may be considered as infinite. A scatterer with infinite acoustic impedance is modeled by infinite mass density or elastic modulus. It is shown, using the plane-wave expansion method, that in both cases the dispersion equation contains singular matrices, with the elements expressed through the form-factor of the crystal lattice. However, in the case of infinite elastic modulus, this singularity is not critical, and the solutions of the dispersion equation turn out to be physically correct. Unlike this, in the case of infinite density the solutions of the dispersion equation become physically incorrect. We explain the mathematical reason of this drastic difference and propose a regularization numerical procedure. While calculating the bandstructure for the solid-fluid compositions using the standard PWE method, a number of authors reported appearance of flat bands associated with the transverse mode in solid scatterer. These bands appear randomly and cannot be justified. We show that the unphysical flat bands are due to the same reason – singularity of the form-factor matrix in the dispersion equation.In all these calculations fluid was considered an inviscid or ideal. In the future work, we are going to take into account the viscosity of the background fluid. This strongly complicates the problem since viscosity-dependent terms lead to nonlinear eigenvalue problem.References:[1] Y. Tanaka et al., 2000, Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch. Phys. Rev. B. 62, 7387.[2] C. Goffaux and J. P. Vigneron, 2001, Theoretical study of a tunable phononic band gap system, Phys. Rev. B 64, 075118.
Funder Acknowledgement(s): Funder Acknowledgement: This work was supported by the National Science Foundation under EFRI Grant No. 1741677.
Faculty Advisor: Dr. Arkadii Krokhin, arkady@unt.edu
Role: I performed most of the analytical and all numerical calculations. I wrote the abstract.