# The resolvent method for shear waves spectra calculation in 2D phononic crystals: dispersion equation, displacement and traction wave fields

## Dipl.-Math. Dr. Maria Korotyaeva

March 22, 2016, 3:30 p.m. S2 416-2

We propose the resolvent method for calculating the shear waves spectra in 2D phononic crystal (PC) waveguides: the free PC plate and the PC plate sandwiched between two substrates.

Since the propagator $M$ over a unit cell approximated by Fourier harmonics in one coordinate can have very large components, we introduce its resolvent $R = (zI-M)^{-1}$ ($z$ is a complex number outside of ${\rm spec}\ M$) as a numerically stable substitute. Another two key tools given in terms of the resolvent, a spectral projector $P_d$ and propagator $M_d$ for the decreasing modes, come into play in the case of a waveguide with a substrate.

The resolvent method providing simple dispersion and wave field equations in terms of $R$, $P_d$ and $M_d$ has several advantages. It is of a good precision due to the exact solution in one direction, computationally cheap due to the reduction of the problem to one unit cell even in a semi-infinite substrate, and versatile since it is applicable to uniform, 1D- or 2D-periodic structures. Moreover, it is extendible to P/SV waves and 3D PC.

In numerical examples, we model low-frequency band gaps and compare them for the mirror-symmetric and perturbed profiles. The displacement and traction wave fields are calculated for the waveguides with highly contrasting matrix/inclusions stiffness values which allows us to reveal the PC geometry.

Keywords: Phononic crystals, Guided waves, Propagator, Resolvent, Spectral projector

Abstract