On the lowest-frequency bandgap of 1D phononic crystals

TL;DR

This paper introduces an analytical method to design one-dimensional phononic crystals with the lowest possible frequency bandgap, verified through numerical optimization, and explores the trade-offs between low-frequency gaps and broadband attenuation.

Contribution

It provides a new proof of the harmonic decomposition of the transfer matrix trace and derives a closed-form asymptotic expression for low frequencies, enabling optimized layer design.

Findings

Analytical approach effectively predicts lowest bandgap frequencies.

Optimal layer configurations can be designed for minimal bandgap frequency.

Trade-off identified between low-frequency gap minimization and broadband attenuation.

Abstract

This manuscript puts forward and verifies an analytical approach to design phononic crystals that feature a bandgap at the lowest possible frequencies. This new approach is verified against numerical optimization. It rests on the exact form of the trace of the cumulative transfer matrix. This matrix arises from the product of N elementary transfer matrices, where N represents the number of layers in the unit cell of the crystal. The paper presents first a new proof of the harmonic decomposition of the trace (which, unlike the original derivation, does not resort to group-theoretical concepts), and then goes to demonstrate that the long-wavelength asymptotics of the function that governs the dispersion relation can be described in closed form for any layering, plus that it possesses a simple and explicit form that facilitates its study. Using this asymptotic result for low frequencies, we then propose an analytical curvature-minimization approach to devise layerings that yield an opening of the first frequency bandgap at the lowest possible frequency value. Finally, we also show that minimizing the frequency at which the first gap opens can be at odds with obtaining broadband attenuation, i.e., maximizing the width of the first frequency bandgap.