# Omindirectional Non-Reciprocity via 2D Modulated Radial Sonic Crystals

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Space-Varying Radial Crystals

#### 3.2. Space–Time-Varying Radial Crystals

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PWEM | Plane Wave Expansion Method |

RSC | Radial Sonic Crystal |

## Appendix A. PWEM for a Spatially Modulated RSC

## Appendix B. PWEM for a Spatiotemporally Modulated RSC

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**Figure 1.**Graphical representation of the one-way effect. (

**a**) Waves are allowed to propagate from the inner to the outer shells ${d}_{i}$ and ${d}_{o}$ of the spatiotemporally modulated radial sonic crystal (RSC) (white region in the drawing) when the modulation travels from the outer to the inner shell. (

**b**) Vice-versa, waves are not allowed to reach the outer shell if the modulation travels from the inner to the outer shell.

**Figure 2.**Schematic of the spatial modulation. (

**a**) Graphical representation of $R(r,\theta )$. (

**b**) Radial density distribution ${\rho}_{r}(r,\theta )$ in space. (

**c**) Representation of the bulk modulus $B(r,\theta )$ in the plane. (

**d**) Tangential density ${\rho}_{\theta}(r,\theta )$ distribution in space. ${\rho}_{\theta}(r,\theta )$ is inversely proportional to the radius, whereas $B(r,\theta )$ is linearly increasing with r.

**Figure 3.**Numerical dispersion (colored contours) with superimposed Plane Wave Expansion Method (PWEM) solution (white dots) for (

**a**) q = 0, (

**b**) q = 1, (

**c**) and q = 2. The Bloch-wave solutions corresponding to $\mu /\pi =\pm 0.5$ and $\mu /\pi =\pm 1.5$ are marked with red dots. The corresponding modes are illustrated below for (

**d**,

**e**,

**j**,

**k**) q = 0, (

**f**,

**g**,

**l**,

**m**) q = 1, and (

**h**,

**i**,

**n**,

**o**) q = 2.

**Figure 4.**Schematic representation of the space–time varying properties. (

**a**) Graphical representation of $R(r,\theta ,t)$. (

**b**) Space–time varying radial tension ${\rho}_{r}(r,\theta ,t)$. The properties are periodic in space and traveling toward the inner shell or the outer shell depending on the modulation direction. (

**c**) Bulk modulus distribution $B\left(\right)open="("\; close=")">r,\theta ,t$. (

**d**) Graphical representation of the spatially varying tangential density ${\rho}_{\theta}(r,\theta ,t)$.

**Figure 5.**(

**a**–

**c**)Numerical dispersion relation (colored contours) for a space–time modulated SRC with superimposed PWEM solution (white dots). (

**a**) q = 0; (

**b**) q = 1; (

**c**) q = 2. The Bloch-wave solutions corresponding to $\mu /\pi =\pm 0.5$ and $\mu /\pi =\pm 1.5$ are marked with red dots. The corresponding modes are illustrated below for (

**d**,

**e**,

**j**,

**h**) q = 0, (

**f**,

**g**,

**l**,

**m**) q = 1, and (

**h**,

**i**,

**n**,

**o**) q = 2.

**Figure 6.**One-way wave propagation for a space–time modulated SRC with excitation in the middle of the ring structure and associated numerical and PWEM dispersion. (

**a**,

**b**) Monopole, (

**c**,

**d**) dipole, and (

**e**,

**f**) quadrupole excitation. The energy mainly propagates from the central portion to the outer shell, while propagation to the inner shell is not allowed.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Enrico Quadrelli, D.; Riva, E.; Cazzulani, G.; Braghin, F.
Omindirectional Non-Reciprocity via 2D Modulated Radial Sonic Crystals. *Crystals* **2020**, *10*, 624.
https://doi.org/10.3390/cryst10070624

**AMA Style**

Enrico Quadrelli D, Riva E, Cazzulani G, Braghin F.
Omindirectional Non-Reciprocity via 2D Modulated Radial Sonic Crystals. *Crystals*. 2020; 10(7):624.
https://doi.org/10.3390/cryst10070624

**Chicago/Turabian Style**

Enrico Quadrelli, Davide, Emanuele Riva, Gabriele Cazzulani, and Francesco Braghin.
2020. "Omindirectional Non-Reciprocity via 2D Modulated Radial Sonic Crystals" *Crystals* 10, no. 7: 624.
https://doi.org/10.3390/cryst10070624