# Parity-Time Symmetry and Exceptional Points for Flexural-Gravity Waves in Buoyant Thin-Plates

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Derivation of Flexural-Gravity Governing Equation

#### 2.2. Transfer and Scattering Matrix Formalism for Flexural-Gravity Waves

#### 2.2.1. Transfer Matrix Formalism

#### 2.2.2. Scattering Matrix Formalism

## 3. Results

## 4. Discussion: CPAL Effect

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure of the gain/loss floating device. The top (

**bottom**) arrays describe the right (

**left**) incidence and show the nonreciprocal behavior due to $\mathcal{PT}$-symmetry phenomena.

**Figure 2.**Scheme of the transfer matrix method formalism, showing the different regions of the problem, i.e., LSW regions in blue color, and floating thin-plate. Also we can see the different plane-wave expansions in the different regions, as well as the ingoing and outgoing components in each layer.

**Figure 3.**(

**a**) Frequency variation of the eigenvalues (${s}_{\pm}$) of the system depicted in the inset, showing a breakdown of $\mathcal{PT}$-symmetry around ${k}_{0}\approx 76.11\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$, for $\Im \left({E}_{g}\right)=1.3$ GPa. The density of the plate is everywhere $900\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ to make it floating atop water and the width and thickness of all three layers is 30 m and 1 m, respectively. (

**b**,

**c**) give the transmittance (T) and reflectance (R) [and total scattering (T+R)] signals in the same spectral domain for right and left incidence, respectively. Asymmetric and Fano-like line-shape reflectance could be observed for the right incidence. The left incidence results in high amplification at the same frequency.

**Figure 4.**Same structure as in Figure 3, but for a broader spectrum spanning frequencies 0 to 80 ${\mathrm{m}}^{-1}$. Multiple resonances could be observed, with the strongest one occurring at the highest frequency, in (

**a**). (

**b**) shows $|{s}_{\pm}|$ and multiple locations for the EPs. The inset is a magnified view around one low frequency EP.

**Figure 5.**Output coefficient $\Xi $ versus ${k}_{0}$ and $\Im \left(E\right)$ in GPa in (

**a**) 3D and (

**b**) 2D (logarithmic scale) showing a localized divergence, around ${k}_{0,\mathrm{max}}=6.4\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$ and ${E}_{g,\mathrm{max}}=2.56$ GPa (yellow color). Amplitude of the eigenvalues of the $\mathcal{PT}$-system (

**c**) $|{s}_{+}|$ and (

**d**) $|{s}_{-}|$ versus the same parameters, showing a distorted semi-ring-like behavior.

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**MDPI and ACS Style**

Farhat, M.; Guenneau, S.; Chen, P.-Y.; Wu, Y.
Parity-Time Symmetry and Exceptional Points for Flexural-Gravity Waves in Buoyant Thin-Plates. *Crystals* **2020**, *10*, 1039.
https://doi.org/10.3390/cryst10111039

**AMA Style**

Farhat M, Guenneau S, Chen P-Y, Wu Y.
Parity-Time Symmetry and Exceptional Points for Flexural-Gravity Waves in Buoyant Thin-Plates. *Crystals*. 2020; 10(11):1039.
https://doi.org/10.3390/cryst10111039

**Chicago/Turabian Style**

Farhat, Mohamed, Sebastien Guenneau, Pai-Yen Chen, and Ying Wu.
2020. "Parity-Time Symmetry and Exceptional Points for Flexural-Gravity Waves in Buoyant Thin-Plates" *Crystals* 10, no. 11: 1039.
https://doi.org/10.3390/cryst10111039