# The Influences of the Hyperbolic Two-Temperatures Theory on Waves Propagation in a Semiconductor Material Containing Spherical Cavity

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## Abstract

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## 1. Introduction

## 2. Basic Equations

## 3. Initial and Boundary Conditions

## 4. Laplace Transform

## 5. Numerical Result and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$t$ | the time, $\mathrm{s}$ |

${T}_{o}$ | the reference temperature, $\mathrm{K}$ |

$T={T}^{*}-{T}_{o}$, ${T}^{*}$ | the temperature variation, $\mathrm{K}$ |

${u}_{i}$ | the displacement components, $\mathrm{m}$ |

$\rho $ | the material density, $\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ |

$k=\partial {n}_{o}/\partial T$ | the coupling parameter of thermal activation, ${\mathrm{m}}^{-3}\xb7{\mathrm{K}}^{-1}$ |

${c}_{e}$ | the specific heating at constant strain, $\mathrm{J}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}$ |

${\gamma}_{n}=\left(2\mu +3\lambda \right){d}_{n}$, ${\mathrm{d}}_{\mathrm{n}}$ | the coefficient of electronic deformation, ${\mathrm{m}}^{3}$ |

$\tau $ | the lifetime of photogenerated carrier, $\mathrm{s}$ |

${\gamma}_{t}=\left(3\lambda +2\mu \right){\alpha}_{t},{\alpha}_{t}$ | the linear thermal expansion coefficient, ${\mathrm{K}}^{-1}$ |

$N=n-{n}_{o}$, ${n}_{o}$ | the carrier concentration at equilibrium, ${\mathrm{m}}^{-3}$ |

${\sigma}_{ij}$ | the stress components, $\mathrm{N}\xb7{\mathrm{m}}^{-2}$ |

$K$ | the thermal conductivity, $\mathrm{W}\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1}$ |

$\lambda ,\mu $ | the Lame’s constants, $\mathrm{N}\xb7{\mathrm{m}}^{-2}$ |

${D}_{e}$ | the coefficient of carrier diffusions, ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ |

${\tau}_{o}$ | the thermal relaxation time, $\mathrm{s}$ |

${q}_{o}$ | constant, $\mathrm{W}\xb7{\mathrm{m}}^{-2}$ |

${t}_{p}$ | the characteristic time of pulsing heat flux, $\mathrm{s}$ |

${s}_{b}$ | the recombination speed on the surface, $\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

$R$ | the internal redial of cavity, $\mathrm{m}$ |

$a$ | the parameter of two-temperature model, ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-2}$ |

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**MDPI and ACS Style**

Hobiny, A.; Abbas, I.; Marin, M.
The Influences of the Hyperbolic Two-Temperatures Theory on Waves Propagation in a Semiconductor Material Containing Spherical Cavity. *Mathematics* **2022**, *10*, 121.
https://doi.org/10.3390/math10010121

**AMA Style**

Hobiny A, Abbas I, Marin M.
The Influences of the Hyperbolic Two-Temperatures Theory on Waves Propagation in a Semiconductor Material Containing Spherical Cavity. *Mathematics*. 2022; 10(1):121.
https://doi.org/10.3390/math10010121

**Chicago/Turabian Style**

Hobiny, Aatef, Ibrahim Abbas, and Marin Marin.
2022. "The Influences of the Hyperbolic Two-Temperatures Theory on Waves Propagation in a Semiconductor Material Containing Spherical Cavity" *Mathematics* 10, no. 1: 121.
https://doi.org/10.3390/math10010121