deletions | additions       diff --git a/Absorption_of_the_laser_beam__.tex b/Absorption_of_the_laser_beam__.tex  index 5bcba49..4d1f05e 100644  --- a/Absorption_of_the_laser_beam__.tex  +++ b/Absorption_of_the_laser_beam__.tex 
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k \nabla ^2 T = \rho C \frac{\partial T}{\partial t} - \gamma I  \label{eq:eqChaleur}  \end{equation}  where $k$ is the thermal conductivity, $\rho$ is the density, $C$ is the heat capacity and $t$ is the time. Calculating the exact solution to this equation is beyond the scope of this article, but we can roughly approximate the first and second terms to be $k T / \gamma^2$ and $\rho C T / \tau$, respectively, during laser emission. Given that $k$ = 0.6 W.m$^{-1}$.K$^{-1}$ (water thermal conductivity), $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 J.kg$^{-1}$.m$^{-3}$ (water heat capacity), $\gamma^{-1} \approx$ 40 $\mu$m and $\tau$ = 10 ns, the first term is negligible compared to the second one; thus, the equation \ref{eq:eqChaleur} can be simplified as:  \begin{equation}  \frac{\partial T}{\partial t} = \frac{\gamma I}{\rho C} = \frac{\gamma}{\rho C S} \frac{dE}{dt}  \label{eq:eqChaleurApprox}  \end{equation} one.  Substituting low-energy experimental parameters ($E$ = 10 mJ, $S$ = 20 mm$^{2}$) leads to a maximum increase in temperature of 3 K, which produces a local dilatation of the medium. The induced displacements can then generate shear waves, which constitutes the \textit{thermoelastic regime}. To estimate the initial displacement amplitude in this regime, we assumed the medium as homogeneous and isotropic. As the depth of absorption (about 40 $\mu$m) is 100 times smaller than the beam diameter (5 mm), we discarded any boundary effects. The stress, $\sigma_{zz}$, is the sum of the axial strain component and the thermal expansion component \cite{scruby1990laser}:  \begin{equation}