deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 054c46f..a20443d 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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Substituting low-energy experimental parameters ($E$ = 10 mJ, $S$ = 20 mm$^{2}$) leads to a maximum increase of temperature of 3 K. This increase of temperature gives rise to a local dilatation of the medium. The induced displacements can then generate to shear waves: this constitutes the \textit{thermoelastic regime}.  To describe physically this regime, we supposed the medium as homogeneous and isotropic. And as As  the depth of absorption (about 40 $\mu$m) is hundred times smaller than the beam diameter (5 mm), we adopted a 1D model. calculated the initial displacement in the middle of the beam and we discarded any boundaries effect.  The stress $\sigma_{zz}$ is the sum between the axial strain component and the thermal expansion component \cite{scruby1990laser}: \begin{equation}  \sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \alpha \frac{ E}{\rho C S \zeta}  \label{eq:stressThermo} 
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u_z = \frac{3 \alpha E}{\rho C S}  \label{eq:deplThermoApprox}  \end{equation}  Substituting same experimental parameters as previously and $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), we obtain a displacement $u_z$= $u_z$ =  0.025 $\mu$m. While slightly higher, this value is in good agreement with experimental displacement (about 0.02 $\mu$m). Note that the theory supposed that the displacement is directed outside the medium, which is seen in the experimental images in the middle of the beam, where as indicated by  the 1D model is white circle in  the most valid. figure \ref{figElastoPVA}.  To calculate the propagation of the displacement along space and time, we have to take into account the transverse dilatation which leads to stronger displacements than along Z. We modeled thus the thermoelastic regime in 2D as two opposite forces directed along Y axis with a depth of 40 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm) \cite{Davies_1993}. The magnitude of the force along space and time is stored in a matrix $H_y^{thermo}(y,z,t)$ (note that X and Z components of the force are supposed null). Propagation as a shear wave along Z axis was calculated using Green operators $G_{yz}$ as calculated by Aki Richards \cite{aki1980quantitative}:  \begin{equation}