deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 1a4b7ad..01b592d 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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\end{equation}  where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ = 10 ns the laser emission duration and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately equal here to 80 nm. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so that equation \ref{eq:eqChaleur} can be simplified as:  \begin{equation}  \frac{\partial T}{\partial t} = \frac{\gamma I}{\rho C} = \frac{1}{\rho \frac{\gamma}{\rho  C S} \frac{dE}{dt} \label{eq:eqChaleurApprox}  \end{equation}  Substituting low-energy experimental parameters ($\gamma^{-1} \approx$ 40 $\mu$m$^{-1}$, $S$ = 20 mm$^{2}$, $E$ = 10 mJ, $\rho$ = 1000 kg.m$^{-3}$, $C$ = 4180 J.kg$^{-1}$.K$^{-1}$) lead to a maximum increase of temperature of 3 K. This local increase of temperature gives rise to a local dilatation of the medium. The induced displacements can then lead generate  to shear waves: this constitutes the \textit{thermoelastic regime}. To describe physically  this regime, we supposed the medium as homogeneous and isotropic; As isotropic. And as  the depth of absorption (about 50 40  $\mu$m) is hundred times smaller than the beam diameter (5 mm), wealso  adopted a 1D model. 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 $u_z$= 0.025 $\mu$m. This While slightly higher, this  value is slightly higher than the in good agreement with  experimental displacement (about 0.02 $\mu$m). Note that as shown theoretically, the theory supposed that  the displacement is directed outside the medium, which is confirmed seen  in the experimental images. images in the middle of the beam, where the 1D model is the most valid.  To calculate the propagation of the displacement along space and time, we have to take into account the dilatation along X and Y axis which lead to stronger displacements than along Z. We modeled thus the thermoelastic regime as two opposite forces during 100 $\mu$s directed along Y axis with a depth of 50 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 $H_y^{thermo}(x,y,z,t)$. Propagation as a shear wave along Z axis was calculated using Green operators $G_{yz}$ as calculated by Aki Richards \cite{aki1980quantitative}: \begin{equation}  G_{yz} (r,\theta,z,t)= \frac{\cos \theta \sin \theta}{4\pi \rho c_p^2 r} \delta(t-\frac{r}{c_p}) - \frac{\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta(t-\frac{r}{c_s}) + \frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau f_{NF}(t) dt}  \label{eq:Gyz}