deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 01b592d..0a57cb9 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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\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 leads  to a maximum increase of temperature of 3 K. Thislocal  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 the depth of absorption (about 40 $\mu$m) is hundred times smaller than the beam diameter (5 mm), we 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} 
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\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. 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 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 transverse  dilatationalong X and Y axis  which lead leads  to stronger displacements than along Z. We modeled thus the thermoelastic regime in 2D  as two opposite forces during 100 $\mu$s 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 $H_y^{thermo}(x,y,z,t)$. $H_y^{thermo}(y,z,t)$ (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}  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} \delta(t-\tau) d\tau}  \label{eq:Gyz}  \end{equation}  where $\theta$ is (r,$\theta$) are  the angle between the applied force and coordinates of  the considered point (r,$\theta$,z), with regards to the force location and direction,  $c_p$ and $c_s$ the compression and shear wave speed respectively, $\tau$ the time, time and  $\delta$a  Dirac distribution and $f_{NF}(t)$ representing near-field effects. distributions.  The three terms correspond respectively to the far-field compression wave, the far-field shear wave and the near-field component. Displacements along Z are then equal to the convolution between $H_y^{thermo} (x,y,z,t)$ (y,z,t)$  with $G_{yz}$. Using a medium density $\rho$ of 1000 kg.m$^{-3}$, a compression wave speed of 1500 m.s$^{-1}$ and a shear wave speed of 5.5 m.s$^{-1}$, we calculated the displacements along space and time. Figure \ref{figGreen}-(A) represents resulting displacement maps along Z axis, 1.0, 1.5, 2.0, 2.5 and 3.0 ms after force application. The normalized displacement maps present many similarities with the low-energy experimental results of the Figure \ref{Figure2}, with a initial central displacement directed outside the medium and the propagation of three half cycles.