deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index a999351..e82a716 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 estimate the initial displacement amplitude in  this regime, we supposed the medium as homogeneous and isotropic. As the depth of absorption (about 40 $\mu$m) is hundred times smaller than the beam diameter (5 mm), wecalculated 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 $u_z$ = 0.025 $\mu$m. While slightly higher, this This  value is in good agreement with very close to the measured  experimental displacement (about 0.02 $\mu$m). Note that the theory supposed that the displacement is both theoretical and experimental central displacements are  directed outside the medium, in agreement with the experimental displacements medium (see arrow  in themiddle of the beam, as indicated by the  white circle in the Figure \ref{figElastoPVA}-(A). \ref{figElastoPVA}-(A)).  To calculate the propagation of the displacement along space and time, displacements as shear waves,  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 $H_y^{thermo}(y,z,t)$. Displacements  along Z axis was calculated using Green operators are then equal to the convolution between $H_y^{thermo} (y,z,t)$ with  $G_{yz}$as calculated by Aki Richards  \cite{aki1980quantitative}: \begin{equation}  G_{yz}(r,\theta,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})  \label{eq:Gyz} 
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+\frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau) d\tau}  \label{eq:Gyz2}  \end{equation}  where (r,$\theta$) are the coordinates of the considered point with regards to the force location and direction, $c_p$ and $c_s$ the compression and shear wave speed respectively, $\tau$ the time and $\delta$ Dirac 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 $H_y^{thermo} (y,z,t)$ 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. application, using $\rho$ = 1000 kg.m$^{-3}$, $c_p$ = 1500 m.s$^{-1}$ and $c_s$ = 5.5 m.s$^{-1}$.  The normalized displacement maps present many similarities with the experimental results in the Figure \ref{figElastoPVA}-(A), with a initial central displacement directed outside the medium and the propagation of three half cycles.