deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index e262eb5..18a1e99 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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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 100 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm) \cite{Davies_1993}. 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} f(t-\frac{r}{c_p}) - \frac{\sin \theta \cos \theta}{4\pi \rho c_s^2 r} f(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}} f_{NF}(t) dt}  \label{eq:Gyz}  \end{equation}  where $\theta$ is the angle between the applied force and the considered point (r,$\theta$,z), $c_p$ and $c_s$ the compression and shear wave speed respectively, $\tau$ the time and $f_{NF}$ $f_{NF}(t)$  representing near-field effects. The three terms correspond respectively to the far-field compression wave, the far-field shear wave and the near-field component. Displacements are then computed by convoluting $G_{yz}$ with time and spatial extent of the force $H_y$ : $u_z=G_{yz}*H_y$.