deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 0a57cb9..ba99687 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 transverse dilatation which 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 $H_y^{thermo}(y,z,t)$ (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)= (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}) + +\\  \frac{3\cos \theta \sin \theta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta(t-\tau) d\tau} \label{eq:Gyz}  \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.