deletions | additions       diff --git a/Absorption_of_the_laser_beam__.tex b/Absorption_of_the_laser_beam__.tex  index de60569..bd81b74 100644  --- a/Absorption_of_the_laser_beam__.tex  +++ b/Absorption_of_the_laser_beam__.tex 
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To calculate the propagation of the displacements as shear waves, we must first consider the transverse dilatation, which leads to stronger displacements than those occurring along the Z axis. We thus modeled the thermoelastic regime in 2D as two opposite forces directed along the Y axis with a depth of 40 $\mu$m and with an amplitude decreasing linearly respectively from 2.5 to 0 mm and from -2.5 to 0 mm \cite{Davies_1993}. The magnitude of the force along space and time is stored in a matrix, $H_y^{thermo}(y,z,t)$. Displacements along the Z axis are then equal to the convolution between $H_y^{thermo} (y,z,t)$ and $G_{yz}$ \cite{aki1980quantitative}:  \begin{equation}  G_{yz} = \frac{\cos \theta \sin \theta}{4\pi \rho r} (\frac{1}{c_p^2} \delta(t-\frac{r}{c_p}) \delta_P  - \frac{1}{c_s^2} \delta(t-\frac{r}{c_s}) \delta_S  +\frac{3}{r^2} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau) \delta_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, $\delta_P = \delta(t-\frac{r}{c_p})$, $\delta_S = \delta(t-\frac{r}{c_s})$, $\delta_\tau = \delta(t-\tau)$,  $c_p$ and $c_s$ are the compression and shear wave speed respectively, $\tau$ is the time, and $\delta$ is a Dirac distribution. The three terms of the equation correspond respectively to the far-field compression wave, the far-field shear wave, and the near-field component. Using constants $\rho$ = 1000 kg.m$^{-3}$, $c_p$ = 1500 m.s$^{-1}$ and $c_s$ = 5.5 m.s$^{-1}$ (similarly to experimental measurement), displacement maps along the Z axis were calculated 1.0, 1.5, 2.0, 2.5, and 3.0 ms after force application, as illustrated in Figure \ref{figGreen}-(A). The normalized displacement maps present many similarities to the experimental results displayed in Figure \ref{figElastoPVA}-(A), with an initial central displacement directed towards the outside of the medium, and a propagation of three half cycles.