deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 0abb3fc..9072c5b 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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\end{equation}  Substituting $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 0.2 J, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= 0.5 $\mu$m. This value is slightly smaller than the experimental displacement (about 3 $\mu$m). This local displacement can lead to shear waves because of the limited size of the source.  To verify this physical model, a numerical calculation was performed. The thermal dilatation was modeled as two opposite forces directed along Y direction with a depth of 50 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm), to simulate an approximate Gaussian shape \cite{Davies_1993}. Propagation as a shear wave along Z axis  was calculated using Green operators $G_{pe}$ ("pe" for "perpendicular") $G_{yz}$  as calculated by Aki Richards \cite{aki1980quantitative}: \begin{equation}  G_{pe} G_{yz}  (r,\theta,z)= \frac{\cos \beta \sin \theta}{4\pi \rho c_p^2 r} \delta_P + \frac{-\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos \theta \sin \beta}{4\pi \rho r^3} \int_{r/c_p}^{r/c_s}{\tau \delta_{NF}} \label{eq:akirichards}  \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, $\delta_S$ and $\delta_P$ Dirac distribution indicating the position of the compression and shear waves along space and time, $\tau$ the time and $\delta_{NF}$ 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.  Displacement can then be computed by convoluting $G_y$ and $G_z$ $G_yz$  with time and spatial extent of the force: \begin{equation}  u_z = G_{pe} G_{yz}  * H(x,y,z,t) H_y(x,y,z,t)  \label{eq:akirichards2}  \end{equation}  where H $H_y$  is a 4-D matrix of the applied force (directed along Y)  along space and time. Results are shown on Figure \ref{figGreenThermo} which represents displacement maps between each frame along Z axis 0.8, 1.6, 2.4, 3.2 and 4.0 ms after force application. The displacement maps present many similarities with the experimental results of the Figure \ref{Figure2}.