deletions | additions       diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex  index 5a98280..51e88c6 100644  --- a/The_absorption_of_the_laser__.tex  +++ b/The_absorption_of_the_laser__.tex 
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u_z = \frac{3 \alpha E}{\rho C S}  \label{eq:deplThermoApprox}  \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.In a 3D model, displacements along X and Y axis would also occurs, as the local expansion acts as dipolar forces parallel to the surface, but calculus is beyond the scope of this article.  With stronger or more focused laser pulse, the local increase of temperature could also vaporize 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 part 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 [REF]. Propagation as a shear wave was calculated using Green operators $G_{pe}$ ("pe" for "perpendicular") as calculated by Aki Richards \cite{aki1980quantitative}:  \begin{equation}  G_{pe} (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), $\rho$ the medium density, $c_p$ and $c_s$ the compression and shear wave speed respectively, $\delta_S$ and $\delta_P$ Dirac distribution indicating  the surface position  of the medium. However in our case, the temperature did not increase enough (about 60 K) compression and shear waves along space and time, $\tau$ the time and $\delta_{NF}$ representing near-field effects. The three terms correspond respectively  tovaporize  the medium. far-field compression wave, the far-field shear wave and the near-field component.  Displacement can then be computed by convoluting $G_y$ and $G_z$ with time and spatial extent of the force:  \begin{equation}  u_z = G_{pe} * H(x,y,z,t)  \label{eq:akirichards2}  \end{equation}  where H is a 4-D matrix of the applied force 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}.