deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index bd764b2..61c7749 100644  --- a/Simu disp maps.tex  +++ b/Simu disp maps.tex 
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%This affirmation is confirmed by the observation of a disk of paler color of the same size as the beam diameter at the impact location of the laser on the phantom, which could correspond to a vaporization of a fraction of the material.  %  This physical phenomenon was then modeled numerically. The thermal dilatation was simulated by calculating the displacement created by two opposite forces decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm), with a depth of 100 $\mu$m. Propagation as a shear wave was calculated using Green operator \cite{aki1980quantitative}, using a medium density $\rho$ of 1000 kg.m$^{-3}$, a compression wave speed of 1500 m.s$^{-1}$ and a shear wave speed of 4 m.s$^{-1}$. Results are shown on Figure \ref{Figure3} which represents displacement maps along Y and 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}. This physical phenomenon was then modeled numerically. The vaporization was modeled as a point force directed along Z direction with a depth of 50 $\mu$m and increasing linearly from -2.5 to 0 mm and decreasing symmetrically from 0 to 2.5 mm, to simulate an approximate Gaussian shape. Propagation as a shear wave was calculated using Green operator as calculated by Aki Richards \cite{aki1980quantitative}:  \begin{equation}  u_z = \frac{1}{4\pi \rho r c_p^2} \cos(\beta)^2 \delta_P + \frac{1}{4\pi r \rho c_s^2} \sin(\beta)^2 \delta_S + \frac{1}{4\pi \rho r^3 c_p^2} (3\cos(\beta)^2-1) \tau Rect.  \label{eq:akirichards}  \end{equation}  It used a medium density $\rho$ of 1000 kg.m$^{-3}$, a compression wave speed of 1500 m.s$^{-1}$ and a shear wave speed of 5.75 m.s$^{-1}$. Results are shown on Figure \ref{Figure3} which represents displacement maps along Y and 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}.