deletions | additions       diff --git a/Simu disp maps.tex b/Simu disp maps.tex  index 2723fdd..d60d3a0 100644  --- a/Simu disp maps.tex  +++ b/Simu disp maps.tex 
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  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 operators $G_y$ and $G_z$ as calculated by Aki Richards \cite{aki1980quantitative}:  \begin{equation}  G_y = \frac{\cos \beta \sin \beta}{4\pi \rho c_p^2 r} \delta_P + \frac{-\sin \beta \cos \beta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos \beta \sin \beta}{4\pi \rho r^3} \tau Rect\\ \int_{r/\alpha}^{r_\beta}{\tau \delta_{NF}}\\  G_z = \frac{\cos^2 \beta}{4\pi \rho c_p^2 r} \delta_P + \frac{\sin^2 \beta}{4\pi \rho c_s^2 r} \delta_S + \frac{3\cos^2 \beta-1}{4\pi \rho r^3} \tau Rect {\tau \delta_{NF}}  \label{eq:akirichards}  \end{equation}  where $\beta$ is the angle between the applied force and the axis y or 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 position of the compression and shear waves along space and time, $\tau$ the time and $Rect$ $\delta_NF$ representing  the near-field term. effects.  The three terms correspond respectively to the far-field compression wave, the far-field shear wave and the near-field term. Displacement can then be computed by convoluting $G_y$ and $G_z$ with time and spatial extent of the force:  \begin{equation}