deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 9370e84..b4f9be7 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\begin{equation}  u_z = \frac{3\alpha E}{\rho C S}  \end{equation}  Taking as an order of magnitude $\alpha = ???$, $E$ = 200 mJ, $\rho$ = 1000 kg.m$^{-3}$ (water density), $C$ = 4180 kg.m$^{-3}$ (water calorific capacity) and $S$ = 20 mm$^2$, we obtain a displacement $u_z$= ??? $\mu$m. This value is still smaller than the typical displacement resolution with ultrasound, of a few micrometers. However, displacements along X and Y axis are higher, as $\delta V = u_x u_y u_z = \frac{3\alpha E}{\rho C}$ where displacement along $z$ is known, so that displacement along $x$ (or, equivalently, along $z$) is equal to ??? $\mu$m.  In the ablative regime, the local increase of temperature is so high that the surface of the medium melts. This phenomenon creates a point-like force $f$ in the medium, given by \cite{scruby1990laser}:  \begin{equation}  f = \frac{S}{\rho}\frac{I^2}{(L+C(T_V-T_0))^2}  \label{eq:ablation}  \end{equation}  where $L$ is the lastent latent  heat required to vaporize the solid,$T_V$ and  $T_0$ and $T_V$  the initial and vaporization temperatures. In both cases, absorption of the laser by the phantom leads to a local displacement which can propagate as elastic waves in the medium. To observe the shear waves, the medium was scanned with a 5 MHz ultrasonic probe made of 128 elements linked to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was used in ultrafast mode \cite{bercoff2004supersonic}, acquiring 1500 ultrasound images per second. Due to the presence of graphite particles, the medium presented a speckle pattern on the ultrasound image. Tracking the speckle spots with an optical flow technique (Lucas-Kanade method) allowed to compute one component of the displacement in the medium (Z-displacement or Y-displacement, depending on the position of the probe on the medium). The laser beam was triggered 10 ms after the first ultrasound acquisition, $t$ = 0 ms being defined as the laser emission.   
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