deletions | additions       diff --git a/Let_s_describe_the_phenomenon__.tex b/Let_s_describe_the_phenomenon__.tex  index 668720a..6953d83 100644  --- a/Let_s_describe_the_phenomenon__.tex  +++ b/Let_s_describe_the_phenomenon__.tex 
...
\frac{\partial T}{\partial t} = \kappa \nabla ^2 T + \frac{q}{\rho C}  \label{eq:eqChaleur}  \end{equation}  where $\rho$ is the density, $C$ the heat capacity and $\kappa$ the thermal diffusivity. The thermal diffusion path, equal to $\sqrt{4\kappa \tau}$, with $\tau$ = 10 ns the laser emission duration and $\kappa$ = 1.43 10$^{-7}$ m$^2$.s$^{-1}$ for water \cite{Blumm_2003}, is approximately equal here to 80 nm. As $\gamma^{-1} \gg \sqrt{4\kappa t}$, propagation of heat is negligible during laser emission, so that equation \ref{eq:eqChaleur} can be simplified as $\frac{\partial as:  \begin{equation}  \frac{\partial  T}{\partial t} = \frac{q}{\rho C}$. C}  \label{eq:eqChaleurApprox}  \end{equation}  Substituting the experimental parameters lead to a maximum increase of temperature of 12 K. This local increase of temperature can lead a local dilatation of the medium occurs. We suppose that the medium is homogeneous and isotropic, and as the depth of absorption is small compared to the beam diameter, we adopt a 1D model. The stress $\sigma_{zz}$ is the sum between the axial strain component and the thermal expansion component \cite{scruby1990laser}:  \begin{equation}  \sigma_{zz} = (\lambda + 2 \mu) \frac{\partial u_z}{\partial z} - 3(\lambda + \frac{2}{3}\mu) \frac{\alpha E}{\rho C S \zeta}  \label{eq:stressThermo}  \end{equation}  where $\lambda + 2 \mu$ is the P-wave modulus and $\lambda + \frac{2}{3}\mu$ the bulk modulus with $\lambda$ and $\mu$ respectively the first and second Lamé's coefficient, $\alpha$ is the thermal dilatation coefficient and $\zeta$ the average depth of laser beam absorption. In the absence of external constraints normal to the surface, the stress across the surface must be zero, i.e. $\sigma_{zz} (z=0) = 0$, so that equation \ref{eq:stressThermo} can be integrated, giving a displacement $u_z$  from the surface $u_z = \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho C S \zeta}$. As in a biological soft tissues, $\mu \ll \lambda$, the displacement can be approximated as: surface:  \begin{equation}  u_z = \frac{3 \alpha E}{\rho \frac{(3\lambda + 2\mu)}{(\lambda + 2\mu)} \frac{\alpha E \zeta}{\rho  C S} S \zeta}  \label{eq:deplThermo}  \end{equation}  . As in a biological soft tissues, $\mu \ll \lambda$, the displacement can be approximated as:  \begin{equation}  u_z = \frac{3 \alpha E}{\rho C S}  \label{eq:deplThermoApprox}  \end{equation}  Taking as an order of magnitude $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), $E$ = 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 $u_z$= 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.  If the laser beam is focused, the local increase of temperature could also vaporize a part of the surface of the medium.