this is for holding javascript data
Pol Grasland-Mongrain edited The_absorption_of_the_laser__.tex
over 8 years ago
Commit id: 01c9841faae7fbf4b1a5e5c655ec9a89cc6238ac
deletions | additions
diff --git a/The_absorption_of_the_laser__.tex b/The_absorption_of_the_laser__.tex
index e601c2f..e3a0d2c 100644
--- a/The_absorption_of_the_laser__.tex
+++ b/The_absorption_of_the_laser__.tex
...
\end{equation}
Substituting same experimental parameters as previously and $\alpha$ = 70.10$^{-6}$ K$^{-1}$ (water linear thermal dilatation coefficient), we obtain a displacement $u_z$= 0.025 $\mu$m. While slightly higher, this value is in good agreement with experimental displacement (about 0.02 $\mu$m). Note that the theory supposed that the displacement is directed outside the medium, which is seen in the experimental images in the middle of the beam, where the 1D model is the most valid.
To calculate the propagation of the displacement along space and time, we have to take into account the transverse dilatation which leads to stronger displacements than along Z. We modeled thus the thermoelastic regime in 2D as two opposite forces
during 100 $\mu$s directed along Y axis with a depth of 40 $\mu$m and decreasing linearly from 2.5 to 0 mm (respectively -2.5 to 0 mm) \cite{Davies_1993}. The magnitude of the force along space and time is stored in a matrix $H_y^{thermo}(y,z,t)$ (note that X and Z components of the force are supposed null). Propagation as a shear wave along Z axis was calculated using Green operators $G_{yz}$ as calculated by Aki Richards \cite{aki1980quantitative}:
\begin{equation}
G_{yz}(r,\theta,t) = \frac{\cos \theta \sin \theta}{4\pi \rho c_p^2 r} \delta(t-\frac{r}{c_p}) - \frac{\sin \theta \cos \theta}{4\pi \rho c_s^2 r} \delta(t-\frac{r}{c_s})
\label{eq:Gyz}