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Pol Grasland-Mongrain edited Simu disp maps.tex
over 8 years ago
Commit id: 2d09fdcab195234634961389ffea3e5c52919c6a
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diff --git a/Simu disp maps.tex b/Simu disp maps.tex
index 57f1b17..ac75e24 100644
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To calculate the propagation of the displacement along space and time, we modeled the ablation regime as a point force directed along Z direction with a depth of 40 $\mu$m during 100 $\mu$s and of constant value from -2.5 to 2.5 mm. The magnitude of the force is stored in a matrix $H_z^{abla}(y,z,t)$. Propagation as a shear wave was calculated using Green operators $G_{zz}$ \cite{aki1980quantitative}:
\begin{equation}
G_{zz} G_{zz}(r,$\theta$,t) = \frac{\cos^2 \theta}{4\pi \rho c_p^2 r} \delta(t-\frac{r}{c_p}) + \frac{\sin^2 \theta}{4\pi \rho c_s^2 r}
\delta(t-\frac{r}{c_s})\\+ \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau)d\tau} \delta(t-\frac{r}{c_s})
\label{eq:Gzz}
\end{equation}
\begin{equation}
+ \frac{3\cos^2 \theta-1}{4\pi \rho r^3} \int\limits_{r/c_p}^{r/c_s}{\tau \delta(t-\tau)d\tau}
\label{eq:Gzz2}
\end{equation}
with same notations as in equation \ref{eq:Gyz}.
Displacements can again be computed by convoluting the applied force 4-D matrix $H_z^{abla}(y,z,t)$ with $G_{zz}$.