deletions | additions       diff --git a/The_magnetic_field_w.tex b/The_magnetic_field_w.tex  index 043f6cb..1c8cfc5 100644  --- a/The_magnetic_field_w.tex  +++ b/The_magnetic_field_w.tex 
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Each sample was observed with a 5 MHz ultrasonic probe made of 128 elements (ATL L7-4, Philips, Amsterdam, Netherlands) coupled to a Verasonics scanner (Verasonics V-1, Redmond, WA, USA). The probe was in contact with the sample with an ultrasound coupling gel but was fixed on a third independent support. It was used in ultrafast mode \cite{bercoff2004supersonic}, to acquire 1000 frames per second using plane waves and Stolt's fk migration algorithm \cite{garcia2013stolt}. The Z component of the displacement in the sample was observed by performing cross-correlations between radiofrequency images with a speckle-tracking technique \cite{montagnon2012real}. The TMS electrical current burst was triggered 10 ms after the beginning of the ultrasound frame acquisition, and the time $t$ = 0 ms was defined as the trigger emission.  \section{Simulation study}  Additionally to the experiments, a three dimensional simulation of the experiments was performed using Matlab (Matlab 2010, The MathWorks, Natick, MA, USA). The simulation was performed by (1) calculating the electrical current induced by the coil, (2) simulating the magnetic field created by the permanent magnet, (3) calculating the resulting Lorentz force inside the medium, and finally (4) calculating the displacement along space and time due to the Lorentz force.   Using Equation \ref{Equation1} with two 75 mm diameter coils crossed by an half cycle of 0.4 ms electrical current, representing the butterfly coil used in the experiment, and Ohm's law, the electrical current $\mathbf{j}$ was estimated in a 4x8x8 sample with an electrical conductivity $\sigma$ = 1 S.m$^{-1}$ inside and 0 outside. No border effects have been taken into account. Electrical current in a XY plane at a depth of 2 cm is illustrated in Figure \ref{Figure3}-(A).  A finite element software (Finite Element Magnetic Method \cite{FEMM}) was used to produce a two dimensional simulation of the magnetic field $\mathbf{B}$. The magnetic field was supposed to be approximately constant in the sample along the Y axis. The software simulated a N48 NdFeB permanent magnet of 5x5 cm$^2$ placed in a 19x27 cm$^2$ surface of air, and resulting magnetic field is illustrated in Figure \ref{Figure3}-(B).