Thesis: Beacon: moved tProp into RadioInter chapter

documents/thesis/chapters/beacon_discipline.tex

@@ -56,49 +56,52 @@ Before going in-depth on the synchronisation using either of such beacons, the s
 % <<<<
 % time delay
 An in-band solution for synchronising the detectors is effectively a reversal of the method of interferometry in Section~\ref{sec:interferometry}.
-The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal.
+The distance between the transmitter $T$ and the antenna $A_i$ incur a time delay $(\tProp)_i$ caused by the finite propagation speed of the radio signal (see \eqref{eq:propagation_delay}).
 \\
 
-Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
+\Todo{continuity}
-In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
-However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
 
-%\begin{figure}%<<<
+%Since the signal is an electromagnetic wave, its instantaneous velocity $v$ depends solely on the refractive index~$n$ of the medium as $v = \frac{c}{n}$.
-%	\centering
+%In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through and the frequencies of the signal.
-%	\begin{subfigure}{0.49\textwidth}%<<<
+%However, in many cases, the refractive index can be taken constant over the trajectory to simplify models.
+%
+%%\begin{figure}%<<<
 %%	\centering
-%		\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
+%%	\begin{subfigure}{0.49\textwidth}%<<<
-%		\caption{
+%%		%\centering
-%			Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
+%%		\includegraphics[width=\textwidth,height=\textheight,keepaspectratio]{beacon/antenna_setup_two.pdf}
-%			Each distance incurs a specific time delay $(\tProp)_i$.
+%%		\caption{
-%			The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
+%%			Schematic of two antennas ($A_i$) at different distances from a transmitter ($T$).
-%			\protect \Todo{use `real' transmitter and radio for schematic}
+%%			Each distance incurs a specific time delay $(\tProp)_i$.
-%		}
+%%			The maximum time delay difference for these antennas is proportional to the baseline distance (green line).
-%		\label{fig:beacon_spatial_setup}
+%%			\protect \Todo{use `real' transmitter and radio for schematic}
-%	\end{subfigure}%>>>
+%%		}
-%	\begin{subfigure}{0.49\textwidth}%<<<
+%%		\label{fig:beacon_spatial_setup}
-%		%\centering
+%%	\end{subfigure}%>>>
-%		\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
+%%	\begin{subfigure}{0.49\textwidth}%<<<
-%		\caption{
+%%		%\centering
-%			From Ref~\cite{PierreAuger:2015aqe}.
+%%		\includegraphics[width=\textwidth]{beacon/auger/1512.02216.figure2.beacon_beat.png}
-%			The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
+%%		\caption{
-%			The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
+%%			From Ref~\cite{PierreAuger:2015aqe}.
-%			With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
+%%			The beacon signal that the \gls{Auger} has employed in \gls{AERA}.
-%			\protect \Todo{incorporate into text}
+%%			The beating between 4 frequencies gives a total period of $1.1\us$ (indicated by the arrows).
-%		}
+%%			With a synchronisation uncertainty below $100\ns$ from the \gls{GNSS}, it fully resolves the period degeneracy.
-%		\label{fig:beacon:pa}
+%%			\protect \Todo{incorporate into text}
-%	\end{subfigure}%>>>
+%%		}
-%\end{figure}%>>>
+%%		\label{fig:beacon:pa}
-
+%%	\end{subfigure}%>>>
-As such, the time delay due to the propagation from the transmitter to an antenna can be written as
+%%\end{figure}%>>>
-\begin{equation}\label{eq:propagation_delay}% <<<
+%
-	\phantom{,}
+%As such, the time delay due to the propagation from the transmitter to an antenna can be written as
-	(\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
+%\begin{equation}\label{eq:propagation_delay}% <<<
-	,
+%	\phantom{,}
-\end{equation}% >>>
+%	(\tProp)_i = \frac{ \left|{ \vec{x}_{T} - \vec{x}_{A_i} }\right| }{c} n_\mathrm{eff}
-where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
+%	,
-\\
+%\end{equation}% >>>
+%where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
+%\\
 
+\Todo{continuity}
 If the time of emitting the signal at the transmitter $\tTrueEmit$ is known, this allows to directly synchronise the transmitter and an antenna since
 \begin{equation}\label{eq:transmitter2antenna_t0}%<<<
 	\phantom{,}