diff --git a/B_derivations/fig01/README.md b/B_derivations/fig01/README.md
new file mode 100644
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--- /dev/null
+++ b/B_derivations/fig01/README.md
@@ -0,0 +1,24 @@
+
+
+
+
+minimum and maximum wavenumber vector for circular region -- case 1
+
+Bash:
+```Bash
+$ pdflatex fig1.tex
+```
+
+minimum and maximum wavenumber vector for circular region -- case 2
+
+Bash:
+```Bash
+$ pdflatex fig2.tex
+```
+
+minimum and maximum wavenumber vector for circular region -- case 3
+
+Bash:
+```Bash
+$ pdflatex fig3.tex
+```
diff --git a/B_derivations/fig01/fig1.pdf b/B_derivations/fig01/fig1.pdf
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diff --git a/B_derivations/fig01/fig1.png b/B_derivations/fig01/fig1.png
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diff --git a/B_derivations/fig01/fig1.tex b/B_derivations/fig01/fig1.tex
new file mode 100644
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--- /dev/null
+++ b/B_derivations/fig01/fig1.tex
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+% minimum and maximum wavenumber vector for circular region -- case 1
+
+%*****************************************************************************
+% Copyright (c) 2019 Fiete Winter *
+% Institut fuer Nachrichtentechnik *
+% Universitaet Rostock *
+% Richard-Wagner-Strasse 31, 18119 Rostock, Germany *
+% *
+% This file is part of the supplementary material for Fiete Winter's *
+% PhD thesis *
+% *
+% You can redistribute the material and/or modify it under the terms of the *
+% GNU General Public License as published by the Free Software Foundation *
+% , either version 3 of the License, or (at your option) any later version. *
+% *
+% This Material is distributed in the hope that it will be useful, but *
+% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY *
+% or FITNESS FOR A PARTICULAR PURPOSE. *
+% See the GNU General Public License for more details. *
+% *
+% You should have received a copy of the GNU General Public License along *
+% with this program. If not, see . *
+% *
+% http://github.com/fietew/phd-thesis fiete.winter@gmail.com *
+%*****************************************************************************
+
+\documentclass{standalone}
+
+\usepackage{import}
+\input{../../preamble/standalone}
+
+\begin{document}
+
+\begin{tikzpicture}
+\pgfmathsetmacro{\Rvec}{1.0}
+\pgfmathsetmacro{\Rdashed}{2.5}
+
+\pgfmathsetmacro{\Rls}{5.0}
+
+\pgfmathsetmacro{\Ru}{\Rls-0.1}
+
+\pgfmathsetmacro{\phider}{0}
+\pgfmathsetmacro{\Rder}{2.0}
+
+
+\pgfmathsetmacro{\phips}{-30}
+\pgfmathsetmacro{\Rps}{2}
+
+\pgfmathsetmacro{\rl}{0.7}
+\pgfmathsetmacro{\Rl}{0.85}
+\pgfmathsetmacro{\phil}{70}
+
+\pgfmathsetmacro{\phimin}{-180}
+\pgfmathsetmacro{\phimax}{+0}
+
+\coordinate (origin) at (0,0);
+
+\fill[fill=black!30] (\phil:\rl) circle (\Rl);
+\coordinate (center) at (\phil:\rl);
+\node[circle, fill=black, inner sep=1pt, label={90:$\sfpos[subscript=h]$}]
+ at (center) {};
+
+\draw(center) -- +(180:\Rl) node[midway,above=-3pt,sloped]{\footnotesize
+$\sfR[subscript=h]$};
+
+\node[above right=0.25cm of center]{$\sfcircle[subscript=h]$};
+
+%%\coordinate (xmin) at (\phimin:\Rmax);
+%%\coordinate (xmax) at (\phimax:\Rmax);
+%
+\coordinate (kmax) at (\phimax:\Rvec);
+\coordinate (kmin) at (\phimin:\Rvec);
+
+%\draw(center) -- (xmax) node[midway,above=-3pt,sloped]{\footnotesize $R_l$};
+%\draw(center) -- (xmin);
+
+%\dotMarkRightAngle[size=4pt](center,xmax,origin);
+%\dotMarkRightAngle[size=4pt](origin,xmin,center);
+
+\draw[draw=black] ($(-70:\Rls)+(0,\Rls)$) arc (-70:-110:\Rls);
+\draw[draw=black,-latex'] ($(-75:\Ru)+(0,\Rls)$) arc (-75:-70:\Ru)
+node[above, sloped, midway] {$u$};
+
+\foreach \phils in {-105, -100, -95, -85, -80, -75}
+{
+ \node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=\phils-180,
+ anchor=cone] at ($(\phils:\Rls)+(0,\Rls)$) {};
+}
+
+\node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=90,
+ anchor=cone,
+ label={180:$\sfpossec$}] (ls) {};
+
+\coordinate[label={-90:$\sftangentsec$}] (derivative) at (\phider:\Rder);
+\draw[-latex'] (0,0) -- (derivative);
+
+\coordinate[label={90:$\sfnormalsec$}] (nls) at (0,\Rder);
+\draw[-latex'] (0,0) -- (nls);
+
+\coordinate (kmaxu) at ($(derivative)!(kmax)!(0,0)$);
+
+\draw[draw=red, thick] (kmaxu) -- (0,0)
+node[below=0.75cm, pos=0.25,anchor=south]
+{\color{red}$\sfknorm^{\max}_{\sfgreens,\sftangentsec}$};
+\draw[dashed] (kmaxu) -- (kmax);
+
+\coordinate (kminu) at ($(derivative)!(kmin)!(0,0)$);
+
+\draw[draw=blue, thick] (kminu) -- (0,0)
+node[below=0.75cm, pos=0.35, anchor=south]
+{\color{blue}$\sfknorm^{\min}_{\sfgreens,\sftangentsec}$};
+\draw[dashed] (kminu) -- (kmin);
+
+\draw[red, thick] (0,0) -- (kmax);
+\draw[blue, thick] (0,0) -- (kmin);
+%
+\end{tikzpicture}
+%
+\end{document}
diff --git a/B_derivations/fig01/fig2.pdf b/B_derivations/fig01/fig2.pdf
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index 0000000..baff74c
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diff --git a/B_derivations/fig01/fig2.png b/B_derivations/fig01/fig2.png
new file mode 100644
index 0000000..ba7bc79
Binary files /dev/null and b/B_derivations/fig01/fig2.png differ
diff --git a/B_derivations/fig01/fig2.tex b/B_derivations/fig01/fig2.tex
new file mode 100644
index 0000000..0a537e9
--- /dev/null
+++ b/B_derivations/fig01/fig2.tex
@@ -0,0 +1,127 @@
+% minimum and maximum wavenumber vector for circular region -- case 2
+
+%*****************************************************************************
+% Copyright (c) 2019 Fiete Winter *
+% Institut fuer Nachrichtentechnik *
+% Universitaet Rostock *
+% Richard-Wagner-Strasse 31, 18119 Rostock, Germany *
+% *
+% This file is part of the supplementary material for Fiete Winter's *
+% PhD thesis *
+% *
+% You can redistribute the material and/or modify it under the terms of the *
+% GNU General Public License as published by the Free Software Foundation *
+% , either version 3 of the License, or (at your option) any later version. *
+% *
+% This Material is distributed in the hope that it will be useful, but *
+% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY *
+% or FITNESS FOR A PARTICULAR PURPOSE. *
+% See the GNU General Public License for more details. *
+% *
+% You should have received a copy of the GNU General Public License along *
+% with this program. If not, see . *
+% *
+% http://github.com/fietew/phd-thesis fiete.winter@gmail.com *
+%*****************************************************************************
+
+\documentclass{standalone}
+
+\usepackage{import}
+\input{../../preamble/standalone}
+
+\begin{document}
+
+\begin{tikzpicture}
+\pgfmathsetmacro{\Rvec}{1.0}
+\pgfmathsetmacro{\Rdashed}{2.5}
+
+\pgfmathsetmacro{\Rls}{5.0}
+
+\pgfmathsetmacro{\Ru}{\Rls-0.1}
+
+\pgfmathsetmacro{\phider}{0}
+\pgfmathsetmacro{\Rder}{2.0}
+
+
+\pgfmathsetmacro{\phips}{-30}
+\pgfmathsetmacro{\Rps}{2}
+
+\pgfmathsetmacro{\rl}{1.4}
+\pgfmathsetmacro{\Rl}{0.85}
+\pgfmathsetmacro{\phil}{30}
+
+\pgfmathsetmacro{\Rmax}{sqrt(\rl*\rl - \Rl*\Rl)}
+\pgfmathsetmacro{\phimin}{\phil+asin(\Rl/\rl)}
+\pgfmathsetmacro{\phimax}{0}
+
+\coordinate (origin) at (0,0);
+
+\fill[fill=black!30] (\phil:\rl) circle (\Rl);
+\coordinate (center) at (\phil:\rl);
+\node[circle, fill=black, inner sep=1pt, label={90:$\sfpos[subscript=h]$}]
+ at (center) {};
+
+\node[above right=0.25cm of center]{$\sfcircle[subscript=h]$};
+
+\draw[dashed] (0,0) -- (\phimin:\Rdashed);
+
+\coordinate (xmin) at (\phimin:\Rmax);
+\coordinate (xmax) at (\phimax:\Rmax);
+
+\coordinate (kmax) at (\phimax:\Rvec);
+\coordinate (kmin) at (\phimin:\Rvec);
+
+\draw(center) -- (xmin) node[midway,below=-3pt,sloped]{\footnotesize $
+\sfR[subscript=h]$};
+
+\dotMarkRightAngle[size=4pt](origin,xmin,center);
+
+\draw[draw=black] ($(-70:\Rls)+(0,\Rls)$) arc (-70:-110:\Rls);
+\draw[draw=black,-latex'] ($(-75:\Ru)+(0,\Rls)$) arc (-75:-70:\Ru)
+node[above, sloped, midway] {$u$};
+
+\foreach \phils in {-105, -100, -95, -85, -80, -75}
+{
+ \node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=\phils-180,
+ anchor=cone] at ($(\phils:\Rls)+(0,\Rls)$) {};
+}
+
+\node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=90,
+ anchor=cone,
+ label={180:$\sfpossec$}] (ls) {};
+
+\coordinate[label={-90:$\sftangentsec$}] (derivative) at (\phider:\Rder);
+\draw[-latex'] (0,0) -- (derivative);
+
+\coordinate[label={90:$\sfnormalsec$}] (nls) at (0,\Rder);
+\draw[-latex'] (0,0) -- (nls);
+
+\coordinate (kmaxu) at ($(derivative)!(kmax)!(0,0)$);
+
+\draw[draw=red, thick] (kmaxu) -- (0,0)
+node[below=0.75cm, pos=0.25,anchor=south]
+{\color{red}$\sfknorm^{\max}_{\sfgreens,\sftangentsec}$};
+\draw[dashed] (kmaxu) -- (kmax);
+
+\coordinate (kminu) at ($(derivative)!(kmin)!(0,0)$);
+\coordinate (kmint) at ($(nls)!(kmin)!(0,0)$);
+
+%\draw[red, thick] (0,0) -- (kmax);
+%
+\draw[draw=blue, thick] (kmint) -- (kmin) node[above=0.75cm,
+pos=0,anchor=north] {\color{blue}$\sfknorm^{\min}_{\sfgreens,\sftangentsec}$};
+
+\draw[dashed] (kminu) -- (kmin);
+
+
+\draw[-latex',blue, thick] (origin) -- (kmin);
+%
+\end{tikzpicture}
+%
+\end{document}
diff --git a/B_derivations/fig01/fig3.pdf b/B_derivations/fig01/fig3.pdf
new file mode 100644
index 0000000..37be0e7
Binary files /dev/null and b/B_derivations/fig01/fig3.pdf differ
diff --git a/B_derivations/fig01/fig3.png b/B_derivations/fig01/fig3.png
new file mode 100644
index 0000000..5175e12
Binary files /dev/null and b/B_derivations/fig01/fig3.png differ
diff --git a/B_derivations/fig01/fig3.tex b/B_derivations/fig01/fig3.tex
new file mode 100644
index 0000000..1149c6d
--- /dev/null
+++ b/B_derivations/fig01/fig3.tex
@@ -0,0 +1,127 @@
+% minimum and maximum wavenumber vector for circular region -- case 3
+
+%*****************************************************************************
+% Copyright (c) 2019 Fiete Winter *
+% Institut fuer Nachrichtentechnik *
+% Universitaet Rostock *
+% Richard-Wagner-Strasse 31, 18119 Rostock, Germany *
+% *
+% This file is part of the supplementary material for Fiete Winter's *
+% PhD thesis *
+% *
+% You can redistribute the material and/or modify it under the terms of the *
+% GNU General Public License as published by the Free Software Foundation *
+% , either version 3 of the License, or (at your option) any later version. *
+% *
+% This Material is distributed in the hope that it will be useful, but *
+% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY *
+% or FITNESS FOR A PARTICULAR PURPOSE. *
+% See the GNU General Public License for more details. *
+% *
+% You should have received a copy of the GNU General Public License along *
+% with this program. If not, see . *
+% *
+% http://github.com/fietew/phd-thesis fiete.winter@gmail.com *
+%*****************************************************************************
+
+\documentclass{standalone}
+
+\usepackage{import}
+\input{../../preamble/standalone}
+
+\begin{document}
+
+\begin{tikzpicture}
+\pgfmathsetmacro{\Rvec}{1.0}
+\pgfmathsetmacro{\Rdashed}{2.2}
+
+\pgfmathsetmacro{\Rls}{5.0}
+
+\pgfmathsetmacro{\Ru}{\Rls-0.1}
+
+\pgfmathsetmacro{\phider}{0}
+\pgfmathsetmacro{\Rder}{2.0}
+
+
+\pgfmathsetmacro{\phips}{-30}
+\pgfmathsetmacro{\Rps}{2}
+
+\pgfmathsetmacro{\rl}{1.4}
+\pgfmathsetmacro{\Rl}{0.85}
+\pgfmathsetmacro{\phil}{80}
+
+\pgfmathsetmacro{\Rmax}{sqrt(\rl*\rl - \Rl*\Rl)}
+\pgfmathsetmacro{\phimin}{\phil+asin(\Rl/\rl)}
+\pgfmathsetmacro{\phimax}{\phil-asin(\Rl/\rl)}
+
+\coordinate (origin) at (0,0);
+
+\fill[fill=black!30] (\phil:\rl) circle (\Rl);
+\coordinate (center) at (\phil:\rl);
+\node[circle, fill=black, inner sep=1pt, label={90:$\sfpos[subscript=h]$}]
+ at (center) {};
+
+\node[above right=0.25cm of center]{$\sfcircle[subscript=h]$};
+
+\draw[dashed] (0,0) -- (\phimin:\Rdashed);
+\draw[dashed] (0,0) -- (\phimax:\Rdashed);
+
+\coordinate (xmin) at (\phimin:\Rmax);
+\coordinate (xmax) at (\phimax:\Rmax);
+
+\coordinate (kmax) at (\phimax:\Rvec);
+\coordinate (kmin) at (\phimin:\Rvec);
+
+\draw(center) -- (xmax) node[midway,above=-3pt,sloped]{\footnotesize
+$\sfR[subscript=h]$};
+\draw(center) -- (xmin);
+
+\dotMarkRightAngle[size=4pt](center,xmax,origin);
+\dotMarkRightAngle[size=4pt](origin,xmin,center);
+
+\draw[draw=black] ($(-70:\Rls)+(0,\Rls)$) arc (-70:-110:\Rls);
+\draw[draw=black,-latex'] ($(-75:\Ru)+(0,\Rls)$) arc (-75:-70:\Ru)
+node[above, sloped, midway] {$u$};
+
+\foreach \phils in {-105, -100, -95, -85, -80, -75}
+{
+ \node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=\phils-180,
+ anchor=cone] at ($(\phils:\Rls)+(0,\Rls)$) {};
+}
+
+\node[loudspeaker,
+ fill=activecolor,
+ transform shape,
+ rotate=90,
+ anchor=cone,
+ label={180:$\sfpossec$}] (ls) {};
+
+\coordinate[label={-90:$\sftangentsec$}] (derivative) at (\phider:\Rder);
+\draw[-latex'] (0,0) -- (derivative);
+
+\coordinate[label={90:$\sfnormalsec$}] (nls) at (0,\Rder);
+\draw[-latex'] (0,0) -- (nls);
+
+\coordinate (kmaxu) at ($(derivative)!(kmax)!(0,0)$);
+
+\draw[draw=red, thick] (kmaxu) -- (0,0)
+node[below=0.75cm, pos=0,anchor=south]
+{\color{red}$\sfknorm^{\max}_{\sfgreens,\sftangentsec}$};
+\draw[dashed] (kmaxu) -- (kmax);
+
+\coordinate (kminu) at ($(derivative)!(kmin)!(0,0)$);
+
+\draw[draw=blue, thick] (kminu) -- (0,0)
+node[below=0.75cm, pos=0,anchor=south]
+{\color{blue}$\sfknorm^{\min}_{\sfgreens,\sftangentsec}$};
+\draw[dashed] (kminu) -- (kmin);
+
+\draw[-latex',red, thick] (0,0) -- (kmax);
+\draw[-latex',blue, thick] (0,0) -- (kmin);
+%
+\end{tikzpicture}
+%
+\end{document}