Wavelength for de Broglie waves

Question

Starting from this old question How to draw a sine wave on a circular path in tikz I have modified the source code according for my interest:

    \documentclass[tikz]{standalone}
    \usepackage{pgfplots}
    \usepackage{amsmath}
    \begin{document}

    \foreach \n in{3,4}{%
    \begin{tikzpicture}
    \begin{axis}[axis equal,
      xmin=-3,xmax=3,
      ymin=-3,ymax=3,
      axis lines=none]
    \addplot[samples=400,domain=0:2*pi,very thick,red] ({(2+.3*cos(deg(\n*x)))*cos(deg(x))},{(2+.3*cos(deg(\n*x)))*sin(deg(x))});
    \addplot[samples=40,domain=0:2*pi,dashed] ({2*cos(deg(x))},{2*sin(deg(x))});
    \node at (axis cs:0,0){$\color{blue}{\bullet}$};
    \node at (axis cs:0,-1){$n=\n$};
    \end{axis}
    \end{tikzpicture}
    }
\end{document}

I have raised the following questions, hoping very much for your help:

• how to create wavelengths automatically with labels when the number \n changes.

(see figure below)

• how you can create the waves of de Broglie (colored in violet) always with the same nucleus as from previous image.

Thank you very much for your patience and cooperation. My greetings and thanks.

Answer 1

Here is a proposal. Of course, one can further tune it. Note that I redefined your loop variable to \nn since otherwise there are problems with the calc syntax, in which you use \n1 etc.

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath}
\usetikzlibrary{decorations.markings,calc}
\begin{document}
\tikzset{mark two maxima/.style n args={3}{%
postaction=decorate,decoration={markings,
mark=at position #1 with {\draw[purple] (0,0) -- (0,-12pt) coordinate[midway] (x0);},
mark=at position #2 with {\draw[purple] (0,0) -- (0,-12pt) coordinate[midway](x1);
\draw let
\p1=($(x1)-(x0)$),\n1={atan2(\y1,\x1)},\n2={veclen(\x1,\y1)*(1/(2*sin(360*#2/2)))} 
in [purple,rotate=-90+2*\n1,latex-latex] (x1)
arc({#2*360}:0:{(\n2)}) node[midway,fill=white]{#3};
;}}}}
\foreach \nn in{3,4}{%
\begin{tikzpicture}
\begin{axis}[axis equal,
  xmin=-3,xmax=3,
  ymin=-3,ymax=3,
  axis lines=none]
\addplot[samples=400,domain=0:2*pi,very thick,red,
mark two maxima={0}{1/\nn}{$\lambda_{\nn}$}]
({(2+.3*cos(deg(\nn*x)))*cos(deg(x))},{(2+.3*cos(deg(\nn*x)))*sin(deg(x))});
\addplot[samples=40,domain=0:2*pi,dashed] ({2*cos(deg(x))},{2*sin(deg(x))});
\node at (axis cs:0,0){$\color{blue}{\bullet}$};
\node at (axis cs:0,-1){$n=\nn$};
\end{axis}
\end{tikzpicture}
}
\end{document}

Special service:

\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{subcaption}
\usepackage{floatrow}
\usepackage{pgfplots}
\usetikzlibrary{decorations.markings,calc}
\tikzset{mark two maxima/.style n args={3}{%
postaction=decorate,decoration={markings,
mark=at position #1 with {\draw[purple] (0,0) -- (0,-12pt) coordinate[midway] (x0);},
mark=at position #2 with {\draw[purple] (0,0) -- (0,-12pt) coordinate[midway](x1);
\draw let
\p1=($(x1)-(x0)$),\n1={atan2(\y1,\x1)},\n2={veclen(\x1,\y1)*(1/(2*sin(360*#2/2)))} 
in [purple,rotate=-90+2*\n1,latex-latex] (x1)
arc({#2*360}:0:{(\n2)}) node[midway,fill=white]{#3};}}}}

\newcommand{\SebastianoPic}[1]{%
\begin{tikzpicture}
\begin{axis}[axis equal,
  xmin=-3,xmax=3,
  ymin=-3,ymax=3,
  axis lines=none]
\addplot[samples=400,domain=0:2*pi,very thick,red,
mark two maxima={0}{1/#1}{$\lambda_{#1}$}]
({(2+.3*cos(deg(#1*x)))*cos(deg(x))},{(2+.3*cos(deg(#1*x)))*sin(deg(x))});
\addplot[samples=40,domain=0:2*pi,dashed] ({2*cos(deg(x))},{2*sin(deg(x))});
\node at (axis cs:0,0){$\color{blue}{\bullet}$};
\end{axis}
\end{tikzpicture}}
\begin{document}

\begin{figure}[htb]
\floatsetup{valign=t, heightadjust=all}
\ffigbox{%
\begin{subfloatrow}
\ffigbox{\SebastianoPic{3}}{\caption{$n=3$.\label{fig:n=3}}}
\ffigbox{\SebastianoPic{4}}{\caption{$n=4$.\label{fig:n=4}}}
\end{subfloatrow}}
{\caption{De Broglie waves.}\label{fig:DeBroglie}}
\end{figure}
\end{document}