mjhjmtu

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
 usepackagepgfplots
 usepackageamsmath
 begindocument

 foreach n in3,4%
 begintikzpicture
 beginaxis[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)$colorbluebullet$;
 node at (axis cs:0,-1)$n=n$;
 endaxis
 endtikzpicture
 
enddocument

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.

Here is a proposal. Of course, one can further tune it.

documentclass[tikz]standalone
usepackagepgfplots
usepackageamsmath
usetikzlibrarydecorations.markings,calc
begindocument
tikzsetmark 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);
path let p1=($(x1)-(x0)$),n1=atan2(y1,x1) in pgfextraxdefmyangn1;
draw [purple,rotate=-90+2*myang,latex-latex] (x1) arc(#2*360:0:2.15cm+6pt) node[midway,fill=white]#3
-- (x0);
foreach n in3,4%
begintikzpicture
beginaxis[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=01/n$lambda_n$] ((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)$colorbluebullet$;
node at (axis cs:0,-1)$n=n$;
endaxis
endtikzpicture

enddocument