How to add ticks for mesh?

Question

I'm trying to reprocedue this plot with Mathematica and got some problems.

x[t_] := 4 + 3 Cos[\[Pi] t] + 2 Cos[2 \[Pi] t] + Cos[3 \[Pi] t];
xa[t_] := 5 + 5 Cos[\[Pi] t]; Plot[{x[t], xa[t]}, {t, 0, 5}, 
 GridLines -> Automatic, AxesOrigin -> {0, 0},
 PlotStyle -> {Automatic, Directive[Dashed]}, Mesh -> {{0}}, 
 MeshFunctions -> {x[#] - xa[#] &}, 
 MeshStyle -> Directive[PointSize[0.03], Red]]

1. Why isn't a point at {0, 10}?
   
2. How would you add ticks 0, T, 2T, 3T, etc as in the image?

I believe that I can solve for the coordinate of points and add the ticks but I'm wondering if there is a simple way or that is the way to go?

Answer 1

Use Solve or NSolve to find all the intersection points and use Epilog to add all the lines and points and ticks.

x[t_] = 4 + 3 Cos[π t] + 2 Cos[2 π t] + Cos[3 π t];
xa[t_] = 5 + 5 Cos[π t];
sol = DeleteDuplicates[Solve[{x[t] == xa[t], 0 <= t <= 5}, t]]
T = 2/3;
Plot[{x[t], xa[t]}, {t, 0, 5}, GridLines -> Automatic, 
 AxesOrigin -> {0, 0}, PlotStyle -> {Automatic, Directive[Dashed]}, 
 Ticks -> None, 
 Epilog -> {Table[
    Text[Style[ToString[i] <> "T", Bold], {i*T, 0}, {0, 2}], {i, 1, 
     7}], {Red, AbsolutePointSize[8], Point[{t, x[t]} /. sol]}, Green,
    Line[{{t, x[t]}, {t, 0}}] /. sol}, PlotRangePadding -> {.1, .8}]