Hi,

this is not a nice solution, but a slightly different approach.

and

Show[Plot[f[n], {n, 0, 2 Pi}, AspectRatio -> Automatic, Mesh -> #, MeshStyle -> Directive[Red]] & /@ {None, {10}} /. Point[x : {__Integer}] :> Map[Inset[Style["5", 20], #] &, x]]

Cheers,

Marco

Nice plot. Is this for hot film sensors by any chance? To make them more distinguishable, you can also make one line solid Dashing[{}} and one line as Dotted. It is always tricky to make lines distinguishable. However now that color printers are far more common and a lot of people view papers on screens (telephone, tablets, laptops), the 'gray-scale' requirement is less strict than before...

Note that I do it in now in an exact manner, for a general function the arc length of a curve is very complicated, a numerical approach is better suited then. Using Interpolation and some points you can quite easily cook up a function... Good Luck.

You can also place them based on the arc length of the curve with some extra work:

ClearAll[f]
f[n_] := Sin[2 n]
alen[t_] = ArcLength[f[n], {n, 0, t}]

lens = MovingAverage[Subdivide[0, alen[2.0 Pi], 20], 2];
sols = FindRoot[alen[t] == #, {t, Pi}] & /@ lens;
sols = t /. sols;
pos = {#, f[#]} & /@ sols;
labels = Text["1", #, Background -> White] & /@ pos;
p1 = Plot[f[n], {n, 0, 2 Pi}, Epilog -> labels, AspectRatio -> Automatic]

Depending on the function you plot this might work easily or not. If not, one has to do some numerical approximations...