plotting a trig functions along with its envelope

Note: This questions is quite different from the ones referred to in the comments. Those deal with numerical questions, while this one is algebraic.

I have plots of the following type:

Plot[Cos[50 t] + Cos[51 t], {t, 0, 10}]

enter image description here

I would like to plot a envelope over this plot, i.e. another plot that joins all of maxima and minima of this plot respectively. Here is my attempt, but it's not exactly what I'd like:

Plot[{Cos[50 t] + Cos[51 t], Cos[t] + 1.5, -Cos[t] - 1.5}, {t, 0, 10}]

enter image description here

How can I generate the actual envelope?

Answer

Playing with the manipulate below might help. It's based on the the acoustics of beats.

Manipulate[Plot[
  {Cos[a*t] + Cos[b*t], 2*Cos[(b - a) t/2], -2*Cos[(b - a) t/2]}, {t, 0, 10},
  PlotStyle -> {
   Directive[Opacity[0.7]], 

   Directive[Black, Thick], 
   Directive[Black, Thick]}],
 {{a, 20}, 1, 50}, {{b, 21}, 1, 50}]

enter image description here

Comments

plotting - Plot 4D data with color as 4th dimension

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plotting - Filling between two spheres in SphericalPlot3D

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plotting - Mathematica: 3D plot based on combined 2D graphs

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