Group and Phase Velocity

Manipulate[
Plot[{Cos[(6 - k)*x - (6 - ω)*t] + Cos[(6 + k)*x - (6 + ω)*t], 
    2*Cos[(-k)*x + ω*t], -2*Cos[(-k)*x + ω*t]}, {x, 0, 30}, 
   ImageSize -> {500,200}, AspectRatio -> 1/5, Frame -> True, 
   PlotRange -> {{0, 30}, {-2, 2}}, PlotStyle -> {Automatic, Dashed, Dashed}, 
   PlotLabel -> Row[{"(Subscript[v, g]) = ", NumberForm[ω/k, 3]}], 
   Epilog -> {Green, PointSize[0.01], Table[Point[{(Pi*n + ω*t + Pi/2)/k, 0}], 
      {n, -15, 15}], Red, Point[{t, Cos[(6 - k)*t - (6 - ω)*t] + 
         Cos[(6 + k)*t - (6 + ω)*t]}]}], 
  {t, 0., 10., Appearance ->"Labeled"}, {{k, 0.2, "Δk"}, 0.1, 0.4,Appearance ->"Labeled"}, 
{{ω, 0.2, "Δω"}, -3, 3, Appearance ->"Labeled"}]

CAPTION

The group velocity (Subscript[v, g]) is the speed of the overall shape of a modulated wave (called the envelope). This is defined by dω/dk (chosen here to equal 1), where ω is the angular velocity and k is the wave number. The phase velocity (Subscript[v, p]) of a wave is the speed at which a given phase of a wave travels through space, equal to ω/k.

DETAILS

The wave shown here is a sum of two waves that differ by a small phase (called a beat) of the form cos((k-Δk) x-(ω-Δω)t)+cos((k+Δk) x-(ω+Δω)t).

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Group and Phase Velocity" from the Wolfram Demonstrations Project
 https://demonstrations.wolfram.com/GroupAndPhaseVelocity/

Contributed by: Enrique Zeleny

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