(*Find the dispersion relation for the Telegrapher's equation*)
f = E^(I (k x - \[Omega] t));
FullSimplify[D[f, {t, 2}] - v^2 D[f, {x, 2}] + b D[f, t] + c f]

Solve[c + k^2 v^2 + (-I b - \[Omega]) \[Omega] == 0, \[Omega]]

(*Generate the animation*)
\[Sigma] = 1; k0 = 5; b = 0.01; c = 5; v = 1;
g = Sum[(E^(I k x) E^(-(k - k0)^2/(2 \[Sigma]^2)) E^(-I \[Omega] t)) /. {\[Omega] -> k v}, {k, 0, 15, 0.1}];
g2 = Sum[(E^(I k x) E^(-(k - k0)^2/(2 \[Sigma]^2)) E^(-I \[Omega] t)) /. {\[Omega] -> 1/2 (-I b + Abs[Sqrt[-b^2 + 4 c + 4 k^2 v^2]])}, {k, 0, 15, 0.1}];
p1 = Table[
   Show[
    Plot[Re[g], {x, -5, 50}, PlotStyle -> {Orange, Thick}, PlotRange -> {-25, 25}]
    ,
    Plot[Re[g2], {x, -5, 50}, PlotStyle -> {Black, Thick}, PlotRange -> {-25, 25}]
    , Axes -> False, PlotLegends -> {"Wave eq.", "Telegrapher's eq."}
    ]
   , {t, 0, 50, 0.5}];
ListAnimate[p1, 10]

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