Summary

Mathematica 11.0 code

(*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]]

(*Plot a pulse both with and without dispersion*)
g = 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]])} /. {\[Sigma] ->1, k0 -> 4, b -> 0, c -> 0, v -> 1, t -> 15}, {k, 0, 15, 
    0.1}];
Show[
 Plot[Re[g], {x, -10, 20}, PlotRange -> All, PlotStyle -> {Orange, Thick}]
 ,
 Plot[{Abs[g], -Abs[g]}, {x, -10, 20}, PlotRange -> All, PlotStyle -> {Black, Black}]
 ]

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