{"id":5573,"date":"2025-08-08T15:22:31","date_gmt":"2025-08-08T20:22:31","guid":{"rendered":"https:\/\/commons.princeton.edu\/josephhenry\/?page_id=5573"},"modified":"2025-08-08T15:25:33","modified_gmt":"2025-08-08T20:25:33","slug":"the-first-transatlantic-cable","status":"publish","type":"page","link":"https:\/\/commons.princeton.edu\/josephhenry\/the-first-transatlantic-cable\/","title":{"rendered":"Mathematica Code"},"content":{"rendered":"

Modified from Jacopo Bertolotti:<\/a><\/p>\n

(*Find the dispersion relation for the Telegrapher’s equation*)
\nf = E^(I (k x – \\[Omega] t));
\nFullSimplify[D[f, {t, 2}] – v^2 D[f, {x, 2}] + b D[f, t] + c f]<\/p>\n

Solve[c + k^2 v^2 + (-I b – \\[Omega]) \\[Omega] == 0, \\[Omega]]<\/p>\n

(*Generate the animation*)
\nR0 = 1; L0 = 1; G0 = 0.1; C0 = 1;<\/p>\n

\\[Sigma] = 1; k0 = 5; b = R0\/L0 + G0\/C0; c = R0*G0\/(L0*C0); v =
\n1\/Sqrt[ L0* C0];
\ng = Sum[(E^(I k x) E^(-(k –
\nk0)^2\/(2 \\[Sigma]^2)) E^(-I \\[Omega] t)) \/. {\\[Omega] ->
\nk v}, {k, 0, 15, 0.1}];
\ng2 = Sum[(E^(I k x) E^(-(k –
\nk0)^2\/(2 \\[Sigma]^2)) E^(-I \\[Omega] t)) \/. {\\[Omega] ->
\n1\/2 (-I b + Abs[Sqrt[-b^2 + 4 c + 4 k^2 v^2]])}, {k, 0, 15, 0.1}];
\np1 = Table[
\nShow[Plot[Re[g], {x, -5, 50}, PlotStyle -> {Orange, Thick},
\nPlotRange -> {-25, 25}],
\nPlot[Re[g2], {x, -5, 50}, PlotStyle -> {Black, Thick},
\nPlotRange -> {-25, 25}], Axes -> False,
\nPlotLegends -> {“Wave eq.”, “Telegrapher’s eq.”}], {t, 0, 50, 0.5}];
\nListAnimate[p1, 10]<\/p>\n

*Still need to find proper RLGC values and length of simulation in proper units to be a good model<\/p>\n

From Mathematica Documentation<\/a>:<\/strong><\/p>\n

a = 0.7; b = 0.6; c = 1.1;<\/p>\n

f[x_] = D1<\/a><\/sup>, x];
\nvInit[x_] = -c*D[f[x], x] – (a + b) f[x]\/2;<\/p>\n

Plot[f[x], {x, -1, 1}, PlotRange -> All]
\nPlot[vInit[x], {x, -1, 1}, PlotRange -> All]<\/p>\n

uif = NDSolveValue[{D[u[t, x], {t, 2}] + (a + b) D[u[t, x], {t, 1}] –
\nc^2 D[u[t, x], {x, 2}] + a b u[t, x] == 0, u[0, x] == f[x],
\nDerivative[1, 0][u][0, x] == vInit[x],
\nDirichletCondition[u[t, x] == f[0], x == 0],
\nDirichletCondition[u[t, x] == f[1], x == 1]},
\nu, {t, 0, 2}, {x, 0, 1}];<\/p>\n

framesTEQ =
\nTable[Plot[uif[t, x], {x, 0, 1}, PlotRange -> {-1, 1.3}], {t, 0, 2,
\n0.01}];<\/p>\n

Manipulate[framesTEQ[[i]], {{i, 3, “time”}, 1, Length[framesTEQ], 1},
\nSaveDefinitions -> True]<\/p>\n

ListAnimate[framesTEQ, AnimationRate -> 25, SaveDefinitions -> True]<\/p>\n

 <\/p>\n

\n
<\/div>\n
    \n
  1. 125 Erf[(x – 0.5)\/0.125 ↩<\/a><\/span><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

    Modified from Jacopo Bertolotti: (*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*) R0 = 1; L0 … Continue reading “Mathematica Code”<\/span><\/a><\/p>\n","protected":false},"author":6922,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-5573","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/pages\/5573","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/users\/6922"}],"replies":[{"embeddable":true,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/comments?post=5573"}],"version-history":[{"count":5,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/pages\/5573\/revisions"}],"predecessor-version":[{"id":5624,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/pages\/5573\/revisions\/5624"}],"wp:attachment":[{"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/media?parent=5573"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/categories?post=5573"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/commons.princeton.edu\/josephhenry\/wp-json\/wp\/v2\/tags?post=5573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}