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plotting - How to graph a series of coupled oscillators and watch the wave move along them


Here are the differential equations that set's up the 11 coupled oscillators.



new = Join[
Table[x[i]''[t] == - x[i][t] + 
 0.1*(x[i + 1][t] - 2*x[i][t] + x[i - 1][t]), {i, 1, 
9}], {x[0]''[t] == -x[0][t], x[10]''[t] == x[9][t], x[0][0] == 1, 
x[0]'[0] == 1, x[1]'[0] == 0, x[1][0] == 0}, 
Table[x[i][0] == 0, {i, 2, 10}], Table[x[i]'[0] == 0, {i, 2, 10}]]

Here are the solutions.


Solt = NDSolve[new, Table[x[i], {i, 0, 10}], {t, 25}]


Here are the individual plots.


Table[Plot[Evaluate[x[i][t] /. Solt], {t, 0, 25}, 
PlotRange -> All], {i, 0, 10}]

I am trying to figure out how to make a graph so along the x-axis are my i's from 0 to 10, and I can watch the wave move along each oscillator as time moves on. I keep getting errors in which it floods my notebook and doesn't stop unless I close the kernel.


This is what I have so far, and I'm not sure how to incorporate time into this.


Plot[Evaluate[x[i][t] /. Solt], {i, 0, 10}]

EDIT Coupled in a circle


Stew = Join[

 Table[x[i]''[t] == - x[i][t] + 
  0.1*(x[i + 1][t] - 2*x[i][t] + x[i - 1][t]), {i, 1, 
 9}], {x[10]''[t] == - x[10][t] + 
  0.1*(x[0][t] - 2*x[10][t] + x[9][t]), 
x[0]''[t] == - x[0][t] + 
  0.1*(x[1][t] - 2*x[0][t] + x[10][t])}, {x[0][0] == 0, 
x[0]'[0] == 0, x[1][0] == 1, x[1]'[0] == 0.5}, 
Table[x[i][0] == 0, {i, 2, 10}], Table[x[i]'[0] == 0, {i, 2, 10}]];

The Dsolve



Loin = NDSolve[Stew, Table[x[i], {i, 0, 10}], {t, 6.28}]

The individual graphs


Table[Plot[Evaluate[x[i][t] /. Loin], {t, 0, 6.28}, 
PlotRange -> All], {i, 0, 10}]

How would I go about putting the i=0 to 10 around in a circle?



Answer



After edit


I think oscillation directions should be parallel.



g[t_] = Table[{Cos[i*2 Pi/11], Sin[i*2 Pi/11], x[i][t]} /. Loin[[1]], {i, 0, 10}]; 

Animate[
 Show[
  ListPointPlot3D[g[t], PlotRange -> 1.5, BoxRatios -> 1, Filling -> Axis,
            PlotStyle -> Directive@AbsolutePointSize@7, Boxed -> False],
  ParametricPlot3D[{Cos@t, Sin@t, 0}, {t, 0, 2 Pi}, PlotStyle -> {Dashed, Black}]
  ,
  ImageSize -> 500, ViewVector -> {{Cos[t/15], Sin[t/15], 1} 11, {0, 0, 0}}, 
  AxesOrigin -> {0, 0, 0}, Ticks -> None, Axes -> True, AxesStyle -> {Red, Green, Blue},

  SphericalRegion -> True],
  {t, 0, 50}]

enter image description here


Before edit


f[t_] = Table[{i, x[i][t]} /. Solt[[1]], {i, 0, 10}];

Animate[ 
 ListPlot[f[t], PlotRange -> {{0, 11}, {-1.5, 1.5}}, 
  Joined -> True, PlotMarkers -> Automatic]

 , {t, 0, 25}
 ]

Good to notice: In f[t] definition := is intentionally replaced by =.


enter image description here


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