File:Diatomic chain.gif
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This is a file from the Wikimedia Commons. The description on its description page there is shown below.
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Summary
| Description |
English: For single frequency oscillation of an infinite chain of two alternate masses connected by identical springs (diatomic chain), the relative phase of oscillation of adjacent masses will depend non-linearly with frequency, resulting in travelling or stationary waves. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1640649301603885056 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 13.1 code
a = 1; k = 1; m1 = 1; m2 = 3/2; \[Mu] = (1/m1 + 1/m2)^-1;
\[Omega][q_] := {Sqrt[k/\[Mu] (1 - Sqrt[1 - (4 \[Mu]^2 Sin[q a]^2)/(m1 m2)])], Sqrt[k/\[Mu] (1 + Sqrt[1 - (4 \[Mu]^2 Sin[q a]^2)/(m1 m2)])]};
\[Omega]list = FullSimplify[{0, 1/4 \[Omega][\[Pi]/(2 a)][[1]], 1/2 \[Omega][\[Pi]/(2 a)][[1]], 3/4 \[Omega][\[Pi]/(2 a)][[1]], \[Omega][\[Pi]/(2 a)][[1]], \[Omega][\[Pi]/(2 a)][[2]], Mean[{\[Omega][\[Pi]/(2 a)][[2]], \[Omega][0][[2]]}], \[Omega][0][[2]]}]
qlist = Chop[ DeleteCases[Join[Max /@ (q /. Table[Chop@NSolve[\[Omega][q][[1]] == \[Omega]list[[j]], q ], {j, 1, Length[\[Omega]list]}]), Max /@ (q /. Table[Chop@NSolve[\[Omega][q][[2]] == \[Omega]list[[j]], q], {j, 1, Length[\[Omega]list]}]) ], q], 10^-6] /. {C[1] -> 0}
amplitude = 0.04;
U1[j_] := If[\[Omega]list[[j]] <= \[Omega][\[Pi]/(2 a)][[1]], amplitude*(((2 k)/m1 Cos[qlist[[j]]])/((2 k)/m1 - \[Omega]list[[j]]^2)), amplitude];
U2[j_] := If[\[Omega]list[[j]] <= \[Omega][\[Pi]/(2 a)][[1]], amplitude, amplitude*(((2 k)/m1 Cos[qlist[[j]]])/((2 k)/m1 - \[Omega]list[[j]]^2))^-1];
frames = Table[
Show[
Plot[\[Omega][q], {q, 0, \[Pi]/(2 a)}, PlotRange -> {{-\[Pi]/2, \[Pi]/2}, {-0.1, 2}}, PlotStyle -> {Black, Thickness[0.006]}, Ticks -> None, AxesLabel -> {"k", "\[Omega]"}, LabelStyle -> {Black, Bold, FontSize -> 20}, AxesStyle -> Directive[Black, Thickness[0.003]], ImageSize -> Large]
,
Graphics[{
Table[{Red, Thick, Line[{{-\[Pi], \[Omega]list[[j]]}, {\[Pi], \[Omega]list[[j]]}}], Black, PointSize[0.007]
,
Point[Table[{\[Pi]/2 (n/50 - 1), U1[j]*Sin[qlist[[j]] 2 a n - \[Omega]list[[j]] t] + \[Omega]list[[j]]}, {n, 0, 50, 1}]]
, Gray, PointSize[0.008], Point[Table[{\[Pi]/2 ((n + 0.5)/50 - 1), U2[j]*Sin[qlist[[j]] (a + 2 a n) - \[Omega]list[[j]] t] + \[Omega]list[[j]]}, {n, 0, 49, 1}]]
}, {j, 1, Length[\[Omega]list]}]
}]
]
, {t, 0, (2 \[Pi])/\[Omega]list[[2]], (2 \[Pi])/(50*\[Omega]list[[2]])}];
ListAnimate[frames]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
 
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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File history
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| Date/Time | Dimensions | User | Comment | |
|---|---|---|---|---|
| current | 01:35, 30 March 2023 | 768 × 502 (1.43 MB) | Berto | Uploaded own work with UploadWizard |
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