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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 53152, 1576]*) (*NotebookOutlinePosition[ 54343, 1613]*) (* CellTagsIndexPosition[ 54299, 1609]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Dispersion of Gaussian Wave Packets", "Title", TextAlignment->Center], Cell[TextData[{ "As you know, the dispersion relation for deep water waves is\n\t\t", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^2\)]], "= gk. \nThe phase speed is\n\t\t", Cell[BoxData[ FormBox[ RowBox[{"\t", SuperscriptBox[ StyleBox["c", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False}], "2"]}], TraditionalForm]]], "=g/k.\nThus long waves (small k) move faster than short waves.\nHere we \ consider a \"wavetrain\", or \"periodic group\" of deep\nwater waves. The \ system can be scaled to be dimensionless so that period of the wave is \ 2\[Pi]. and the dispersion relation is \[Omega](k)=", Cell[BoxData[ FormBox[ SqrtBox[ StyleBox["k", FontSlant->"Plain"]], TraditionalForm]]], " . We make some pretty plots of:\n\t\t", Cell[BoxData[ \(TextForm \`\(\[CapitalSigma]\+1\%k\_m\) \(a\_k\) cos\ \((kx - \[Omega] \((k)\) t)\)\)]] }], "Text", TextAlignment->Left], Cell[CellGroupData[{ Cell["Initialization Cells:", "Section", InitializationCell->True], Cell[TextData[ "cs[k_,nm_] gives a list of the values of cos(kx) at nm_ points of x between \ -\[Pi] and \[Pi]"], "Text", InitializationCell->True], Cell[BoxData[ \(cs[k_, nm_] := Module[{del, j}, \n\t\tdel = k*2*Pi/nm // N; \n\t\tj = \((nm - 1)\)/2; \n\t\tTable[Cos[del*i], {i, \(-j\), j}]]\)], "Input", InitializationCell->True], Cell[TextData[ "sn[k_,nm_] gives a list of the values of sin(kx) at nm points of x between -\ \[Pi] and \[Pi]"], "Text", InitializationCell->True], Cell[BoxData[ \(sn[k_, nm_] := Module[{del, j}, \n\t\tdel = k*2*Pi/nm // N; \n\t\tj = \((nm - 1)\)/2; \n\t\tTable[Sin[del*i], {i, \(-j\), j}]]\)], "Input", InitializationCell->True], Cell[TextData[{ "The next gives a list of lists of cos(kx) and sin(kx) for k=0 to k=", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["k", FontSlant->"Plain"], StyleBox["m", FontSlant->"Plain"]], TraditionalForm]]] }], "Text", InitializationCell->True], Cell[BoxData[ \(manycs[km_, nm_] := Table[cs[k, nm], {k, 0, km}]\)], "Input", InitializationCell->True], Cell[BoxData[ \(manysn[km_, nm_] := Table[sn[k, nm], {k, 0, km}]\)], "Input", InitializationCell->True], Cell[BoxData[{ FormBox[ \(gauss[c_, k0_, km_]\ \ gives\ a\ list\ \ of\ the\ Fourier\ coefficients \ \(a\_k\) in\), TextForm], FormBox[ RowBox[{"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t", RowBox[{ \(\[CapitalSigma]\+1\%k\_m\), \(a\_k\), "cos", " ", \((kx)\), FormBox[\(\t\t\t\t\ \ \n\t\t that\ make\ a\ \ pretty\ good\ approximation\ to\ the\ Gaussian\ \(wavepacket\ : \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\ e\^\(-\(x\^2\/\(4\ c\)\)\)\ \ cos \(\((k\_0\ x)\) . \)\)\), "TextForm"]}], " "}], TextForm], FormBox[ \(Note\ that\ k = 0\ gets\ half\ the\ magnitude\ of\ the\ normal\ \(formula . \)\), TextForm]}], "Text", InitializationCell->True], Cell[BoxData[ \(gauss[c_, k0_, km_] := {\n\t\t \((\(Sqrt[c]/Sqrt[Pi]\)/2)\)* \((Exp[\(-c\)*\((0 - k0)\)^2] + Exp[\(-c\)*\((\(-0\) - k0)\)^2]) \) // N, Table[\((Sqrt[c]/Sqrt[Pi])\)* \((Exp[\(-c\)*\((k - k0)\)^2] + Exp[\(-c\)*\((\(-k\) - k0)\)^2])\) // N, {k, 1, km}]} // Flatten\)], "Input", InitializationCell->True], Cell[BoxData[ \(TextForm \`setup\ is\ a\ module\ that\ sets\ up\ the\ lists\ of\ cosine\ and\ sine \ functions . \nkm\ should\ first\ be\ defined\ as\ an\ integer\ \((of\ course)\)\)], "Text", InitializationCell->True], Cell[BoxData[ \(setup := Module[{}, \n\t\tnm = 2*km + 1; \n\t\tca = manycs[km, nm]; \n\t\t sa = manysn[km, nm]; \n\t\t]\)], "Input", InitializationCell->True], Cell[TextData[{ "Superimposes the waves ", Cell[BoxData[ \(TextForm \`\(\[CapitalSigma]\+1\%k\_m\) \(a\_k\) cos\ \((kx - \[Omega]t)\)\)]], " to make a\ngroup. Uses identity \ cos(kx-\[Omega]t)=cos(kx)cos(\[Omega]t)+sin(kx)sin(\[Omega]t)" }], "Text", InitializationCell->True], Cell[BoxData[ \(grp[t_] := Module[{pc, ps}, \n\t\tpc = Table[Cos[\[Omega][k] t], {k, 0, km}]; \n \t\t\tps = Table[Sin[\[Omega][k] t], {k, 0, km}]; \n\t\t Sum[ak[\([k]\)]* \((pc[\([k]\)]*ca[\([k]\)] + ps[\([k]\)]*sa[\([k]\)])\), \n \t\t{k, 1, km + 1}]]\)], "Input", InitializationCell->True], Cell["plotgrp[t_] plots the group at time t_ ", "Text", InitializationCell->True], Cell[BoxData[ \(plotgrp[t_] := ListPlot[grp[t], PlotJoined -> True, Axes -> False, PlotRange -> {\(-1.1\), 1.1}]\)], "Input", InitializationCell->True] }, Closed]], Cell[CellGroupData[{ Cell["Why it works:", "Section"], Cell[TextData[{ "Note", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(E\^\(\(-c\)\ \((k - k0)\)\^2\)\ Cos[k\ x] \[DifferentialD]k\)}]]], "=", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(E\^\(\(-c\)\ \((j)\)\^2\)\ Cos[\((j + k0)\)\ x] \[DifferentialD]j\)}]]], "\nFor the latter integral, ", StyleBox["Mathematica", FontSlant->"Italic"], " (see below) tells us that:\n", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(E\^\(\(-c\)\ \((j)\)\^2\)\ Cos[\((j + k0)\)\ x] \[DifferentialD]j\)}]]], "=", Cell[BoxData[ \(\(E\^\(-\(x\^2\/\(4\ c\)\)\)\ \@\[Pi]\ Cos[k0\ x]\)\/\@c\)]], "\nNow we also have \n", Cell[BoxData[ RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(E\^\(\(-c\)\ \((k - k0)\)\^2\)\ Cos[k\ x]\) \[DifferentialD]k\)}]]], "\n=", Cell[BoxData[ \(\[Integral]\+0\%\[Infinity] E\^\(\(-c\)\ \((k - k0)\)\^2\)\ Cos[k\ x] \[DifferentialD]k\)]], " +", Cell[BoxData[ \(\[Integral]\+\(-\[Infinity]\)\%0 E\^\(\(-c\)\ \((k - k0)\)\^2\)\ Cos[k\ x] \[DifferentialD]k\)]], " \n=", Cell[BoxData[ \(\[Integral]\+0\%\[Infinity] E\^\(\(-c\)\ \((k - k0)\)\^2\)\ Cos[k\ x] \[DifferentialD]k\)]], "+", Cell[BoxData[ \(\[Integral]\+0\%\[Infinity] E\^\(\(-c\)\ \((\(-k\) - k0)\)\^2\)\ Cos[k\ x] \[DifferentialD]k\)]], "\nOur Fourier series over the positive k approximates the above \ integrals." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ Exp[\(-c\)\ j^2] Cos[\((j + k0)\)\ x], {j, \(-Infinity\), Infinity}] \)], "Input"], Cell[BoxData[ RowBox[{"If", "[", RowBox[{ \(Im[x] == 0 && Re[c] > 0\), ",", \(\(E\^\(-\(x\^2\/\(4\ c\)\)\)\ \@\[Pi]\ Cos[k0\ x]\)\/\@c\), ",", RowBox[{ SubsuperscriptBox["\[Integral]", InterpretationBox[\(-\[Infinity]\), DirectedInfinity[ -1]], InterpretationBox["\[Infinity]", DirectedInfinity[ 1]]], \(\(E\^\(\(-c\)\ j\^2\)\ Cos[\((j + k0)\)\ x]\) \[DifferentialD]j\)}]}], "]"}]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Now lets use it:", "Section"], Cell["\<\ Now let's use it! Here is a dispersion relation appropriate for \ deep water waves:\ \>", "Text"], Cell[BoxData[ \(\[Omega][k_] := Sqrt[k] // N\)], "Input"], Cell["\<\ Let's use km=40 wavenumbers. c=.01 gives a nice, isolated hump, \ but not too isolated to abuse our truncated Fourier series.\ \>", "Text"], Cell[BoxData[ \(\(km = 40; \)\)], "Input"], Cell[BoxData[ \(setup\)], "Input"], Cell[BoxData[ \(\(ak = gauss[ .01, 0, km]; \)\)], "Input"], Cell["Here is the group at t=0:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(plotgrp[0]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0117578 0.309017 0.280925 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash .03557 .30902 m .04733 .30902 L .05908 .30902 L .07084 .30902 L .0826 .30902 L .09436 .30902 L .10611 .30902 L .11787 .30902 L .12963 .30902 L .14139 .30902 L .15315 .30902 L .1649 .30902 L .17666 .30902 L .18842 .30902 L .20018 .30902 L .21193 .30902 L .22369 .30902 L .23545 .30902 L .24721 .30902 L .25897 .30902 L .27072 .30902 L .28248 .30902 L .29424 .30902 L .306 .30902 L .31775 .30902 L .32951 .30902 L .34127 .30902 L .35303 .30902 L .36479 .30902 L .37654 .30902 L .3883 .30902 L .40006 .30902 L .41182 .30904 L .42357 .30919 L .43533 .31027 L .44709 .31555 L .45885 .33433 L .47061 .38156 L .48236 .46293 L .49412 .55071 L .50588 .58994 L .51764 .55071 L .52939 .46293 L .54115 .38156 L .55291 .33433 L .56467 .31555 L .57643 .31027 L .58818 .30919 L .59994 .30904 L .6117 .30902 L Mistroke .62346 .30902 L .63521 .30902 L .64697 .30902 L .65873 .30902 L .67049 .30902 L .68225 .30902 L .694 .30902 L .70576 .30902 L .71752 .30902 L .72928 .30902 L .74103 .30902 L .75279 .30902 L .76455 .30902 L .77631 .30902 L .78807 .30902 L .79982 .30902 L .81158 .30902 L .82334 .30902 L .8351 .30902 L .84685 .30902 L .85861 .30902 L .87037 .30902 L .88213 .30902 L .89389 .30902 L .90564 .30902 L .9174 .30902 L .92916 .30902 L .94092 .30902 L .95267 .30902 L .96443 .30902 L .97619 .30902 L Mfstroke % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{10, 10}, {10, 10}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgOol00`00Oomoo`02Ool00`00Oomoo`2:Ool008ioo`03001oogoo009oo`03001oogoo08Yoo`00 Sgoo00D007ooOomoo`0008aoo`00Sgoo00@007ooOol008eoo`00Sgoo00@007ooOol008eoo`00T7oo 00<007oo0000SGoo002@Ool00`00Ool0002=Ool0091oo`8008ioo`00TGoo00<007ooOol0S7oo003o OolQOol00?moob5oo`00ogoo8Goo003oOolQOol00?moob5oo`00ogoo8Goo003oOolQOol00?moob5o o`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-2.13048, -1.10001, 0.297076, 0.0124338}}] }, Open ]], Cell["\<\ Now let's animate the dispersing wave. What you will first see is a sequence of frames being formed. That is not yet the animation. After all 41 frames are made, highlight them all and go the the \"Cell\" menu \ and \"Animate Selected Graphics\". Or close the sequence first, highlight the second vertical bar to the right, and animate that.\ \>", "Text"], Cell[BoxData[ \(\(Animate[plotgrp[t], {t, 0, 8, .2}]; \)\)], "Input"], Cell["\<\ Now let's make a wave packet. Here it will be quite clear that the group velocity is about half the phase velocity of the dominant wavenumber \ k0=10.\ \>", "Text"], Cell[BoxData[ \(\(ak = gauss[ .1, 10, km]; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(plotgrp[0]; \)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.0117578 0.309017 0.280925 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash .03557 .30902 m .04733 .30902 L .05908 .30902 L .07084 .30902 L .0826 .30902 L .09436 .30902 L .10611 .30902 L .11787 .30902 L .12963 .30902 L .14139 .30902 L 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things to do. 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