(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 18367, 551]*) (*NotebookOutlinePosition[ 19114, 577]*) (* CellTagsIndexPosition[ 19070, 573]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Rapidly varying potential", "Section"], Cell["\<\ Once we get well past the point of breakdown of the slowly varying potential \ theory, we have in essence a rapidly varying potential. The key behavior in \ this regime is best demonstrated by a plot of pulse velocity.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(avv\ = \ Part[Centroid\ /. \ \((SolutionData\ /. \ rapid)\), All, \(-1\)]/ 1200;\)\), "\[IndentingNewLine]", \(\(ave\ = \ \((1200\ avv\ - \ Part[Centroid\ /. \ \((SolutionData\ /. rapid)\), All, \(-800\)])\)/200;\)\), "\[IndentingNewLine]", \(\(fmv\ = \ 2\ Part[Momentum\ /. \ \((SolutionData\ /. \ rapid)\), \ All, \ \(-1\)]/Part[Mass\ /. \ \((SolutionData\ /. \ rapid)\), \ All, \(-1\)];\)\), "\[IndentingNewLine]", \(\(ww\ = \ Omega\ /. \ rapid;\)\), "\[IndentingNewLine]", \(\(Show[ Block[{$DisplayFunction\ = \ Identity}, {ListPlot[Transpose[{ww, avv}], \ PlotStyle \[Rule] PointSize[0.015]], \[IndentingNewLine]ListPlot[ Transpose[{ww, ave}], \ PlotStyle \[Rule] {RGBColor[0, 0, 1], PointSize[0.015]}], ListPlot[Transpose[{ww, fmv}], \ PlotStyle \[Rule] {RGBColor[0, 1, 0], PointSize[0.015]}]}]]; \)\)}], "Input", CellOpen->False], 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 -1.05556 0.253968 0.0786612 0.432014 [ [.0873 .06616 -9 -9 ] [.0873 .06616 9 0 ] [.34127 .06616 -9 -9 ] [.34127 .06616 9 0 ] [.46825 .06616 -3 -9 ] [.46825 .06616 3 0 ] [.59524 .06616 -9 -9 ] [.59524 .06616 9 0 ] [.72222 .06616 -3 -9 ] [.72222 .06616 3 0 ] [.84921 .06616 -9 -9 ] [.84921 .06616 9 0 ] [.97619 .06616 -3 -9 ] [.97619 .06616 3 0 ] [.20179 .16506 -18 -4.5 ] [.20179 .16506 0 4.5 ] [.20179 .25147 -18 -4.5 ] [.20179 .25147 0 4.5 ] [.20179 .33787 -18 -4.5 ] [.20179 .33787 0 4.5 ] [.20179 .42427 -18 -4.5 ] [.20179 .42427 0 4.5 ] [.20179 .51067 -6 -4.5 ] [.20179 .51067 0 4.5 ] [.20179 .59708 -18 -4.5 ] [.20179 .59708 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .0873 .07866 m .0873 .08491 L s [(4.5)] .0873 .06616 0 1 Mshowa .34127 .07866 m .34127 .08491 L s [(5.5)] .34127 .06616 0 1 Mshowa .46825 .07866 m .46825 .08491 L s [(6)] .46825 .06616 0 1 Mshowa .59524 .07866 m .59524 .08491 L s [(6.5)] .59524 .06616 0 1 Mshowa .72222 .07866 m .72222 .08491 L s [(7)] .72222 .06616 0 1 Mshowa .84921 .07866 m .84921 .08491 L s [(7.5)] .84921 .06616 0 1 Mshowa .97619 .07866 m .97619 .08491 L s [(8)] .97619 .06616 0 1 Mshowa .125 Mabswid .1127 .07866 m .1127 .08241 L s .1381 .07866 m .1381 .08241 L s .16349 .07866 m .16349 .08241 L s .18889 .07866 m .18889 .08241 L s .23968 .07866 m .23968 .08241 L s .26508 .07866 m .26508 .08241 L s .29048 .07866 m .29048 .08241 L s .31587 .07866 m .31587 .08241 L s .36667 .07866 m .36667 .08241 L s .39206 .07866 m .39206 .08241 L s .41746 .07866 m .41746 .08241 L s .44286 .07866 m .44286 .08241 L s .49365 .07866 m .49365 .08241 L s .51905 .07866 m .51905 .08241 L s .54444 .07866 m .54444 .08241 L s .56984 .07866 m .56984 .08241 L s .62063 .07866 m .62063 .08241 L s .64603 .07866 m .64603 .08241 L s .67143 .07866 m .67143 .08241 L s .69683 .07866 m .69683 .08241 L s .74762 .07866 m .74762 .08241 L s .77302 .07866 m .77302 .08241 L s .79841 .07866 m .79841 .08241 L s .82381 .07866 m .82381 .08241 L s .8746 .07866 m .8746 .08241 L s .9 .07866 m .9 .08241 L s .9254 .07866 m .9254 .08241 L s .95079 .07866 m .95079 .08241 L s .0619 .07866 m .0619 .08241 L s .03651 .07866 m .03651 .08241 L s .01111 .07866 m .01111 .08241 L s .25 Mabswid 0 .07866 m 1 .07866 L s .21429 .16506 m .22054 .16506 L s [(0.2)] .20179 .16506 1 0 Mshowa .21429 .25147 m .22054 .25147 L s [(0.4)] .20179 .25147 1 0 Mshowa .21429 .33787 m .22054 .33787 L s [(0.6)] .20179 .33787 1 0 Mshowa .21429 .42427 m .22054 .42427 L s [(0.8)] .20179 .42427 1 0 Mshowa .21429 .51067 m .22054 .51067 L s [(1)] .20179 .51067 1 0 Mshowa .21429 .59708 m .22054 .59708 L s [(1.2)] .20179 .59708 1 0 Mshowa .125 Mabswid .21429 .10026 m .21804 .10026 L s .21429 .12186 m .21804 .12186 L s .21429 .14346 m .21804 .14346 L s .21429 .18666 m .21804 .18666 L s .21429 .20827 m .21804 .20827 L s .21429 .22987 m .21804 .22987 L s .21429 .27307 m .21804 .27307 L s .21429 .29467 m .21804 .29467 L s .21429 .31627 m .21804 .31627 L s .21429 .35947 m .21804 .35947 L s .21429 .38107 m .21804 .38107 L s .21429 .40267 m .21804 .40267 L s .21429 .44587 m .21804 .44587 L s .21429 .46747 m .21804 .46747 L s .21429 .48907 m .21804 .48907 L s .21429 .53228 m .21804 .53228 L s .21429 .55388 m .21804 .55388 L s .21429 .57548 m .21804 .57548 L s .21429 .05706 m .21804 .05706 L s .21429 .03546 m .21804 .03546 L s .21429 .01386 m .21804 .01386 L s .25 Mabswid .21429 0 m .21429 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .015 w .02381 .07985 Mdot .0873 .08019 Mdot .15079 .08057 Mdot .21429 .08137 Mdot .34127 .08565 Mdot .46825 .10544 Mdot .59524 .19334 Mdot .72222 .33675 Mdot .84921 .47446 Mdot .97619 .60332 Mdot 0 0 1 r .02381 .07837 Mdot .0873 .07893 Mdot .15079 .07867 Mdot .21429 .07866 Mdot .34127 .07874 Mdot .46825 .07826 Mdot .59524 .11771 Mdot .72222 .23193 Mdot .84921 .37269 Mdot .97619 .50636 Mdot 0 1 0 r .02381 .15261 Mdot .0873 .15237 Mdot .15079 .01472 Mdot .21429 .0332 Mdot .34127 .07328 Mdot .46825 .10122 Mdot .59524 .11156 Mdot .72222 .22138 Mdot .84921 .36256 Mdot .97619 .49636 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{380, 234.75}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHggoo00@007ooOol000Qoo`04001oogoo0006Ool00`00Oomoo`2k Ool5001XOol003]oo`04001oogoo0008Ool01000Oomoo`001Woo0`00^goo1@00J7oo000kOol01000 Oomoo`0027oo0`001goo00<007ooOol0_7oo0`00JGoo000kOol01000Oomoo`002Goo00<007ooOol0 1Woo00<007ooOol0ogoo:Goo000lOol2000:Ool30006Ool00`00Oomoo`3oOolYOol0055oo`03001o ogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`03001o ogoo0>aoo`<3h3Uoo`00DGoo0P00k7oo1@?P>7oo001AOol00`00Oomoo`3[Ool50n0hOol0055oo`03 001oogoo0>]oo`D3h3Qoo`00DGoo00<007ooOol0k7oo0`?P>Goo001AOol00`00Oomoo`3[Ool501lh Ool0055oo`03001oogoo0>]oo`D07cQoo`00DGoo00<007ooOol0jgoo1@0O>7oo001AOol00`00Oomo o`3/Ool301liOol0055oo`800?moobYoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3o OolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3o OolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol2003oOolZOol0 055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0 055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo000lOol20004Ool20004Ool20007 Ool00`00Oomoo`3oOolYOol003]oo`04001oogoo0008Ool01000Oomoo`001Woo00<007ooOol0ogoo :Goo000kOol01000Oomoo`0027oo00@007ooOol000Ioo`<00?moobUoo`00>goo00@007ooOol000Uo o`8000Moo`03001oogoo0?moobUoo`00>goo00@007ooOol000Qoo`04001oogoo0006Ool00`00Oomo o`3oOolYOol003aoo`8000Yoo`8000Moo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo 001AOol00`00Oomoo`3oOolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo 001AOol2003oOolZOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol0 0`00Oomoo`3oOolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol0 0`00Oomoo`3oOolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol2 003oOolZOol0055oo`03001oogoo0>aoo`<003Uoo`00DGoo00<007ooOol0jgoo1@00>7oo001AOol0 0`00Oomoo`3[Ool5000hOol0055oo`03001oogoo0>]oo`D003Qoo`00DGoo00<007ooOol0k7oo0`00 >Goo001AOol00`00Oomoo`3oOolYOol0055oo`03001oogoo0?moobUoo`00DGoo0P00ogoo:Woo001A Ool00`00Oomoo`3oOolMOol30n09Ool0055oo`03001oogoo0?mooaaoo`D3h0Qoo`00DGoo00<007oo Ool0ogoo77oo1@?P27oo001AOol00`00Oomoo`3oOolLOol50n08Ool0055oo`03001oogoo0?mooaeo o`<3h0Uoo`00Agoo1@001Goo00<007ooOol0ogoo77oo1@0O27oo0019Ool00`00Oomoo`05Ool00`00 Oomoo`3oOolLOol501l8Ool004Uoo`03001oogoo00Eoo`<00?mooaaoo`D07`Qoo`00BGoo00<007oo Ool01Goo00<007ooOol0ogoo7Goo0`0O2Goo0019Ool00`00Oomoo`05Ool00`00Oomoo`3oOolYOol0 04Qoo`8000Moo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3o OolYOol0055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol2003oOolZOol0 055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0 055oo`03001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0 055oo`03001oogoo0?moobUoo`00DGoo0P00ogoo:Woo001AOol00`00Oomoo`3oOolYOol0055oo`03 001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`03 001oogoo0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`80 0?moobYoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`03001oogoo 0?moobUoo`00DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`03001oogoo 0?moobUoo`00>goo1@000Woo0P000goo10001Woo00<007ooOol0ogoo:Goo000mOol00`00Oomoo`07 Ool00`00Oomoo`07Ool00`00Oomoo`3oOolYOol003eoo`03001oogoo00Qoo`03001oogoo00Ioo`<0 0?mooaeoo`<000Uoo`00?Goo00<007ooOol02Goo00<007ooOol01Goo00<007ooOol0ogoo77oo1@00 27oo000mOol00`00Oomoo`07Ool01000Oomoo`001Woo00<007ooOol0ogoo77oo1@0027oo000lOol2 000:Ool20007Ool00`00Oomoo`3oOolLOol50008Ool0055oo`03001oogoo0?mooaeoo`<000Uoo`00 DGoo00<007ooOol0ogoo:Goo001AOol00`00Oomoo`3oOolYOol0055oo`03001oogoo0?moobUoo`00 \ \>"], ImageRangeCache->{{{0, 379}, {233.75, 0}} -> {4.15218, -0.182087, 0.0104108 , 0.00612018}}] }, Open ]], Cell["\<\ The black shows the average velocity over the whole solution evolution, the \ blue, over the last 200 units of time, and the green shows the \"momentum\" \ divided by the \"mass\", which gives a velocity analogue.\ \>", "Text"], Cell[TextData[{ "For the higher frequencies, we seem to have a pulse that moves across the \ potential with relatively small loss. A look at the ", ButtonBox["evolution", ButtonData:>{"FinalEvolution8.nb", None}, ButtonStyle->"Hyperlink"], " of the solution with \[Omega] = 8 after t = 1200 gives an idea as to what \ is happening." }], "Text"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, ScreenStyleEnvironment->"Working", WindowSize->{697, 664}, WindowMargins->{{2, Automatic}, {Automatic, 1}}, Magnification->1, StyleDefinitions -> "ArticleClassic.nb" ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 44, 0, 56, "Section"], Cell[1786, 53, 243, 4, 44, "Text"], Cell[CellGroupData[{ Cell[2054, 61, 1109, 23, 19, "Input", CellOpen->False], Cell[3166, 86, 14571, 445, 243, 4528, 316, "GraphicsData", "PostScript", "Graphics"] }, Open ]], Cell[17752, 534, 237, 4, 44, "Text"], Cell[17992, 540, 359, 8, 44, "Text"] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)