(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. 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). 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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[ 5280, 203]*) (*NotebookOutlinePosition[ 5977, 227]*) (* CellTagsIndexPosition[ 5933, 223]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["MATH 342 Advanced Math for Applications", "Title"], Cell["\<\ Assignment 3 Due Friday, April 25, 2008\ \>", "Subtitle"], Cell["Name: ", "Subtitle"], Cell[CellGroupData[{ Cell["Background", "Section"], Cell[TextData[{ "Consider a string of length ", StyleBox["L", FontSlant->"Italic"], " with wave speed ", StyleBox["c", FontSlant->"Italic"], " and initial displacement ", StyleBox["f", FontSlant->"Italic"], "[", StyleBox["x", FontSlant->"Italic"], "]." }], "Text"], Cell[BoxData[{ \(Clear[f, x]\), "\n", \(\(L = 5;\)\), "\n", \(\(c\ = \ 0.5;\)\), "\n", \(\(f[x_]\ = \ \[ExponentialE]\^\(\(-8\) \((x - 2)\)\^2\);\)\)}], "Input"], Cell[BoxData[ \(\(Plot[f[x], {x, 0, L}, PlotRange \[Rule] All];\)\)], "Input"], Cell[TextData[{ "If the ends of the string are fixed at ", StyleBox["y", FontSlant->"Italic"], "(0, ", StyleBox["t", FontSlant->"Italic"], ") = ", StyleBox["y", FontSlant->"Italic"], "(", StyleBox["L", FontSlant->"Italic"], ", ", StyleBox["t", FontSlant->"Italic"], ") = 0, then the solution can be obtained by extending ", StyleBox["f", FontSlant->"Italic"], "[", StyleBox["x", FontSlant->"Italic"], "] oddly and with period 2", StyleBox["L", FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[{ \(fodd[x_] := Sign[x]\ f[Sign[x]\ x]\), "\[IndentingNewLine]", \(fext[x_]\ := \ fodd[Mod[x + L, 2 L] - L]\)}], "Input"], Cell[BoxData[ \(\(Plot[fext[x], \ {x, \(-2\) L, 2 L}, PlotRange \[Rule] All];\)\)], "Input"], Cell[TextData[{ "Then the solution is ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "(fext(", StyleBox["x", FontSlant->"Italic"], " + ", StyleBox["c t", FontSlant->"Italic"], ") + fext(", StyleBox["x", FontSlant->"Italic"], " - ", StyleBox["c t", FontSlant->"Italic"], "))." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 1", "Section"], Cell[TextData[{ "Create an animation of the solution given at the end of the Background \ section. The animation should run for one period, so that if you run the \ animation as a loop, you will see the periodic solution. Create another \ animation with a different initial condition. Be sure that your initial \ function, ", StyleBox["f", FontSlant->"Italic"], "[", StyleBox["x", FontSlant->"Italic"], "], satisfies the boundary conditions.", " Describe the behavior of a wave when it hits the boundary at x = 0 or at \ x = L." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Exercise 2", "Section"], Cell[TextData[{ "If you change the boundary conditions to ", Cell[BoxData[ \(TraditionalForm\`\(\[PartialD]u\/\[PartialD]x\) \((0, t)\)\ = \ \(\(\[PartialD]u\/\[PartialD]x\) \((L, t)\)\ = \ 0\)\)]], ", you need to extend ", StyleBox["f", FontSlant->"Italic"], " evenly rather than oddly. The solution is still ", Cell[BoxData[ \(TraditionalForm\`1\/2\)]], "(fext(", StyleBox["x", FontSlant->"Italic"], " + ", StyleBox["c t", FontSlant->"Italic"], ") + fext(", StyleBox["x", FontSlant->"Italic"], " - ", StyleBox["c t", FontSlant->"Italic"], ")). Create animations of the solution for two different initial \ conditions. The initial conditions must also satisfy the boundary \ conditions. Describe the behavior of a wave when it hits the boundary." }], "Text"] }, Closed]] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{860, 602}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, PrintingCopies->1, PrintingPageRange->{1, Automatic} ] (******************************************************************* 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[1776, 53, 56, 0, 95, "Title"], Cell[1835, 55, 67, 3, 80, "Subtitle"], Cell[1905, 60, 26, 0, 51, "Subtitle"], Cell[CellGroupData[{ Cell[1956, 64, 29, 0, 73, "Section"], Cell[1988, 66, 298, 14, 33, "Text"], Cell[2289, 82, 182, 4, 93, "Input"], Cell[2474, 88, 82, 1, 30, "Input"], Cell[2559, 91, 548, 26, 33, "Text"], Cell[3110, 119, 145, 2, 50, "Input"], Cell[3258, 123, 106, 2, 30, "Input"], Cell[3367, 127, 330, 17, 36, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[3734, 149, 29, 0, 43, "Section"], Cell[3766, 151, 564, 14, 71, "Text"] }, Closed]], Cell[CellGroupData[{ Cell[4367, 170, 29, 0, 43, "Section"], Cell[4399, 172, 853, 27, 74, "Text"] }, Closed]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)