(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 20829288, 338027] NotebookOptionsPosition[ 20709962, 335486] NotebookOutlinePosition[ 20821799, 337797] CellTagsIndexPosition[ 20821756, 337794] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["polarVibes_BesselFns", "Title", CellChangeTimes->{{3.444109944671875*^9, 3.444109953234375*^9}}], Cell["\<\ This notebook contains code to illustrate Bessel functions and their \ application to solving problems involving the Laplacian inpolar coordinates.\ \>", "Subtitle", CellChangeTimes->{{3.44410995590625*^9, 3.444109992609375*^9}}], Cell["\<\ Once again, define a style for plots so they project visibly.\ \>", "Text", 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StyleBox[ SuperscriptBox[ RowBox[{ SubscriptBox["Y", "3"], "(", "x", ")"}], "2"], FontColor -> RGBColor[0.5, 0, 0.5], FontSize -> 24, StripOnInput -> False]}], "}"}], TraditionalForm]}, AxesOrigin->{0, 0}, ImageSize->400, PlotRange->{{0, 25}, {0., 1.5304915692338796`}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]}], "}"}]], "Output", CellChangeTimes->{{3.444111251921875*^9, 3.444111273578125*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Use Bessel functions as eigenfunctions for solving PDEs.", "Section"], Cell[TextData[{ "In the example of the circular drumhead, the ODE for the radial dependence \ turned out to be Bessel's equation where the order n is determined by the \ periodicity requirement in the \[Theta] direction. Specifically, if the trig \ functions in the \[Theta] direction have n cycles around the drum (i.e. in 2\ \[Pi] radians), then the relevant radial solutions are ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["J", "n"], "(", "kr", ")"}], TraditionalForm]]], ". (Remember that the other solution, ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["Y", "n"], "(", "kr", ")"}], TraditionalForm]]], "would blow up at r=0, so it cannot contribute to describe the finite \ displacement of the drumhead.) " }], "Text"], Cell[TextData[{ "For the moment, let's simplify things a bit and consider initial conditions \ that are independent of \[Theta], so the solution becomes independent of \ \[Theta]. With this assumption, n=0 and the angular solution solution \ simplifies to \[CapitalTheta](\[Theta])~ Cos[0*\[Theta]]~1. The time \ dependence is harmonic with frequency c*k, but we still do not know what k \ is. We need to determine it from the radial boundary condition, i.e. \ R(\[Rho]) = 0, which implies ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["J", "0"], "(", "k\[Rho]", ")"}], "=", "0"}], TraditionalForm]]], ". The eigenfunctions are then Bessel functions with arguments chosen so \ that the outer rim of the drumhead corresponds to a zero of the Bessel \ function.To have a hope of fitting arbitrary functions, we need to have an \ infinite number of eigenfunctions and an infinite number of associated roots \ of the Bessel function. The good news is that the Bessel functions oscillate \ (it can be shown that for large values of the argument, ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["J", "n"], "(", "x", ")"}], "~", SqrtBox[ FractionBox["2", "\[Pi]x"]]}], RowBox[{"cos", "(", RowBox[{"x", "-", FractionBox["\[Pi]", "4"], "-", FractionBox["n\[Pi]", "2"]}], ")"}]}], TraditionalForm]]], ") and provide an infinite number of zeros. Are they evenly spaced like the \ zeros of sin and cos? Not quite (although they become evenly spaced for large \ arguments), so they will need to be found numerically. The good news is that \ this has been done for us automatically by BesselJZero. Use it to construct a \ list of the first several eigenvalues:" }], "Text", CellChangeTimes->{{3.444113139875*^9, 3.444113187078125*^9}, { 3.444113286796875*^9, 3.444113322390625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Alpha]", " ", "=", " ", RowBox[{ RowBox[{"Table", "[", RowBox[{ RowBox[{"BesselJZero", "[", RowBox[{"0", ",", "m"}], "]"}], ",", RowBox[{"{", RowBox[{"m", ",", "1", ",", "5"}], "}"}]}], "]"}], " ", "//", "N"}]}]], "Input", CellChangeTimes->{{3.444113204546875*^9, 3.444113216734375*^9}, { 3.444113249265625*^9, 3.444113259625*^9}, {3.444113325109375*^9, 3.444113355765625*^9}, {3.4441134586875*^9, 3.44411346553125*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "2.404825557695773`", ",", "5.5200781102863115`", ",", "8.653727912911013`", ",", "11.791534439014281`", ",", "14.930917708487787`"}], "}"}]], "Output",\ CellChangeTimes->{{3.444113206296875*^9, 3.444113217296875*^9}, { 3.4441132556875*^9, 3.444113260359375*^9}, 3.444113357984375*^9, 3.44411346821875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "-", RowBox[{"RotateRight", "[", "%", "]"}]}]], "Input", CellChangeTimes->{{3.44411337771875*^9, 3.444113384109375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", "12.526092150792014`"}], ",", "3.1152525525905386`", ",", "3.133649802624701`", ",", "3.137806526103269`", ",", "3.1393832694735053`"}], "}"}]], "Output", CellChangeTimes->{3.44411338478125*^9}] }, Closed]], Cell[TextData[{ "So now we have the first 5 roots of ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["J", "0"], "(", "x", ")"}], "."}], TraditionalForm]]], "Let's pick a value, say \[Rho]=1, for the radius of the drumhead so we can \ generate some plots numerically. Radial solutions then look like ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["J", "0"], "(", RowBox[{ SubscriptBox["\[Alpha]", "n"], "r"}], ")"}], TraditionalForm]]], ". Let's take a look at these functions." }], "Text", CellChangeTimes->{{3.444113439171875*^9, 3.444113439296875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Table", "[", RowBox[{ RowBox[{"BesselJ", "[", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"\[Alpha]", "[", RowBox[{"[", "n", "]"}], "]"}], "*", "r"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", "1", ",", "5"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.444113475328125*^9, 3.444113491171875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"BesselJ", "[", RowBox[{"0", ",", RowBox[{"2.404825557695773`", " ", "r"}]}], "]"}], ",", RowBox[{"BesselJ", "[", RowBox[{"0", ",", RowBox[{"5.5200781102863115`", " ", "r"}]}], "]"}], ",", RowBox[{"BesselJ", "[", RowBox[{"0", ",", RowBox[{"8.653727912911013`", " ", "r"}]}], "]"}], ",", RowBox[{"BesselJ", "[", RowBox[{"0", ",", RowBox[{"11.791534439014281`", " ", "r"}]}], "]"}], ",", RowBox[{"BesselJ", "[", RowBox[{"0", ",", RowBox[{"14.930917708487787`", " ", "r"}]}], "]"}]}], "}"}]], 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Let's look at \ the inner products (rembering the weighting factor r in the inner product \ that we obtain from Sturm-Liouville):\ \>", "Text", CellChangeTimes->{{3.44411354915625*^9, 3.444113566375*^9}, { 3.444113618359375*^9, 3.444113626671875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Table", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Integrate", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"r", "*", RowBox[{"BesselJ", "[", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"\[Alpha]", "[", RowBox[{"[", RowBox[{"m", "+", "1"}], "]"}], "]"}], "*", "r"}]}], "]"}], "*", RowBox[{"BesselJ", "[", RowBox[{"0", ",", " ", RowBox[{ RowBox[{"\[Alpha]", "[", RowBox[{"[", RowBox[{"n", "+", "1"}], "]"}], "]"}], "*", "r"}]}], "]"}]}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"r", ",", "0", ",", "1"}], "}"}]}], "]"}], ",", "\[IndentingNewLine]", RowBox[{"{", RowBox[{"m", ",", "0", ",", "4"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"n", ",", "0", ",", "4"}], "}"}]}], "]"}], ";"}]], "Input", CellChangeTimes->{{3.444113574203125*^9, 3.44411357828125*^9}, { 3.44411374659375*^9, 3.444113757078125*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"MatrixForm", "[", RowBox[{"Chop", "[", "%", "]"}], "]"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0.13475706197095846`", "0", "0", "0"}, {"0", "0.05789006929110183`", "0", "0"}, {"0", "0", "0.0368431755682041`", "0"}, {"0", "0", "0", "0.027018786599058146`"} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output", CellChangeTimes->{3.444113598578125*^9}] }, Open ]], Cell["\<\ So we have verified the orthogonality relation and we are in position to deal \ with finding coefficients to satisfy initial conditions. Again let's start \ simple and let the initial displacement be u(r,0)=(1-r) with zero initial \ velocity. 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