(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 1388259, 23734] NotebookOptionsPosition[ 1384495, 23613] NotebookOutlinePosition[ 1384978, 23631] CellTagsIndexPosition[ 1384935, 23628] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["A Vibrating Circular Membrane", FontSize->36]], "Title", CellChangeTimes->{{3.4699647146922503`*^9, 3.4699647240203753`*^9}, { 3.4700539089891253`*^9, 3.4700539142078753`*^9}, {3.4730905593877497`*^9, 3.4730905637158747`*^9}, {3.47370336040625*^9, 3.47370336521875*^9}, 3.473703662875*^9, {3.47406136871875*^9, 3.4740613931875*^9}, { 3.4755146515421247`*^9, 3.4755146601202497`*^9}, {3.4758425340108747`*^9, 3.4758425394483747`*^9}, {3.4758693876046247`*^9, 3.4758694009952497`*^9}, {3.4761219287995*^9, 3.476121932112*^9}, { 3.476633434263625*^9, 3.47663344084175*^9}, {3.477824404542375*^9, 3.477824405792375*^9}}, FontSize->16], Cell[TextData[StyleBox["Prepared by Dennis W. Brewer", FontSize->24]], "Subtitle", CellChangeTimes->{{3.4699647311922503`*^9, 3.4699647356610003`*^9}, { 3.474061618578125*^9, 3.474061621*^9}, {3.474061994578125*^9, 3.474061997515625*^9}}, FontSize->16], Cell[CellGroupData[{ Cell["Introduction", "Subsection", CellChangeTimes->{{3.4699649946766253`*^9, 3.4699649979266253`*^9}, { 3.4699658101922503`*^9, 3.4699658118641253`*^9}, {3.4699841041766253`*^9, 3.4699841107860003`*^9}}, FontSize->16], Cell[TextData[{ "The wave equation in polar coordinates is ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "tt"], "=", RowBox[{ SuperscriptBox["a", "2"], "(", RowBox[{ SubscriptBox["u", "rr"], "+", RowBox[{ FractionBox["1", "r"], SubscriptBox["u", "r"]}], "+", RowBox[{ FractionBox["1", SuperscriptBox["r", "2"]], SubscriptBox["u", "\[Theta]\[Theta]"]}]}], ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". If we assume radial symmetry of the initial conditions then ", Cell[BoxData[ FormBox[ RowBox[{"u", " "}], TraditionalForm]]], " is independent of ", Cell[BoxData[ FormBox[ RowBox[{"\[Theta]", " ", "and", " "}], TraditionalForm]]], "the problem reduces to ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "tt"], "=", RowBox[{ SuperscriptBox["a", "2"], "(", RowBox[{ SubscriptBox["u", "rr"], "+", RowBox[{ FractionBox["1", "r"], SubscriptBox["u", "r"]}]}], ")"}]}], TraditionalForm]]], " with boundary condition ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"c", ",", "t"}], ")"}], "=", "0"}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"t", ">", "0"}], TraditionalForm]]], " and initial conditions ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"r", ",", "0"}], ")"}], "=", RowBox[{"f", "(", "r", ")"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["u", "t"], "(", RowBox[{"r", ",", "0"}], ")"}], "=", RowBox[{"g", "(", "r", ")"}]}], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "r", "<", "c"}], TraditionalForm]]], ". \nProceeding by the product method, we seek a solution of the form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"u", "(", RowBox[{"r", ",", "t"}], ")"}], "=", RowBox[{ RowBox[{"R", "(", "r", ")"}], RowBox[{"T", "(", "t", ")"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". Substituting into the PDE and separating the variables yields ", Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ RowBox[{"T", "''"}], RowBox[{ SuperscriptBox["a", "2"], "T"}]], "=", RowBox[{ RowBox[{ FractionBox[ RowBox[{"R", "''"}], "R"], "+", FractionBox[ RowBox[{"R", "'"}], "rR"]}], "=", RowBox[{"-", "\[Lambda]"}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], " which gives to ordinary differential equations ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"T", "''"}], "+", RowBox[{ SuperscriptBox["a", "2"], "\[Lambda]T"}]}], "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"rR", "''"}], "+", RowBox[{"R", "'"}], "+", "\[Lambda]rR"}], "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], ". Since we expect the solutions to oscillate in time, we take ", Cell[BoxData[ FormBox[ RowBox[{"\[Lambda]", "=", RowBox[{ SuperscriptBox["\[Alpha]", "2"], ">", "0"}]}], TraditionalForm]], FormatType->"TraditionalForm"], " so that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"T", "(", "t", ")"}], "=", RowBox[{ RowBox[{ SubscriptBox["c", "1"], RowBox[{"cos", "(", "a\[Alpha]t", ")"}]}], "+", RowBox[{ SubscriptBox["c", "2"], RowBox[{"sin", "(", "a\[Alpha]t", ")"}]}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ". \nNow the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"rR", "''"}], "+", RowBox[{"R", "'"}], "+", RowBox[{ SuperscriptBox["\[Alpha]", "2"], "rR"}]}], "=", "0"}], TraditionalForm]], FormatType->"TraditionalForm"], " is related to Bessel's equation of order zero. 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