(* 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[ 44207, 975] NotebookOptionsPosition[ 42059, 901] NotebookOutlinePosition[ 42480, 917] CellTagsIndexPosition[ 42437, 914] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Simple 2-D Doppler Shift", "Title", CellChangeTimes->{{3.391482157370798*^9, 3.3914821847401533`*^9}, { 3.391482691979528*^9, 3.391482692860795*^9}, {3.3914862632447553`*^9, 3.3914862641260223`*^9}}], Cell[CellGroupData[{ Cell["Motivation", "Subsubsection", CellChangeTimes->{{3.391484852245838*^9, 3.391484853317379*^9}}], Cell["\<\ Develop a simple formula for Doppler shift where the trajectory of the sound \ source is not in the direction of the (stationary) 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Assume it is \ on a straight track some distance from the (stationary) listener's position.\ \>", "Text", CellChangeTimes->{{3.391485457205728*^9, 3.3914855383123536`*^9}, { 3.391486332043683*^9, 3.3914863323641443`*^9}}], Cell[TextData[{ "The component of velocity in the direction of the listener is proportional \ to the hypotenuse of a right triangle with sides:\n ", Cell[BoxData[ FormBox["d", TraditionalForm]]], " -- distance of listener to track\n ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " -- position of the train whistle along the track" }], "Text", CellChangeTimes->{{3.391482191329629*^9, 3.391482205279688*^9}, 3.3914822577952013`*^9, {3.391482306425128*^9, 3.391482361524357*^9}, { 3.3914824627599263`*^9, 3.3914827160641603`*^9}, {3.3914828133740845`*^9, 3.3914828161681023`*^9}, {3.3914832131289043`*^9, 3.3914832162734256`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", RowBox[{"x_", ",", "d_"}], "]"}], ":=", RowBox[{ FractionBox[ RowBox[{"2", RowBox[{"ArcTan", "[", RowBox[{ RowBox[{"-", "x"}], ",", "d"}], "]"}]}], "\[Pi]"], "-", "1"}]}]], "Input", CellChangeTimes->{{3.391482390876563*^9, 3.3914824098137937`*^9}, { 3.39148279492756*^9, 3.391482798232312*^9}, {3.3914828384200993`*^9, 3.3914828425961037`*^9}}], Cell[TextData[{ "In the far field (", Cell[BoxData[ FormBox["d", TraditionalForm]]], " is large), the Doppler shift is gradual. 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