(************** 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[ 205208, 3209]*) (*NotebookOutlinePosition[ 213024, 3392]*) (* CellTagsIndexPosition[ 211317, 3354]*) (*WindowFrame->Normal*) Notebook[{ Cell["Periodic Phenomena and Fourier Series", "Title"], Cell[CellGroupData[{ Cell["\<\ Playing with Familiar Periodic Phenomena\ \>", "Subtitle"], Cell[TextData[{ StyleBox["Sounds will not be available on PDF or HTML versions", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], "\nLet's begin by \"looking\" at a familiar periodic phenomena:" }], "Text"], Cell[BoxData[{ \(c\ = 0; \ d\ = 1; \ e = 2; \ f = 3; \ g = 4; \ a = 5; \ b = 6;\), "\[IndentingNewLine]", \(\(freq[c]\ = 261.6;\)\), "\[IndentingNewLine]", \(\(freq[d]\ \ = 293.7;\)\), "\[IndentingNewLine]", \(\(freq[e]\ = 329.6;\)\), "\[IndentingNewLine]", \(\(freq[f]\ = \ 349.2;\)\), "\[IndentingNewLine]", \(\(freq[g]\ = 392.0;\)\), "\[IndentingNewLine]", \(\(freq[a]\ = \ 440.0;\)\), "\[IndentingNewLine]", \(\(\(freq[b]\ = \ 493.9;\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Note[note_]\ := \ Sin[\ 2\ Pi\ freq[note]\ t];\)\), "\[IndentingNewLine]", \(Play[Note[c], {t, 0, 2}]\)}], "Input", CellTags->{ "mmtag:17:notes_and_frequencies", "mmtag:17:middle_c", "mmtag:17:Play[]", "mmtag:17:sound_generation"}], Cell[BoxData[ \(Note[{c, e}]\)], "Input"], Cell[BoxData[ \(SetAttributes[Note, Listable]\)], "Input"], Cell[BoxData[ \(Note[{c, e, f}]\)], "Input"], Cell[BoxData[ \(Play[{Note[c], Note[e]}, {t, 0, 2}]\)], "Input"], Cell[BoxData[ \(Play[Note[{c, e}], {t, 0, 2}]\)], "Input"], Cell[BoxData[ \(Play[Evaluate[Note[{c, e}]], {t, 0, 2}]\)], "Input"], Cell[BoxData[ \(Chord[note_]\ := \ Table[Note[Mod[note\ + \ i, 6]], {i, 0, 4, 2}]\)], "Input"], Cell[BoxData[ \(Play[Evaluate[Chord[e]], {t, 0, 2}]\)], "Input"], Cell[BoxData[ \(Play[ If[t > 0.25\ && \ t\ < \ 1.25, Note[a], Note[c]], {t, 0, 1.5}]\)], "Input"], Cell[TextData[{ "Let's see if we can play this: l", Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .11227 %%ImageSize: 482 55 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 -2.77556e-17 0.002079 -0.002079 0.002079 [ [ 0 0 0 0 ] [ 1 .11227 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .11227 L 0 .11227 L closepath clip newpath % Start of colorimage (RGB) p 0 -0.00208 translate 1.00208 .11435 scale 1446 string 482 55 8 [482 0 0 55 0 0] { 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24*0.75}]\)], "Input"], Cell[BoxData[ \(Play[ Sin[1000 x\ Sin[Exp[x/3]\ \ + \ Sin[x]/x]]\ + \ \ Exp[x/10]\ Sin[ x]\ Sin[1500\ x], {x, \(-20\), 10}]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Perioidic Functions of wavelength \[Lambda] using the modulus \ function:\ \>", "Subtitle"], Cell["\<\ Boomerang uses Mod to force a function, f, with a single argument, \ x, to be periodic with wavelength \[Lambda]\ \>", "Text", CellTags->{ "mmtag:17:periodicity__constraining_a_function_to_be_periodic", "mmtag:17:Mod[]"}], Cell[BoxData[ \(Boomerang[f_\ , \ x_\ , \ \[Lambda]_\ ]\ := \ f[Mod[x, \[Lambda]]]\)], "Input"], Cell[BoxData[ \(AFunction[x_\ ]\ := \ \((\((3 - x)\)^3)\)/27\)], "Input"], Cell[TextData[{ "The following step uses ", StyleBox["Boomerang", FontWeight->"Bold"], " to produce a periodic repetition of ", StyleBox["AFunction", FontWeight->"Bold"], " over the range 0 < ", StyleBox["x", FontSlant->"Italic"], " < 6:" }], "Text"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(AFunction\\)\\\" is similar to existing \ symbol \\\"\\!\\(Function\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(Plot[Boomerang[AFunction, x, 6], {x, \(-12\), 12}, PlotRange \[Rule] All]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Demonstration of the orthogonality of the trigonometric functions\ \ \>", "Subtitle", CellTags->"mmtag:17:orthogonality_of_trignometric_functions"], Cell[BoxData[ \(coscos\ = \ Integrate[ Cos[\(2\ \[Pi]\ Minteger\ x\)\/\[Lambda]]\ Cos[\(2\ \[Pi]\ Ninteger\ \ x\)\/\[Lambda]]\ , {x, xo, xo\ + \ \[Lambda]}, \ Assumptions \[Rule] {Minteger\ \[Epsilon]\ Integers, \ Ninteger\ \[Element] \ Integers, xo\ \[Element] \ Reals, \ \[Lambda]\ > \ 0}]\)], "Input"], Cell[BoxData[ \(Simplify[ coscos /. {Minteger \[Rule] 4\ , \ Ninteger \[Rule] 34}]\)], "Input"], Cell[BoxData[ \(Assuming[ Minteger\ \[Epsilon]\ Integers\ && \ Ninteger\ \[Element] \ Integers\ && xo\ \[Element] \ Reals\ && \ \[Lambda]\ \[Element] \ Reals, \ Limit[coscos, Minteger \[Rule] Ninteger]]\)], "Input", CellTags->{"mmtag:17:Assuming[]", "mmtag:17:Limit[]__and_Assuming[]"}], Cell[BoxData[ \(cossin\ = \ Integrate[ Cos[\(2\ \[Pi]\ Minteger\ x\)\/\[Lambda]]\ Sin[\(2\ \[Pi]\ Ninteger\ \ x\)\/\[Lambda]]\ , {x, xo, xo\ + \ \[Lambda]}, \ Assumptions \[Rule] {Minteger\ \[Epsilon]\ Integers, \ Ninteger\ \[Element] \ Integers, xo\ \[Element] \ Reals, \ \[Lambda]\ > \ \ 0}]\)], "Input"], Cell[BoxData[ \(Simplify[ cossin /. {Minteger \[Rule] \(-7\)\ , \ Ninteger \[Rule] 35}]\)], "Input"], Cell[BoxData[ \(Assuming[ Minteger\ \[Epsilon]\ Integers\ && \ Ninteger\ \[Element] \ Integers\ && xo\ \[Element] \ Reals\ && \ \[Lambda]\ \[Element] \ Reals, \ Limit[cossin, Minteger \[Rule] Ninteger]]\)], "Input"], Cell[BoxData[ \(sinsin\ = \ Integrate[ Sin[\(2\ \[Pi]\ Minteger\ x\)\/\[Lambda]]\ Sin[\(2\ \[Pi]\ Ninteger\ \ x\)\/\[Lambda]]\ , {x, xo, xo\ + \ \[Lambda]}, \ Assumptions \[Rule] {Minteger\ \[Epsilon]\ Integers, \ Ninteger\ \[Element] \ Integers, xo\ \[Element] \ Reals, \ \[Lambda]\ > \ 0}]\)], "Input"], Cell[BoxData[ \(Simplify[ sinsin /. {Minteger \[Rule] 10\ , \ Ninteger \[Rule] 9}]\)], "Input"], Cell[BoxData[ \(Assuming[ Minteger\ \[Epsilon]\ Integers\ && \ Ninteger\ \[Element] \ Integers\ && xo\ \[Element] \ Reals\ && \ \[Lambda]\ \[Element] \ Reals, \ Limit[sinsin, Minteger \[Rule] Ninteger]]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Fourier Series: Example of Determination of Fourier \ Foefficients\ \>", "Subtitle"], Cell[TextData[{ "First we will \"do it the hard way\" and write short programs that \ evaluate Fourier coefficients; then we will demonstrate how to make use of \ built-in functions in ", StyleBox["Mathematica", FontSlant->"Italic"], "'s ", StyleBox["FourierTransform", FontWeight->"Bold"], " package\[Ellipsis]" }], "Text"], Cell["\<\ Define functions based on the formulas derived for the fourier \ amplitudes\ \>", "Text"], Cell["The constant term:", "Text"], Cell[BoxData[ \(EvenTerms[0, function_\ , \ wavelength_]\ := \ \(1\/wavelength\) \(\[Integral]\_0\%wavelength function[dummy] \[DifferentialD]dummy\)\)], "Input"], Cell["\<\ A function that defines each even amplitude individually (this is \ not very efficient, it would be better to evaluate the integral once and use \ that result)\ \>", "Text"], Cell[BoxData[ \(EvenTerms[SP_Integer, function_\ , \ wavelength_]\ := \ \(EvenTerms[SP, function\ , \ wavelength] = \(2\/wavelength\) \(\[Integral]\_0\%wavelength function[ dummy]\ Cos[\(2\ SP\ \[Pi]\ dummy\)\/wavelength] \ \[DifferentialD]dummy\)\)\)], "Input", CellTags-> "mmtag:17:Fourier_series__example_function_for_calculating_cosine_terms"], Cell["\<\ Define the zeroth odd term as zero for symmetry with the even \ terms:\ \>", "Text"], Cell[BoxData[ \(OddTerms[0, function_\ , \ wavelength_]\ := \ 0\)], "Input"], Cell[BoxData[ \(OddTerms[SP_Integer, function_\ , \ wavelength_]\ := \ \(OddTerms[SP, function\ , \ wavelength] = \(2\/wavelength\) \(\[Integral]\_0\%wavelength function[ dummy]\ Sin[\(2\ SP\ \[Pi]\ dummy\)\/wavelength] \ \[DifferentialD]dummy\)\)\)], "Input", CellTags-> "mmtag:17:Fourier_series__example_function_for_calculating_sine_terms"], Cell["\<\ A function to create a vector of amplitudes for the odd terms and \ one for the even terms\ \>", "Text"], Cell[BoxData[ \(OddAmplitudeVector[NTerms_Integer, \ function_, \ wavelength_]\ := \[IndentingNewLine]Table[ OddTerms[i, \ function, \ wavelength], \ {i, 0, NTerms}]\)], "Input"], Cell[BoxData[ \(EvenAmplitudeVector[NTerms_Integer, \ function_, \ wavelength_]\ := \[IndentingNewLine]Table[ EvenTerms[i, \ function, \ wavelength], \ {i, 0, NTerms}]\)], "Input"], Cell["Define a function to use as an example:", "Text"], Cell[BoxData[ \(myfunction[x_\ ]\ := \((x*\((2 - x)\)*\((1 - x)\)^2)\)\)], "Input"], Cell[BoxData[ \(OriginalPlot = \ Plot[myfunction[x], {x, 0, 2}, PlotStyle \[Rule] {Hue[1], Thickness[0.015]}]\)], "Input"], Cell["\<\ Now calculate Fourier amplitudes for this function\[Ellipsis]\ \>", \ "Text"], Cell[BoxData[ \(\(\(\ \)\(EvenAmplitudeVector[6, \ myfunction, 2]\)\)\)], "Input"], Cell[TextData[{ "The odd terms should all vanish because the function is even around the \ center x=", Cell[BoxData[ \(TraditionalForm\`\[Lambda]\/2\)]] }], "Text"], Cell[BoxData[ \(OddAmplitudeVector[6, \ myfunction, 2]\)], "Input"], Cell[BoxData[ \(OddBasisVector[NTerms_Integer, var_, \ \ wavelength_]\ := \ Table[Sin[\(2\ \[Pi]\ \ i\ var\)\/wavelength], \ {i, 0, NTerms}]\)], "Input", CellTags->"mmtag:17:Fourier_series__basis_functions"], Cell[BoxData[ \(OddBasisVector[6, x, 2]\)], "Input"], Cell[BoxData[ \(EvenBasisVector[NTerms_Integer, var_, \ \ wavelength_]\ := \ Table[Cos[\(2\ \[Pi]\ \ i\ var\)\/wavelength], \ {i, 0, NTerms}]\)], "Input"], Cell[BoxData[ \(EvenBasisVector[6, x, 2]\)], "Input"], Cell["\<\ Make the Fourier series representation by taking the dot product of \ the AmplitudeVectors with the BasisVectors:\ \>", "Text"], Cell[BoxData[ \(FourierTruncSeries[n_, \ function_, var_\ , \ wavelength_] := \ EvenAmplitudeVector[n, function, wavelength] . EvenBasisVector[n, var, wavelength]\ + \ \[IndentingNewLine]OddAmplitudeVector[n, function, wavelength] . OddBasisVector[n, var, wavelength]\)], "Input", CellTags->"mmtag:17:Fourier_series__function_to_create_truncated_series"], Cell[TextData[{ "For ", StyleBox["myfunction", FontWeight->"Bold"], " we write the Fourier series out to six terms as:" }], "Text"], Cell[BoxData[ \(FourierTruncSeries[6, myfunction, x, 2]\)], "Input"], Cell[BoxData[ \(FourierPlot\ = Plot[FourierTruncSeries[3, myfunction, x, 2], {x, \(-2\), 4}]\)], "Input"], Cell[TextData[{ "Note the periodicity of the plot above. The wavelength is 2. Only three \ terms were calculated in the Fourier series representation of ", StyleBox["myfunction", FontWeight->"Bold"], ". Below the Fourier series and the exact function are shown superposed." }], "Text"], Cell[BoxData[ \(Show[OriginalPlot, FourierPlot]\)], "Input"], Cell[TextData[{ "Now the calculation is repeated but retaining 6 terms in the Fourier \ series representation and hopefully obtaining a more accurate representation \ of ", StyleBox["myfunction", FontWeight->"Bold"], "." }], "Text"], Cell[BoxData[ \(FourierPlot\ = Plot[FourierTruncSeries[6, myfunction, x, 2], {x, \(-2\), 4}]\)], "Input", CellTags->"mmtag:17:Fourier_series__example_plot"], Cell[BoxData[ \(Show[OriginalPlot, FourierPlot]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Fourier Series Coefficients using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s FourierTransform package: Preparing the Input Function" }], "Subtitle"], Cell[BoxData[ \(<< Calculus`FourierTransform`\)], "Input", CellTags->{ "mmtag:17:FourierTransform_Package", "mmtag:17:Package__FourierTransform"}],\ Cell[BoxData[ \(AFunction[x_]\ := \ \((x - 3)\)^3\/27\)], "Input"], Cell[BoxData[ \(Plot[AFunction[x], {x, 0, 6}]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], "'s Fourier Series functions are defined for function that are periodic in \ the domain x \[Element] (-1/2,1/2). So we need to map the periodic functions \ to this domain" }], "Text", CellTags->"mmtag:17:Fourier_series__changing_domain_of_periodicity"], Cell[BoxData[ \(ReduceHalfHalf[f_\ , \ x_\ , \ \[Lambda]_\ ]\ := \ f[\((x + 1/2)\)*\[Lambda]\ ]\)], "Input"], Cell[BoxData[ \(ReducedFunction\ = \ ReduceHalfHalf[AFunction, x, 6] // Simplify\)], "Input"], Cell[BoxData[ \(Plot[ReducedFunction, {x, \(-1\)/2, 1/2}, PlotRange \[Rule] All]\)], "Input"], Cell[BoxData[ \(FourierCosCoefficient[ReducedFunction, x, n]\)], "Input", CellTags->"mmtag:17:FourierCosCoefficient[]"], Cell[BoxData[ \(FourierSinCoefficient[ReducedFunction, x, n]\)], "Input", CellTags->"mmtag:17:FourierSinCoefficient[]"], Cell[BoxData[ \(FourierTrigSeries[ReducedFunction, x, 5]\)], "Input", CellTags->"mmtag:17:FourierTrigSeries[]"] }, Closed]], Cell[CellGroupData[{ Cell["Convergence of Fourier Series: Gibbs Phenomenon", "Subtitle"], Cell[BoxData[ \(\(AnimateTruncatedFourierSeries[ function_, \ {truncationstart_, \ truncationend_, \ truncjump_}]\ := \ Table[Plot[ Evaluate[FourierTrigSeries[function, x, itrunc]], {x, \(-1\), 1}, PlotRange \[Rule] \ {\(-2\), 2}], {itrunc, truncationstart, truncationend, \ truncjump}];\)\)], "Input"], Cell["\<\ The function above will work, but it is horribly inefficient! \ Because it asks FourierTrigSeries to calculate one more term each time, it is \ doing some redundant work. We can fix this up by having it calculate one new term and adding to the sum calculated previously. Here \ it is:\ \>", "Text", CellTags-> "mmtag:17:Fourier_series__example_function_for_calculating_and_efficiency"], Cell[BoxData[ \(AnimateTruncatedFourierSeries[ function_, \ {truncationstart_, \ truncationend_}]\ := \ Module[{coscof, sincof, \ currentappx\ , n, \ TwoPi = \ 2\ \[Pi]}, \ \[IndentingNewLine]currentappx\ = \ FourierCosCoefficient[function, x, 0]; \[IndentingNewLine]coscof[ n_]\ = \ Simplify[ FourierCosCoefficient[function, x, n]]; \[IndentingNewLine]sincof[ n_]\ = \ Simplify[ FourierSinCoefficient[function, x, n]]; \[IndentingNewLine]\[IndentingNewLine]Table[ Plot[Evaluate[ currentappx\ += \ coscof[itrunc]*Cos[TwoPi\ itrunc\ x]\ + \ \ sincof[itrunc]* Sin[TwoPi\ itrunc\ \ x]], {x, \(-1\), 1}, PlotRange \[Rule] \ {\(-2\), 2}], {itrunc, truncationstart, truncationend}];]\)], "Input", CellTags->{ "mmtag:17:Module[]", "mmtag:17:Fourier_series__example_animation"}], Cell["\<\ The following will demonstrate how convergence is difficult where \ the function changes rapidly---this is known as Gibbs' Phenomenon\ \>", "Text"], Cell[BoxData[ \(AnimateTruncatedFourierSeries[ReducedFunction, {1, 60}]\)], "Input", CellTags->{ "mmtag:17:Fourier_series__Gibbs_phenomena", "mmtag:17:Gibbs_phenomena", "mmtag:17:Fourier_series__visualization_of_convergence"}], Cell["Recall the definition used above for myfunction", "Text"], Cell[BoxData[ \(myfunction[x]\)], "Input"], Cell[TextData[{ "Reduce myfunction to a domain so that we can use ", StyleBox["Mathematica'", FontSlant->"Italic"], "s Fourier Package on it." }], "Text"], Cell[BoxData[ \(ReducedMyFunction\ = \ ReduceHalfHalf[myfunction, x, 2] // Simplify\)], "Input"], Cell["General form of the even amplitudes", "Text"], Cell[BoxData[ \(myfunccos\ = FourierCosCoefficient[4\ x\^2 - 16\ x\^4, x, n]\)], "Input"], Cell[BoxData[ \(FourierSinCoefficient[4\ x\^2 - 16\ x\^4, x, n]\)], "Input"], Cell["\<\ Illustrate how amplitude of each term behaves. 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