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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[ 490117, 11806]*) (*NotebookOutlinePosition[ 491188, 11842]*) (* CellTagsIndexPosition[ 491144, 11838]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["\nExperiment 01\n\n", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["Sound Waves\n", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Title", TextAlignment->Center, FontVariations->{"Underline"->True}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Starting Up", "Section", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}]], "Subtitle", TextAlignment->Center, FontSize->18, FontWeight->"Bold", FontSlant->"Plain"], Cell[TextData[{ StyleBox["Prevent warnings about the use of symbols similar to those used \ by ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" because it will not let us change their definitions.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[ \(Off[General::spell]; Off[General::spell1];\)], "Input"], Cell[TextData[StyleBox["Initialize and load the needed packages.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[BoxData[{ \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\), "\n", \(Needs["\"]\), "\[IndentingNewLine]", \(Needs["\"]\), "\[IndentingNewLine]", \(\(ndataTotal = 0;\)\)}], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "\n", StyleBox["Velocity Of Sound from Resonance Tube Measurements", "Section", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Subtitle", TextAlignment->Center, FontSize->18, FontWeight->"Bold", FontSlant->"Plain"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Raw Experimental Data", "Section", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Section", TextAlignment->Left, TextJustification->0, FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["The experimental data consist of various lengths of \ air column. 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"\[IndentingNewLine]", \(frequency[ndata] = {421. , 606. , 848. , 1091. , \ 1335. , \ 1576. };\), "\[IndentingNewLine]", RowBox[{\(nfreq = Length[frequency[ndata]];\), "\[IndentingNewLine]", RowBox[{"(*", StyleBox[\(If\ line\ drawn\ through\ data\ points\ does\ not\ go\ \ through\ the\ origin, increase\ or\ decrease\ nfirst\ below\ by\ 1\ or\ 2. \), FontColor->RGBColor[1, 0, 0]], "*)"}]}], "\[IndentingNewLine]", \(nfirst = 2;\), "\n", \(order[ndata] = Range[nfirst, nfreq + nfirst - 1];\), "\n", \(harmonic[ndata] = 2*order[ndata] - 1;\), "\[IndentingNewLine]", \(rawData[ndata] = Transpose[{harmonic[ndata], frequency[ndata]}];\), "\[IndentingNewLine]", \(ListPlot[ rawData[ndata], Frame\ \[Rule] \ True, Axes\ \[Rule] \ False, PlotRange\ \[Rule] \ {{0, 12}, {0, 1750}}, \ \n\t\tFrameLabel\ \[Rule] \ {"\<2n-1\>", \ "\"}, \n\t\tPlotLabel\ \[Rule] \ "\", DefaultFont\ \[Rule] \ {"\", 11}, Prolog\ \[Rule] \ AbsolutePointSize[5]];\)}], "Input"], Cell[BoxData[ \(0.2`\)], "Output"], 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\(\(\[Sigma]\_lengths = Table[\[Sigma][n], {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(orders = Table[order[n], {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(harmonics = Table[harmonic[n], {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(frequencies = Table[frequency[n], {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(frequencyData = Table[rawData[n], {n, 1, ndataTotal}];\)\)}], "Input"], Cell[TextData[StyleBox["Show a sample frequencyData for, say, length #3.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(frequencyData[\([3]\)]\)], "Input"], Cell[BoxData[ \({{3, 444.`}, {5, 731.`}, {7, 1027.`}, {9, 1322.`}, {11, 1618.`}, {13, 1908.`}}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Plot Data", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Section", FontColor->RGBColor[0, 0, 1]], Cell[TextData[StyleBox["Now plot 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FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["Next, let's fit the frequency vs. harmonics data for a given air \ column length to", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[" f = ", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\(\(v\/\(4 L\)\) \((2 n - 1)\)\(\ \)\)\)], FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["with ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" function ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Regress", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[". The constant (or intercept) term is excluded from the \ regression (using the option ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["IncludeConstant -> False", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" below). Since the slope equals ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`v\/\(4 L\)\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[", the speed of sound in air can be obtained by multiplying the \ fitted slope by four times of the length of air column.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[{ \(Clear[n, x]\), "\[IndentingNewLine]", \(fitFrequencyData[n_Integer] := Regress[frequencyData[\([n]\)], {x}, x, IncludeConstant\ \[Rule] \ False, RegressionReport\ \[Rule] \ {BestFit, ParameterCITable}]\)}], "Input"], Cell[TextData[StyleBox["A sample fit result for, say, length #3 can be viewed \ as follows:", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(fitFrequencyData[3]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{\(BestFit \[Rule] 146.86343612334804`\ x\), ",", RowBox[{"ParameterCITable", "\[Rule]", TagBox[GridBox[{ {"\<\"\"\>", "\<\"Estimate\"\>", "\<\"SE\"\>", "\<\"CI\"\>"}, {"x", "146.86343612334804`", "0.11786096424688612`", \({146.56046486952442`, 147.16640737717165`}\)} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableDepth -> 2, TableHeadings -> {{x}, {"Estimate", "SE", "CI"}}]]]}]}], "}"}]], "Output"] }, Open ]], Cell[TextData[StyleBox["Extract the parameters from the fit. The results of \ interest are the slopes.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"], Cell[BoxData[ \(\(frequencyDataFits = Table[Transpose[\((ParameterCITable /. fitFrequencyData[n])\)[\([1]\)]], {n, 1, ndataTotal}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(slopes = Flatten[Table[ frequencyDataFits[\([n, 1]\)], {n, 1, ndataTotal}]]\)], "Input"], Cell[BoxData[ \({105.98596491228075`, 121.63215859030839`, 146.86343612334804`, 201.17192982456146`, 341.47878787878767`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Sigma]\_slopes = Flatten[Table[ frequencyDataFits[\([n, 2]\)], {n, 1, ndataTotal}]]\)], "Input"], Cell[BoxData[ \({0.0834763316025959`, 1.1898192371096774`, 0.11786096424688612`, 0.12058133528027705`, 1.9451477148768288`}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Generate plots of the fits. The plots are temporarily suppressed \ by using the ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Plot", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" option ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["DisplayFunction -> Identity.", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[{ \(\(frequencyDataBestFits = Table[\((BestFit /. fitFrequencyData[n])\), {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(frequencyDataBestFitsPlots = Table[Plot[frequencyDataBestFits[\([n]\)], {x, 0, 12}, PlotRange\ \[Rule] \ {{0, 12}, {0, 3000}}, DisplayFunction \[Rule] \ Identity], {n, 1, ndataTotal}];\)\)}], "Input"], Cell[TextData[StyleBox["Now overlay the fit results with the data.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"], 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The fractional error in speed is equal to ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\@\(\((\[Sigma]\_slope/slope)\)\^2\ + \ \((\[Sigma]\_length/length)\)\^2\)\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[".", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(speeds = Table[4\ slopes[\([n]\)] lengths[\([n]\)]/100. , {n, 1, ndataTotal}];\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_speeds = speeds\ \@\(\((\[Sigma]\_slopes/slopes)\)\^2\ + \ \ \((\[Sigma]\_lengths/lengths)\)\^2\);\)\), "\[IndentingNewLine]", \(PaddedForm[ TableForm[{lengths, speeds, \[Sigma]\_speeds} // Transpose, TableHeadings \[Rule] \ {None, {"\", "\", "\"}}], {5, 2}]\)}], "Input"], Cell[BoxData[ TagBox[ TagBox[GridBox[{ {"\<\"Length (cm)\"\>", "\<\"Speed (m/sec)\"\>", "\<\"Error \ (m/sec)\"\>"}, { InterpretationBox["\<\" 80.20\"\>", 80.200000000000003, AutoDelete->True], InterpretationBox["\<\" 340.00\"\>", 340.00297543859665, AutoDelete->True], InterpretationBox["\<\" 0.89\"\>", .88917162474159606, AutoDelete->True]}, { InterpretationBox["\<\" 70.30\"\>", 70.299999999999997, AutoDelete->True], InterpretationBox["\<\" 342.03\"\>", 342.0296299559472, AutoDelete->True], InterpretationBox["\<\" 3.48\"\>", 3.4843978936998656, AutoDelete->True]}, { InterpretationBox["\<\" 57.60\"\>", 57.600000000000001, AutoDelete->True], InterpretationBox["\<\" 338.37\"\>", 338.37335682819389, AutoDelete->True], InterpretationBox["\<\" 1.21\"\>", 1.205880554867867, AutoDelete->True]}, { InterpretationBox["\<\" 41.90\"\>", 41.899999999999999, AutoDelete->True], InterpretationBox["\<\" 337.16\"\>", 337.16415438596499, AutoDelete->True], InterpretationBox["\<\" 1.62\"\>", 1.6220146163635945, AutoDelete->True]}, { InterpretationBox["\<\" 25.30\"\>", 25.300000000000001, AutoDelete->True], InterpretationBox["\<\" 345.58\"\>", 345.57653333333315, AutoDelete->True], InterpretationBox["\<\" 3.37\"\>", 3.3671720577922448, AutoDelete->True]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], Function[ BoxForm`e$, TableForm[ BoxForm`e$, TableHeadings -> {None, {"Length (cm)", "Speed (m/sec)", "Error (m/sec)"}}]]], (PaddedForm[ #, {5, 2}]&)]], "Output"] }, Open ]], Cell[TextData[StyleBox["The result above is equivalent to making five \ separate measurements of the speed of sound in air using five different \ lengths of air column. The final result is the weighted average. 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As one can see, the speed increases as the length \ increases. This is because there is a systematic error caused by assuming \ that the pressure node (or displacement antinode) at the open end of the tube \ is at the physical end of the tube. This is not true. In fact, the pressure \ node is located roughly at a distance R (the radius of the tube) beyond the \ physical end. 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The speed of sound with end correction \ included can now be obtained by dividing 4 by the new fitted slope and the \ end correction can be obtained by dividing the new fitted intercept by ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["(", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`4\/v\)], FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[")", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" which is just", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], " ", StyleBox["the new slope.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(slopeLengthData = Transpose[{lengths, 1/slopes}];\)\), "\[IndentingNewLine]", \(fitSlopeLengthData = 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\(\(slopeLengthDataFits = Transpose[\((ParameterCITable /. fitSlopeLengthData)\)[\([1]\)]];\)\), "\[IndentingNewLine]", \(\(newIntercept = slopeLengthDataFits[\([1, 1]\)];\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_newIntercept = slopeLengthDataFits[\([2, 1]\)];\)\), "\[IndentingNewLine]", \(\(newSlope = slopeLengthDataFits[\([1, 2]\)];\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_newSlope = slopeLengthDataFits[\([2, 2]\)];\)\), "\[IndentingNewLine]", \(\(speedWithEndCorrection = \(4/newSlope\)/ 100;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_speedWithEndCorrection = speedWithEndCorrection\ \((\[Sigma]\_newSlope/ newSlope)\);\)\), "\[IndentingNewLine]", \(\(\[CapitalDelta]L = newIntercept/newSlope;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_\[CapitalDelta]L = \[CapitalDelta]L\ \ \@\(\((\[Sigma]\_newIntercept/newIntercept)\)\^2\ + \ \ \((\[Sigma]\_newSlope/newSlope)\)\^2\);\)\), "\[IndentingNewLine]", \(Print[ StyleForm["\"\ <> \ ToString[ PaddedForm[ speedWithEndCorrection, {5, 2}]]\ <> \ "\< \[PlusMinus] \>"\ <> \ ToString[ PaddedForm[\[Sigma]\_speedWithEndCorrection, {3, 2}]]\ <> \ \ "\<) m/sec\>", FontFamily\ \[Rule] \ "\", FontWeight\ \[Rule] \ "\", FontColor\ \[Rule] \ RGBColor[1, 0, 1], FontSize\ \[Rule] \ 12]]\), "\[IndentingNewLine]", \(Print[ StyleForm["\"\ <> \ ToString[ PaddedForm[\[CapitalDelta]L, {3, 2}]]\ <> \ "\< \[PlusMinus] \>"\ <> \ ToString[ PaddedForm[\[Sigma]\_\[CapitalDelta]L, {3, 2}]]\ <> \ \ "\<) cm\>", FontFamily\ \[Rule] \ "\", FontWeight\ \[Rule] \ "\", FontColor\ \[Rule] \ RGBColor[1, 0, 1], FontSize\ \[Rule] \ 12]]\)}], "Input"], Cell[BoxData[ TagBox[ StyleBox["\<\"Speed of sound with end correction = ( 339.68 \ \[PlusMinus] 3.25) m/sec\"\>", FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], (StyleForm[ #, FontFamily -> "Helvetica", FontWeight -> "Bold", FontColor -> RGBColor[ 1, 0, 1], FontSize -> 12]&)]], "Print"], Cell[BoxData[ TagBox[ StyleBox["\<\"End correction = (-0.09 \[PlusMinus] -0.56) cm\"\>", FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], (StyleForm[ #, FontFamily -> "Helvetica", FontWeight -> "Bold", FontColor -> RGBColor[ 1, 0, 1], FontSize -> 12]&)]], "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "\n", StyleBox["Velocity Of Sound from Atmospheric Data\n", "Section", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->True}] }], "Subtitle", TextAlignment->Center, FontSize->18, FontWeight->"Bold", FontSlant->"Plain"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Experimental Data", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Section", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ StyleBox["In this part of the experiment, the speed of sound in air will be \ determined indirectly (actually computed) by measuring the ambient or \ background values of the pressure (", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`p\_0\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[") and density (", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\[Rho]\_0\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[") of the air using the formula: ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`v\_s\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[" = ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`\@\(\((\[Gamma]\ p\_0)\)/\[Rho]\_0\)\)], FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox[", where \[Gamma] is the ratio of the specific heats of air at \ constant pressure and constant volume and is equal to 1.4.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[BoxData[{ \(\(T\_web = 71\ \ \ \ \ \ \ (*\ temperature\ of\ web\ bulb\ in\ degrees\ F\ *) ;\)\), "\ \[IndentingNewLine]", \(\(T\_dry = \ 73\ \ \ \ \ \ (*\ temperature\ of\ dry\ bulb\ in\ degrees\ F\ *) ;\)\), "\ \[IndentingNewLine]", \(\(\[Sigma]\_\(T\_dry\) = 0.3;\)\), "\[IndentingNewLine]", \(\(relativeHumidity = 0.91\ \ \ \ \ \ \ (*\ in\ desimal, \ not\ %\ , \ obtained\ from\ nearby\ tables\ *) ;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_relativeHumidity = 0.0025;\)\), "\[IndentingNewLine]", \(\(p\_svapor = 21.043\ \ \ \ \ \ \ (*\ in\ cm\ Hg, \ obtained\ from\ nearby\ tables\ *) ;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_\(p\_svapor\) = 0.327;\)\), "\[IndentingNewLine]", \(\(h\_0 = 77.16\ \ \ \ \ \ \ \ (*\ air\ pressure\ in\ cm\ Hg\ *) ;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_\(h\_0\) = 0.03;\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["The Computation", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Section", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(p\_vapor = relativeHumidity\ p\_svapor;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_\(p\_vapor\) = p\_vapor\ \@\(\((\[Sigma]\_relativeHumidity/relativeHumidity)\)\^2\ \ + \ \((\[Sigma]\_\(p\_svapor\)/p\_svapor)\)\^2\);\)\), "\[IndentingNewLine]", \ \(\(p\_0 = 13.52*980\ h\_0\ \ \ \ \ \ \ \ (*\ p\_0 = \[Rho]\ g\ h\_0, \ \[Rho]\ is\ density\ of\ Hg\ *) ;\)\), "\ \[IndentingNewLine]", \(\(\[Sigma]\_\(p\_0\) = p\_0\ \((\[Sigma]\_\(h\_0\)/h\_0)\);\)\), "\[IndentingNewLine]", \(\(\[Rho]\_0 = 1.2929\ \((273.13\/\(\(5\/9\) \((T\_dry\ - \ 32)\)\ + \ 273.13\))\)\ \ \((\(h\_0 - 0.3783\ p\_vapor\)\/76)\)/1000;\)\[IndentingNewLine] (*\ We\ will\ use\ \[Sigma]\_q\ = \ \@\(\((\(\[PartialD]q\/\[PartialD]x\) \ \[Sigma]\_x)\)\^2\ + \ \((\(\[PartialD]q\/\[PartialD]y\) \[Sigma]\_y)\)\^2\ \ + \ ... \)\ to\ compute\ \[Sigma]\_\(\[Rho]\_0\)\ *) \[IndentingNewLine] (*\ We\ will\ first\ define\ a\ function\ for\ \[Rho]\_0\ and\ use\ \ Mathematica\ function\ D[y, x]\ to\ take\ the\ derivative\ of\ y\ with\ respect\ to\ x\ *) \), \ "\[IndentingNewLine]", \(\(y = 1.2929\ \((273.13\/\(\(5\/9\) \((F\ - \ 32)\)\ + \ 273.13\))\)\ \((\ \(h\ - \ 0.3783\ p\)\/76)\)/1000;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_y = \@\(\((\[Sigma]\_F\ D[y, F])\)\^2\ + \ \((\[Sigma]\_h\ \ D[y, h])\)\^2\ + \ \((\[Sigma]\_p\ D[y, p])\)\^2\);\)\), \ "\[IndentingNewLine]", \(\(\[Sigma]\_\(\[Rho]\_0\) = \[Sigma]\_y /. {F\ \[Rule] \ T\_dry, h\ \[Rule] \ h\_0, p\ \[Rule] \ p\_vapor, \[Sigma]\_F \[Rule] \ \[Sigma]\_\(T\_dry\), \ \[Sigma]\_h \[Rule] \ \[Sigma]\_\(h\_0\), \[Sigma]\_p \[Rule] \ \[Sigma]\_\(p\ \_vapor\)};\)\), "\[IndentingNewLine]", \(\(\[Gamma] = 1.4;\)\), "\[IndentingNewLine]", \(\(speedWithAtmos = \@\(\(\[Gamma]\ p\_0\)\/\[Rho]\_0\)/ 100;\)\), "\[IndentingNewLine]", \(\(\[Sigma]\_speedWithAtmos = speedWithAtmos \@\(\((\((\[Sigma]\_\(p\_0\)\/2)\)/p\_0)\)\^2\ + \ \ \((\((\[Sigma]\_\(\[Rho]\_0\)\/2)\)/\[Rho]\_0)\)\^2\);\)\), "\ \[IndentingNewLine]", \(Print[ StyleForm["\"\ <> \ ToString[ PaddedForm[ speedWithAtmos, {5, 2}]]\ <> \ "\< \[PlusMinus] \>"\ <> \ ToString[ PaddedForm[\[Sigma]\_speedWithAtmos, {3, 2}]]\ <> \ "\<) m/sec\>", FontFamily\ \[Rule] \ "\", FontWeight\ \[Rule] \ "\", FontColor\ \[Rule] \ RGBColor[1, 0, 1], FontSize\ \[Rule] \ 12]]\)}], "Input"], Cell[BoxData[ TagBox[ StyleBox["\<\"Speed of sound in air using atmospheric data = ( 361.07 \ \[PlusMinus] 0.33) m/sec\"\>", FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], (StyleForm[ #, FontFamily -> "Helvetica", FontWeight -> "Bold", FontColor -> RGBColor[ 1, 0, 1], FontSize -> 12]&)]], "Print"] }, Open ]], Cell[TextData[StyleBox["The speed of sound in air obtained from resonance \ tube method is in good agreement with the value calculated from the \ atmospheric data.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]]], "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[TextData[{ StyleBox["ABSTRACT\n\tThis experiment was done in order to help students \ find the speed of sound using a column of water of varying height and a sonic \ generator with a microphone to measure the height of the column, with \ different frequencies of sound. The distance was determined by the harmonics \ that were set up in the open ended tube with the water in it, this is how the \ frequencies were determined and how the length was determined, after using \ these values and the height of the tube on a linear regression line. The \ results came out to be very close to the actual speed of sound. The value we \ obtained was 361.07 +/- 0.33 m/sec when the corrections for the placement of \ the microphone were made.\n\t\nINTRODUCTION\n\tThe experiment was conducted \ using ordinary tap water, an sonic ranger and a tall cylinder with length \ markings down its side. These were the primary items used, in addition to \ the eletronic equipment such as the geerator for the sonic ranger and the \ oscilliscope used to monitor the response in the microphone. It is a rather \ simple experiment in which the sonic ranger, using a sinosoidal wave from the \ generator, is used to sent up harmonics in a column which is filled with \ water at the bottom, so one end is closed off, the other is open. The \ genetor is scrolled through the frequencies, after the height of the water \ column has been determined and set, and several harmonics are found. Some of \ the lowest harmonics are difficult to find because the mircophone can't pick \ the frequencies out finely enough and so they had to be ignored. But for \ each column height, atleatst 4 harmonics were found. The frequences of these \ harmonics was graphed against the number of the harmonic to determine how \ many harmonics were missing, from the first one to the nth one, n being the \ number of the first measured harmonic. ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" is then used to do the number crunching inorder to find the \ speed of sound in air, taking into account the vapor pressure of the water at \ the surface, the difficulty reading the level of the water in the cylinder \ and the fact that the microphone is not exactly at the antinode but slightly \ ahead of it.\n\t\nEXPERIMENTAL SETUP AND PROCEDURES\n\tThe set up consisted \ of a genertor, a microphone placed ontop of a one meter tall vertical \ cylinder which was to be filled to varying heights with water. The \ microphone then is hooked up to an oscilliscope and is used to determine \ where the harmonics are. There is a problem with this already, the \ microphone flush with the opening of the tube, but since we are not in a \ perfect world, the antinode is a short ways outside of the tube, but this is \ counted for in the numerical analysis. Once these have been connected \ correctly, the cylinder is filled with water to a certain level. This is \ read from the side of the cylinder and the distance to the microphone is then \ measured. This gives one the height of the comlumn in which the harmonics \ are to be set up. The microphone, generator and the sonic ranger are then \ turned on, as is the oscilliscope. The experimentor then begins to scroll \ throught the frequencies, starting where they are able to get a flat line on \ the oscilliscope, without much static or jumpiness. The frequencies are then \ scrolled throught looking for a large increase in the response from the \ microphone. The maximum of this sharp rise one of the hamonics of for a \ standing wave within the tube. This frequency is recorded, along with how \ well is was measured, +/- a certain number of frequencies. Once this is \ recorded, the experimentor continues to scroll up in the frequencies to find \ the next harmonic, and continues similarly until a satisfactory number of \ harmonics with errors have been attained. These values are then entered into \ ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" for numerical analysis.\n\t\nDATA\n\tAll of the data for this \ experiment can be view above in the ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" notebook. The frequencies at which the harmonics occur are in \ lists on lines labeled frequency[ndata]= The heights of the columbs are on \ lines labeled length[ndata]= . To find these values, there were several \ significant sources of errors. One of the largest is in reading the level of \ the water. The water does not sit flat in the tub, but curves to form a \ miniscious because of the surface tension of the water. For reading this, it \ was decided that the water level would be determined to be at the lowest \ point of the curve, when looking at a cross section. This is because there \ is very little water on the sides, curving up, and it is near the flattest \ portion of the water surface. Another major source of error was in how well \ the harmonics could be determined. In each case as a harmonic was approached \ there was no one specific spike in the response the harmonic is not an exact \ delta function in reality. The response fits much better to a gaussian \ distribution in the response and so it was difficult to find the exact top. \ Also, the response, or lack there of, in the microphone to changes of say 1 \ or 2 hertz, was too small to measure. These all added to the significant \ source of error in finding the harmonics exactly. The previous were both \ random errors, but the following is a systematic error. The positioning of \ the microphone almost flush with the top of the tube created some problems. \ Because the microphone covers a small part of the tube and the waves not \ being exact plane waves, the actual antinode is a short ways outside of the \ tube. This introduced a systematic error which brought the speed of sound \ down, by about 20 meters a second! But in the ", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" notebook it is accounted for this in the later sections.\n\t\n\ ANALYSIS\n\tFor the most part, the numbers were in very good agreement with \ what they should be. When plotted out there was a definate trend, except the \ first point was just outside of it. In general, as the column of water was \ shortened, the speed of sound increased, this isn't right, but it can't be \ accounted for with the vapor pressures of water and the fact that the \ antinode is actually just behind the microphone. The longest column produced \ a higher speed for the speed of sound than was expected by it even when error \ bars were counted, but this is only nominally. When the fact that the \ antinode is behind the microphone and all other factors were taken into \ account, the first point raised the measured value insignificantly and also \ did not put the error bars out of wack so it was still a good point being \ only one standard deviation uot from what was expected and so was used.\n\t\n\ DISCUSSION AND CONCLUSION\n\tThe speed of sound in air at 296.9 degrees \ Kelvin is 346.2 m/s +/-0.51 m/sec according to measurements made by NIST in \ conjuction with NASA in April 2000. The value obtained for the speed of \ sound in this experiment at 296.3 degrees Kelvin is 339.68 +/- 3.25 m/sec. \ The value obtained in this experiment is slower than the actual value \ accepted, but it is within reasonal agreement. It can be seen that the \ calculated value from the experiment only differs by 2% from the accepted \ value and this is in very good agreement, dispite the fact that the actual \ values with error bars included do not over lap. Again, one of the major \ reasons for this is believed to be the fact that the antinode is behind the \ microphone, but the exact distance is not known and this can really throw \ things off, but with the approximation for this used as a correction, it \ accounted for it remakably well. One way to improve on this, is to find a \ way of determining the location of the antinode for each frequency better and \ so having that be a better correction. Another way is to use a much wider \ tube, when this is done, the effect of the curving of the surface of the \ water will have much less of an effect.", FontFamily->"Times New Roman", FontSize->12, FontColor->RGBColor[0, 0, 1]] }], "Text"] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowToolbars->"EditBar", CellGrouping->Manual, WindowSize->{1272, 907}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, 1}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{"nuchome", "gcchang", \ "Mathematica", "phys375"}, "Exp01.nb.ps", CharacterEncoding -> \ "WindowsANSI"], "Magnification"->1}, ShowCellLabel->False, Magnification->1.5, StyleDefinitions -> "LabReport.nb" ] (*********************************************************************** Cached data follows. 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