(* 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[ 58535, 1137] NotebookOptionsPosition[ 57059, 1086] NotebookOutlinePosition[ 57458, 1103] CellTagsIndexPosition[ 57415, 1100] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Interference of Waves & Beat frequencies", "Title", CellChangeTimes->{{3.473170329086286*^9, 3.473170367129527*^9}, { 3.5032658611969347`*^9, 3.503265865689735*^9}}], Cell["You do not have to turn this in; keep it for your notes.", "Subsubtitle", CellChangeTimes->{{3.473170348524406*^9, 3.473170356854623*^9}}], Cell[CellGroupData[{ Cell["Example", "Subsection", CellChangeTimes->{{3.473170377012311*^9, 3.473170378121389*^9}}], Cell[TextData[{ StyleBox["Let's see what happens when we add two waves that are in phase. \ We will use ", FontSize->14], StyleBox["Mathematica", FontSize->14, FontSlant->"Italic"], StyleBox["'s plot function to graph the waves, and then we'll tell it to \ show both waves on a single graph. 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{3.5347923177047005`*^9, 3.5347923327062006`*^9}, {3.534853827815362*^9, 3.534853854879362*^9}, { 3.534853885732362*^9, 3.534853904666362*^9}, {3.5350513831901073`*^9, 3.535051423453965*^9}, 3.535051539471909*^9}] }, Open ]], Cell[TextData[{ StyleBox["On top of the plot above, sketch your prediction for what happens \ when one plots the ", FontSize->14], StyleBox["sum of the waves. ", FontSize->14, FontWeight->"Bold"], StyleBox[" Then test your prediction with the command below.", FontSize->14] }], "Text", CellChangeTimes->{{3.473171198761385*^9, 3.473171206919526*^9}, { 3.473173377189575*^9, 3.473173380333939*^9}, {3.5350513651875916`*^9, 3.5350513718020344`*^9}, {3.535051426339984*^9, 3.535051448975729*^9}}], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{"wave1", "[", "t", "]"}], "+", RowBox[{"wave2", "[", "t", "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "6"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "6"}], ",", "6"}], "}"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.473170541285585*^9, 3.473170541978842*^9}, { 3.4731706307024393`*^9, 3.473170642812059*^9}, {3.4731712464054337`*^9, 3.4731712713173323`*^9}, {3.473172460141217*^9, 3.473172466955193*^9}, { 3.534791071148058*^9, 3.5347911393908815`*^9}, {3.534791188809823*^9, 3.5347912533802795`*^9}, {3.534791844261361*^9, 3.5347918663745728`*^9}, { 3.53479196205914*^9, 3.5347919804199758`*^9}, {3.534792210173949*^9, 3.53479223448738*^9}, {3.534792295553486*^9, 3.534792300064937*^9}, 3.534792337610691*^9, {3.5350512234918833`*^9, 3.535051250417656*^9}, { 3.535051525135417*^9, 3.5350515379742994`*^9}}], Cell[TextData[{ StyleBox["....what kind of interference is this?\n\nNow you'll do one.\n", FontSize->14], StyleBox["\nProblem 1.", "Subsection"], StyleBox["\nUse the show command (as in the above example) to plot 2* \ Cos[4*t] and -3* Cos[4*t] on the same graph (note that the second wave has \ been flipped over with a negative sign). Make a sketch showing your \ prediction of what will happen when you add the waves. \n\n\n\n\n\n\n\n\n\n\n\ \nTest your prediction using by plotting the sum of the waves. What kind of \ interference is this?", FontSize->14] }], "Text", CellChangeTimes->{{3.473171289687579*^9, 3.4731714503123207`*^9}, { 3.4731715543362637`*^9, 3.473171559645234*^9}, {3.473173043582844*^9, 3.473173066197283*^9}, {3.503265834037335*^9, 3.503265838592535*^9}, { 3.5350512902447114`*^9, 3.535051343987056*^9}, {3.5350514835767508`*^9, 3.535051505432491*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Adding Waves with Different Frequencies.", FontSize->14]], "Section", CellChangeTimes->{3.473172871982329*^9}], Cell[TextData[{ StyleBox["Problem 2. ", "Subsection"], " ", StyleBox["\nUse the show command to plot 2* Cos[4*t] and ", FontSize->14], StyleBox["3* Cos[7*t]", FontSize->14, FontVariations->{"Underline"->True}], StyleBox[" on the same graph. Make a sketch showing your prediction of what \ will happen when you add the waves. \n\n\n\n\n\n\n\n\n\n\n\n\n\nTest your \ prediction using by plotting the sum of the waves. What happened?", FontSize->14] }], "Text", CellChangeTimes->{{3.473171498572566*^9, 3.473171511843142*^9}, 3.47317157203743*^9, {3.473172798574448*^9, 3.473172828798938*^9}, 3.47317286726171*^9, {3.473173073582712*^9, 3.473173074997813*^9}, { 3.5347731554251175`*^9, 3.534773156049117*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{{3.473173084077876*^9, 3.4731730849630013`*^9}, 3.534773156548317*^9}], Cell[TextData[{ StyleBox["Problem 3. ", "Subsection"], StyleBox["\nUse the show command to plot 2* Cos[4*t] and ", FontSize->14], StyleBox["3* Cos[4.3*t]", FontSize->14, FontVariations->{"Underline"->True}], StyleBox[" on the same graph. ", FontSize->14], StyleBox["Adjust the time range so that you are plotting\n t = 0 to 30 \ instead of t = 0 to 6. ", FontSize->14, FontVariations->{"Underline"->True}], StyleBox[" Make a sketch showing your prediction of what will happen when \ you add the waves. ", FontSize->14], "\n\n\n\n" }], "Text", CellChangeTimes->{{3.473172525380563*^9, 3.4731725440785933`*^9}, { 3.47317262301285*^9, 3.473172628958387*^9}, {3.4731727106738377`*^9, 3.473172787412929*^9}, {3.473173098208362*^9, 3.473173102040834*^9}, { 3.4731733413708963`*^9, 3.4731733625657883`*^9}, {3.503264627163441*^9, 3.503264630252261*^9}, {3.5032651055593348`*^9, 3.503265111908535*^9}, { 3.5032652252409353`*^9, 3.5032652275965347`*^9}, {3.503265258905735*^9, 3.503265298373735*^9}, 3.5347731524455175`*^9}], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", "\[IndentingNewLine]", "\n"}]], "Input", CellChangeTimes->{{3.473173095834996*^9, 3.473173096409093*^9}, { 3.5032642829913516`*^9, 3.5032642869850283`*^9}, {3.503265285956135*^9, 3.5032652972037354`*^9}}], Cell[TextData[{ StyleBox["Based on your results, what is the difference between adding two \ waves with very different frequencies, and adding two waves with only ", FontSize->16], StyleBox["slightly", FontSize->16, FontSlant->"Italic"], StyleBox[" different frequencies?", FontSize->16] }], "Text", CellChangeTimes->{3.503265294551735*^9}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5032653042081347`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 4. ", "Subsection"]], "Subsubtitle", CellChangeTimes->{{3.503265310775735*^9, 3.5032653129129353`*^9}, 3.503265401723735*^9}], Cell[TextData[StyleBox["With your headphones on, try the following:", FontSize->16]], "Text", CellChangeTimes->{{3.5032654065597353`*^9, 3.503265415638935*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"wave1", "[", "t_", "]"}], " ", "=", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "440", "*", "t"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"wave2", "[", "t_", "]"}], " ", "=", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "493", "*", "t"}], "]"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Play", "[", RowBox[{ RowBox[{ RowBox[{"wave1", "[", "t", "]"}], "+", RowBox[{"wave2", "[", "t", "]"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "3"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.503265368948135*^9, 3.503265395577335*^9}, 3.503265826954935*^9, {3.503266775067611*^9, 3.5032667922900114`*^9}, { 3.503266847598611*^9, 3.5032668485034113`*^9}}], Cell[TextData[{ StyleBox["In the command above, what do the numbers 440 and 493 represent?\n\ \nIf you're not sure, it may help to recall that we represent waves in the \ form a*cos(", FontSize->16], StyleBox["w ", FontFamily->"Symbol", FontSize->16], StyleBox["t), where ", FontSize->16], StyleBox["w", FontFamily->"Symbol", FontSize->16], StyleBox[" (angular frequency) is 2 Pi times f (the \"linear\" frequency). \ \nRecall that the formula for the beat frequency is f = |f", FontSize->16], Cell[BoxData[ FormBox[ SubscriptBox["", "1"], TraditionalForm]]], StyleBox[" - f", FontSize->16], Cell[BoxData[ FormBox[ SubscriptBox["", "2"], TraditionalForm]]], StyleBox["|. Calculate the beat frequency that you would predict for this \ beat pattern. (You can use the ", FontSize->16], StyleBox["Mathematica", FontSize->16, FontSlant->"Italic"], StyleBox[" command Abs[ ] to do this if desired.) \n\n\n\nBased on your \ calculation of the frequency, how many beats would you expect to see in .5 \ seconds?", FontSize->16] }], "Text", CellChangeTimes->{{3.503264302460526*^9, 3.5032643570459757`*^9}, { 3.503264390134212*^9, 3.503264598162855*^9}, {3.5032646479271746`*^9, 3.503264649549585*^9}, {3.5032654329237347`*^9, 3.503265492702935*^9}, 3.503265526866935*^9, {3.503265569851335*^9, 3.503265569944935*^9}, { 3.5032660442409353`*^9, 3.503266060168535*^9}, {3.503266090245335*^9, 3.503266140914135*^9}, 3.5032663397308207`*^9, {3.5347731408547173`*^9, 3.5347731429295173`*^9}, {3.5350515754145393`*^9, 3.535051761336531*^9}}], Cell[TextData[{ StyleBox["\n\n\n\nTest your predictions using by plotting the sum of the \ waves:\n", FontSize->16], StyleBox["Plot[wave1[t] + wave2[t], {t, 0, .5}]\n\n", "Input", FontSize->16], StyleBox[" Did you see as many beats as you predicted?", FontSize->16], StyleBox["\n", FontSize->14] }], "Text", CellChangeTimes->{{3.473172907108345*^9, 3.473172931100354*^9}, { 3.503264059937604*^9, 3.503264063182404*^9}, {3.5032646331850796`*^9, 3.5032646746657457`*^9}, 3.503265283506935*^9, {3.503265581239335*^9, 3.503265593688135*^9}, 3.503265822072135*^9, {3.534773135394717*^9, 3.534773138265117*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Problem 5", "Subsubtitle"]], "Subsubsection", CellChangeTimes->{{3.5032641173300037`*^9, 3.503264119841604*^9}, 3.503265317826935*^9}], Cell[TextData[{ StyleBox["(a) Experiment to determine how far apart the frequencies of the \ waves can be and still produce ", FontSize->16], StyleBox["visible", FontSize->16, FontWeight->"Bold"], StyleBox[" beats. (There's no \"right\" answer to this question; you'll \ have to make a judgment call.)\n\n\n\n(b) Experiment to determine how far \ apart the angular frequencies of the waves can be and still produce ", FontSize->16], StyleBox["audible", FontSize->16, FontWeight->"Bold"], StyleBox[" beats. (ditto).\n\n(c) Suppose a piano player plays two notes \ simultaneously--middle A (440 Hz) and middle C (262 Hz). How many beats \ would this produce in one second? \n\nWould you expect these notes to \ produce audible beats? Why or why not?\n\n", FontSize->16] }], "Text", CellChangeTimes->{{3.503263983653604*^9, 3.503264042808804*^9}, { 3.503264135457204*^9, 3.503264136206004*^9}, {3.5032650917845316`*^9, 3.503265092595735*^9}, {3.503265654231735*^9, 3.503265694292535*^9}, { 3.503265814724535*^9, 3.503265814958535*^9}, {3.5350517850018826`*^9, 3.5350517855166864`*^9}, {3.5350518393370314`*^9, 3.5350519310695486`*^9}}], Cell[BoxData[""], "Input", CellChangeTimes->{{3.503265785942535*^9, 3.5032657905445347`*^9}}], Cell[TextData[StyleBox["Have the instructor check off your work when you are \ done.", FontSize->18, FontSlant->"Italic"]], "Text", CellChangeTimes->{{3.5032657955677347`*^9, 3.503265802026135*^9}}] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{846, 682}, WindowMargins->{{46, Automatic}, {Automatic, 79}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (November 10, 2008)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 174, 2, 83, "Title"], Cell[744, 26, 145, 1, 28, "Subsubtitle"], Cell[CellGroupData[{ Cell[914, 31, 95, 1, 36, "Subsection"], Cell[1012, 34, 541, 13, 49, "Text"], Cell[CellGroupData[{ Cell[1578, 51, 1734, 43, 112, "Input"], Cell[3315, 96, 43377, 718, 191, "Output"] }, Open ]], Cell[46707, 817, 510, 12, 50, "Text"], Cell[47220, 831, 999, 20, 31, "Input"], Cell[48222, 853, 897, 16, 391, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[49156, 874, 136, 2, 63, "Section"], Cell[49295, 878, 734, 16, 334, "Text"], Cell[50032, 896, 139, 2, 52, "Input"], Cell[50174, 900, 1062, 24, 140, "Text"], Cell[51239, 926, 256, 4, 92, "Input"], Cell[51498, 932, 352, 10, 55, "Text"], Cell[51853, 944, 89, 1, 52, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[51979, 950, 162, 2, 26, "Subsubtitle"], Cell[52144, 954, 162, 2, 33, "Text"], Cell[52309, 958, 814, 22, 72, "Input"], Cell[53126, 982, 1598, 41, 232, "Text"], Cell[54727, 1025, 633, 15, 204, "Text"], Cell[CellGroupData[{ Cell[55385, 1044, 163, 2, 31, "Subsubsection"], Cell[55551, 1048, 1167, 24, 319, "Text"], Cell[56721, 1074, 94, 1, 31, "Input"], Cell[56818, 1077, 201, 4, 34, "Text"] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)