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}{XPPEDIT 18 0 "Type setting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi (\"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"u\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetti ng:-mi(\"tt\", italic = \"true\", mathvariant = \"italic\")), subscrip tshift = \"0\"), Typesetting:-mo(\"⁡\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Type setting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"t rue\", mathvariant = \"italic\"), Typesetting:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \"true\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em \"), Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic \")), mathvariant = \"normal\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fa lse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"f alse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277 7778em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"c\", \+ italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", m athvariant = \"normal\"), superscriptshift = \"0\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-msub( Typesetting:-mi(\"u\", italic = \"true\", mathvariant = \"italic\"), T ypesetting:-mrow(Typesetting:-mi(\"xx\", italic = \"true\", mathvarian t = \"italic\")), subscriptshift = \"0\"), Typesetting:-mo(\"&ApplyFun ction;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetti ng:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesettin g:-mo(\",\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"true\", stretchy = \"false\", symmetric = \"false\", largeop = \"fal se\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.3333333em\"), Typesetting:-mi(\"t\", italic = \"true \", mathvariant = \"italic\")), mathvariant = \"normal\")), Typesettin g:-mi(\"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+ -F#6%-I%msubGF$6%-F,6%Q\"uF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF '-F#6#-F,6%Q#ttF'F9F " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 209 10 "An Example" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 210 23 "Four ier Series Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 64 "Set some o f the parameters, and define the initial displacement." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "c:=1:\n" }{MPLTEXT 1 0 6 "L:=2:\n" }{MPLTEXT 1 0 6 "h:=1:\n" }{MPLTEXT 1 0 24 "f:=x->piecewise(x<0, 0,\n" }{MPLTEXT 1 0 29 " xL, 0):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 208 29 "Plot the initial displacement" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f(x), x=-L/2..3*L/2, thickness =2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 40 "Let's use Maple to compu te the solution." }}{PARA 0 "" 0 "" {TEXT 208 48 "In class we found, f or general initial position " }{XPPEDIT 18 0 "Typesetting:-mrow(Typese tting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"fF'/%' italicGQ%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 30 " and general \+ initial velocity " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"g\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+m odulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"gF'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'" }{TEXT 208 34 ", that the solution is \+ of the form" }}{PARA 0 "" 0 "" {TEXT 208 5 " " }{XPPEDIT 18 0 "Typ esetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-m i(\"\"), Typesetting:-mrow(Typesetting:-mi(\"u\", italic = \"true\", m athvariant = \"italic\"), Typesetting:-mo(\"⁡\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\",\", math variant = \"normal\", fence = \"false\", separator = \"true\", stretch y = \"false\", symmetric = \"false\", largeop = 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mi(\"cos\", italic = \"false\", mathvariant = \"normal\"), Typesetting :-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting :-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msqrt(Typ esetting:-msub(Typesetting:-mi(\"\1673\", italic = \"false\", mathvari ant = \"normal\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \+ \"true\", mathvariant = \"italic\")), subscriptshift = \"0\")), Typese tting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"t\", it alic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), m athvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mo(\"+ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msub(Typese tting:-mi(\"b\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"i talic\")), subscriptshift = \"0\"), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \" false\", mathvariant = \"normal\"), Typesetting:-mo(\"⁡ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = 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fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\"L\", \+ italic = \"true\", mathvariant = \"italic\")), linethickness = \"1\", \+ denomalign = \"center\", numalign = \"center\", bevelled = \"false\")) , mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi( \"\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F#6%- F,6%Q\"uF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q0&Appl yFunction;F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%)stretchyG FD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%'accentGFD/%'lspace GQ&0.0emF'/%'rspaceGFS-I(mfencedGF$6$-F#6%-F,6%Q\"xF'F6F9-F=6-Q\",F'F@ FB/FFF8FGFIFKFMFOFQ/FUQ,0.3333333emF'-F,6%Q\"tF'F6F9F@-F=6-Q\"=F'F@FBF EFGFIFKFMFO/FRQ,0.2777778emF'/FUFeo-F#6'-I+munderoverGF$6'-F=6-Q&∑ F'F@FBFE/FHF8FI/FLF8/FNF8FOFQ/FUQ,0.1666667emF'-F#6%-F,6%Q\"nF'F6F9Fao -I#mnGF$6$Q\"1F'F@-F,6%Q(∞F'F6F9FO/%,accentunderGFDF+-I'mspaceGF $6&/%'heightGQ&0.0exF'/%&widthGQ$5.0F'/%&depthGFgq/%*linebreakGQ%autoF '-F#6&-FW6$-F#6'F+-F#6&-I%msubGF$6%-F,6%Q\"aF'F6F9-F#6#Ffp/%/subscript shiftGQ\"0F'-F=6-Q1⁢F'F@FBFEFGFIFKFMFOFQFT-F#6%-F,6%Q$c osF'/F7FDF@F<-FW6$-F#6%F+-F#6%-I&msqrtGF$6#-Fir6%-F,6%Q)λF'F[tF @F^sF`sFcsF^oF+F@F+-F=6-Q\"+F'F@FBFEFGFIFKFMFO/FRQ,0.2222222emF'/FUF^u -F#6&-Fir6%-F,6%Q\"bF'F6F9F^sF`sFcs-F#6%-F,6%Q$sinF'F[tF@FF@FBFDFFFHFJFM-I(mfen cedGF$6$-F#6#-F,6%Q\"xF'FVFYF8-F56-Q1⁢F'F8F;F>F@FBFDFFF HFJFM-F#6%-F,6%Q$sinF'/FWF=F8Fao-Feo6$-F#6#-I&mfracGF$6(-F#6'-F,6%Q\"n F'FVFYF\\p-F,6%Q%πF'FdpF8F\\pFio-F#6#FS/%.linethicknessGQ\"1F'/%+de nomalignGQ'centerF'/%)numalignGF[r/%)bevelledGF=F8F+F+-I'mspaceGF$6&/% 'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFer/%*linebreakGQ%autoF'-F 56-Q0ⅆF'F8F;F>F@FBFDFFFHFJFMFioF+" }{TEXT 208 17 " \+ and " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"b\", italic = \" true\", mathvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi( \"n\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:- mfrac(Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal \")), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Types etting:-mi(\"c\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"n\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&Invisib leTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mi(\"\1700\", italic = \"false\" , mathvariant = \"normal\")), Typesetting:-mi(\"\")), linethickness = \+ \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"fa lse\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"bF'/%'itali cGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshi ftGQ\"0F'-I#moGF$6-Q\"=F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGF M/%)stretchyGFM/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accen tGFM/%'lspaceGQ,0.2777778emF'/%'rspaceGFfn-I&mfracGF$6(-F#6#-I#mnGF$6$ Q\"2F'FI-F#6%F+-F#6'-F,6%Q\"cF'F7F:-FF6-Q1⁢F'FIFKFNFPFR FTFVFX/FenQ&0.0emF'/FhnF]pF?Fio-F,6%Q%πF'/F8FMFIF+/%.linethicknessG Q\"1F'/%+denomalignGQ'centerF'/%)numalignGFhp/%)bevelledGFMF+" }{TEXT 208 1 " " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(\"\21053\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mn(\"0\", mathvariant = \"normal\"), Typesetting:-m i(\"L\", italic = \"true\", mathvariant = \"italic\"), superscriptshif t = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting :-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&Appl yFunction;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0e m\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Types etting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathv ariant = \"normal\")), Typesetting:-mo(\"⁢\", mathvaria nt = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", math variant = \"normal\"), Typesetting:-mo(\"⁡\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mfrac(Typesetti ng:-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"ita lic\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\" , fence = \"false\", separator = \"false\", stretchy = \"false\", symm etric = \"false\", largeop = \"false\", movablelimits = \"false\", acc ent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting: -mi(\"\1700\", italic = \"false\", mathvariant = \"normal\"), Typesett ing:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", ital ic = \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetti ng:-mi(\"L\", italic = \"true\", mathvariant = \"italic\")), linethick ness = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Ty pesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0ex\", width = \+ \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo( \"ⅆ\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"tru e\", mathvariant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6)-I(msub supGF$6'-I#moGF$6-Q&∫F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF' /%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelim itsGF=/%'accentGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL-I#mnGF$6$Q\"0F'F8-F, 6%Q\"LF'/%'italicGQ%trueF'/F9Q'italicF'/%1superscriptshiftGQ\"2F'/%/su bscriptshiftGFRF+-F#6'F+-F#6%-F,6%Q\"gF'FVFY-F56-Q0⁡F'F8 F;F>F@FBFDFFFHFJFM-I(mfencedGF$6$-F#6#-F,6%Q\"xF'FVFYF8-F56-Q1&Invisib leTimes;F'F8F;F>F@FBFDFFFHFJFM-F#6%-F,6%Q$sinF'/FWF=F8Fao-Feo6$-F#6#-I &mfracGF$6(-F#6'-F,6%Q\"nF'FVFYF\\p-F,6%Q%πF'FdpF8F\\pFio-F#6#FS/%. linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF[r/%)bevelled GF=F8F+F+-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGF er/%*linebreakGQ%autoF'-F56-Q0ⅆF'F8F;F>F@FBFDFFFHFJFMFio F+" }}{PARA 0 "" 0 "" {TEXT 208 27 "Since the initial velocity " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"g\", it alic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&ApplyFu nction;\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\" , rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesett ing:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), mathvari ant = \"normal\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typ esetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi(\"\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q !F'-F#6&F+-F#6%-F,6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'- I#moGF$6-Q0⁡F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separato rGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%'ac centGFD/%'lspaceGQ&0.0emF'/%'rspaceGFS-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F 6F9F@-F=6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ,0.2777778emF'/FUF\\o-I#mnGF$6$Q\" 0F'F@F+" }{TEXT 208 10 ", we have " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting :-mi(\"b\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic \")), subscriptshift = \"0\"), Typesetting:-mo(\"=\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277777 8em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting :-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6% -I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"bF'/%'italicGQ%trueF'/%,mathvar iantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q \"=F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*sy mmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ,0.27 77778emF'/%'rspaceGFfn-I#mnGF$6$FDFIF+" }{TEXT 208 1 "." }}{PARA 0 "" 0 "" {TEXT 208 18 "Now let's compute " }{XPPEDIT 18 0 "Typesetting:-mr ow(Typesetting:-msub(Typesetting:-mi(\"a\", italic = \"true\", mathvar iant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \+ \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"));" "-I%m rowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF $6%Q\"aF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F/6%Q\"nF'F2 F5/%/subscriptshiftGQ\"0F'" }{TEXT 208 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "assume(n, integer):\n" }{MPLTEXT 1 0 45 "a:=n->(2/L)* int(f(x)*sin(n*Pi*x/L), x=0..L);\n" }{MPLTEXT 1 0 10 "a[n]=a(n);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 208 18 "So the solution is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u:=(x,t)->sum(a(n)*cos(sqrt(lambda[n])*t)*sin( n*Pi*x/L), n=1..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 6 "whe re " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msub(Typesetting:- mi(\"\1673\", italic = \"false\", mathvariant = \"normal\"), Typesetti ng:-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"ita lic\")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,Type settingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q)λF'/%'italicGQ&fa lseF'/%,mathvariantGQ'normalF'-F#6#-F/6%Q\"nF'/F3Q%trueF'/F6Q'italicF' /%/subscriptshiftGQ\"0F'" }{TEXT 208 14 " is defined as" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lambda := n -> (c*n*Pi/L)^2;\n" }{MPLTEXT 1 0 20 "lambda[n]=lambda(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 8 "And t he " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"N\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"NF'/%'italicGQ%trueF'/%,mathvar iantGQ'italicF'" }{TEXT 208 57 "-th partial sum approximation of the s olution is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "u:=(x,t,N)-> sum(a(n)*cos(sqrt(lambda(n))*t)*sin(n*Pi*x/L), n=1..N);\n" }{MPLTEXT 1 0 19 "U[N](x,t)=u(x,t,N);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 64 "W e plot the (partial sum approximation to the) solution at time " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"0 \", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\" tF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'nor malF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%( largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'r spaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 208 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,0,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 35 "Some more plots at different times " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"t\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"tF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot( u(x,1/2,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,1,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(u(x,3/2,20), x=0..L, thickness =2);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 210 12 "An Animation" }} {EXCHG {PARA 0 "" 0 "" {TEXT 208 23 "And now a little movie:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "animate(u(x,t,2), x=0..L, t=0..20, frames =100, color=red, thickness=2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 209 26 "Normal Modes and \+ Overtones" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 29 "The normal modes are given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "mode := (x,t,n) -> a(n )*cos(sqrt(lambda(n))*t)*sin(n*Pi*x/L);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 30 "Here is an animation of mode 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "animate(mode(x,t,1), x=0..L, t=0..20, frames=100, col or=red, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 23 "Now for some overtones." }}{PARA 0 "" 0 "" {TEXT 208 61 "Note that all \"even \" modes are zero (see the solution above)." }}{PARA 0 "" 0 "" {TEXT 208 16 "Therefore mode 3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate (mode(x,t,3), x=0..L, t=0..2, frames=20, color=red, thickness=2);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 208 18 "And here is mode 5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate(mode(x,t,5), x=0..L, t=0..2, frames=20 , color=red, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 209 15 "Traveling Waves" }} {EXCHG {PARA 0 "" 0 "" {TEXT 208 53 "Next we illustrate the traveling \+ wave interpretation." }}{PARA 0 "" 0 "" {TEXT 208 37 "First we need to extend the function " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:- mi(\"f\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/ I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"fF'/%'italicG Q%trueF'/%,mathvariantGQ'italicF'" }{TEXT 208 45 " defining the initia l displacement to an odd " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetti ng:-mi(\"\"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \+ \"normal\"), Typesetting:-mo(\"⁢\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typeset ting:-mi(\"L\", italic = \"true\", mathvariant = \"italic\")), Typeset ting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6#Q!F'-F#6%-I#mnGF$6$Q\"2F'/%,mathvariantGQ'normalF'-I#moG F$6-Q1⁢F'F5/%&fenceGQ&falseF'/%*separatorGF>/%)stretchy GF>/%*symmetricGF>/%(largeopGF>/%.movablelimitsGF>/%'accentGF>/%'lspac eGQ&0.0emF'/%'rspaceGFM-F,6%Q\"LF'/%'italicGQ%trueF'/F6Q'italicF'F+" } {TEXT 208 19 "-periodic function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "fbar := x -> piecewise(x<=-L/2, -h*(x+L),\n" }{MPLTEXT 1 0 35 " \+ x=5*L/ 2, -h*(x-3*L)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 11 "and plot it" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(fbar(x), x=-L..3*L, thicknes s=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 36 "Now we can show the tra veling waves." }}{PARA 0 "" 0 "" {TEXT 208 41 "First the one which tra vels to the right:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "animate(fbar( x-c*t), x=0..L, t=0..2, frames=100, color=red, thickness=2);\n" } {MPLTEXT 1 0 80 "right:=animate(fbar(x-c*t), x=0..L, t=0..2, frames=10 0, color=red, thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 35 "T hen the one traveling to the left:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "animate(fbar(x+c*t), x=0..L, t=0..2, frames=100, color=green, thic kness=2);\n" }{MPLTEXT 1 0 81 "left:=animate(fbar(x+c*t), x=0..L, t=0. .2, frames=100, color=green, thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 30 "And now both of them together:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{right,left\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 105 "And finally an animation of both waves together with th eir average (which is the solution to the problem)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 100 "average:=animate((fbar(x-c*t)+fbar(x+c*t))/2, x=0. .L, t=0..2, frames=100, color=blue, thickness=2):\n" }{MPLTEXT 1 0 30 "display(\{right,left,average\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 209 15 "Another Ex ample" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 210 23 "Fourier Series Solution " }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 64 "Set some of the parameters, a nd define the initial displacement." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "c:=1:\n" }{MPLTEXT 1 0 6 "L:=2:\n" }{MPLTEXT 1 0 6 "h:=1:\n" } {MPLTEXT 1 0 24 "f:=x->piecewise(x<0, 0,\n" }{MPLTEXT 1 0 21 " \+ xL, 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 29 "Plot the initial displacement" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(f(x), x=-L/2..3*L/2, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 40 "Let's use Maple to compute the solution. " }}{PARA 0 "" 0 "" {TEXT 208 48 "In class we found, for general initi al position " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"f\", italic = \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulen ameG6\"I,TypesettingGI(_syslibGF'6#-I#miGF$6%Q\"fF'/%'italicGQ%trueF'/ %,mathvariantGQ'italicF'" }{TEXT 208 30 " and general initial velocity " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \+ \"true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,T ypesettingGI(_syslibGF'6#-I#miGF$6%Q\"gF'/%'italicGQ%trueF'/%,mathvari antGQ'italicF'" }{TEXT 208 34 ", that the solution is of the form" }} {PARA 0 "" 0 "" {TEXT 208 5 " " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typese tting:-mrow(Typesetting:-mi(\"u\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"⁡\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\",\", mathvariant = \"nor mal\", fence = \"false\", separator = \"true\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.3333333em\"), Ty pesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\")), ma thvariant = \"normal\")), Typesetting:-mo(\"=\", mathvariant = \"norma l\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\" ), Typesetting:-mrow(Typesetting:-munderover(Typesetting:-mo(\"∑\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"true\", symmetric = \"false\", largeop = \"true\", movabl elimits = \"true\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.1666667em\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \" true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.27 77778em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), Typeset ting:-mi(\"∞\", italic = \"true\", mathvariant = \"italic\"), ac cent = \"false\", accentunder = \"false\"), Typesetting:-mi(\"\"), Typ esetting:-mspace(height = \"0.0ex\", width = \"5.0\", depth = \"0.0ex \", linebreak = \"auto\"), Typesetting:-mrow(Typesetting:-mfenced(Type setting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-ms ub(Typesetting:-mi(\"a\", italic = \"true\", mathvariant = \"italic\") , Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvari ant = \"italic\")), subscriptshift = \"0\"), Typesetting:-mo(\"&Invisi bleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"cos\", it alic = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&ApplyF unction;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typese tting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msqrt(Typesetting:-msu b(Typesetting:-mi(\"λ\", italic = \"false\", mathvariant = \"no rmal\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"true\", m athvariant = \"italic\")), subscriptshift = \"0\")), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"t\", italic = \"tr ue\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), mathvariant \+ = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mo(\"+\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( \"b\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow (Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Typesetting:-mo(\"⁢\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", m athvariant = \"normal\"), Typesetting:-mo(\"⁡\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-msqrt(Typesetting:-msub(Typesetting:-mi(\" λ\", italic = \"false\", mathvariant = \"normal\"), Typesetting :-mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"itali c\")), subscriptshift = \"0\")), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \+ \"italic\")), Typesetting:-mi(\"\")), mathvariant = \"normal\")), Type setting:-mi(\"\")), Typesetting:-mi(\"\")), mathvariant = \"normal\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow( Typesetting:-mi(\"sin\", italic = \"false\", mathvariant = \"normal\") , Typesetting:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenc ed(Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting: -mi(\"n\", italic = \"true\", mathvariant = \"italic\"), Typesetting:- mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa ce = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"π\", italic \+ = \"false\", mathvariant = \"normal\"), Typesetting:-mo(\"&InvisibleTi mes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", r space = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvari ant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\"L\", italic = \+ \"true\", mathvariant = \"italic\")), linethickness = \"1\", denomalig n = \"center\", numalign = \"center\", bevelled = \"false\")), mathvar iant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")), T ypesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulena meG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F#6%-F,6%Q\"uF '/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q0&ApplyFunction ;F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symm etricGFD/%(largeopGFD/%.movablelimitsGFD/%'accentGFD/%'lspaceGQ&0.0emF '/%'rspaceGFS-I(mfencedGF$6$-F#6%-F,6%Q\"xF'F6F9-F=6-Q\",F'F@FB/FFF8FG FIFKFMFOFQ/FUQ,0.3333333emF'-F,6%Q\"tF'F6F9F@-F=6-Q\"=F'F@FBFEFGFIFKFM FO/FRQ,0.2777778emF'/FUFeo-F#6'-I+munderoverGF$6'-F=6-Q&∑F'F@FBFE/ FHF8FI/FLF8/FNF8FOFQ/FUQ,0.1666667emF'-F#6%-F,6%Q\"nF'F6F9Fao-I#mnGF$6 $Q\"1F'F@-F,6%Q(∞F'F6F9FO/%,accentunderGFDF+-I'mspaceGF$6&/%'hei ghtGQ&0.0exF'/%&widthGQ$5.0F'/%&depthGFgq/%*linebreakGQ%autoF'-F#6&-FW 6$-F#6'F+-F#6&-I%msubGF$6%-F,6%Q\"aF'F6F9-F#6#Ffp/%/subscriptshiftGQ\" 0F'-F=6-Q1⁢F'F@FBFEFGFIFKFMFOFQFT-F#6%-F,6%Q$cosF'/F7FD F@F<-FW6$-F#6%F+-F#6%-I&msqrtGF$6#-Fir6%-F,6%Q)λF'F[tF@F^sF`sFc sF^oF+F@F+-F=6-Q\"+F'F@FBFEFGFIFKFMFO/FRQ,0.2222222emF'/FUF^u-F#6&-Fir 6%-F,6%Q\"bF'F6F9F^sF`sFcs-F#6%-F,6%Q$sinF'F[tF@FF@FBFDFFFHFJFM-I(mfencedGF$6$-F #6#-F,6%Q\"xF'FVFYF8-F56-Q1⁢F'F8F;F>F@FBFDFFFHFJFM-F#6% -F,6%Q$sinF'/FWF=F8Fao-Feo6$-F#6#-I&mfracGF$6(-F#6'-F,6%Q\"nF'FVFYF\\p -F,6%Q%πF'FdpF8F\\pFio-F#6#FS/%.linethicknessGQ\"1F'/%+denomalignGQ 'centerF'/%)numalignGF[r/%)bevelledGF=F8F+F+-I'mspaceGF$6&/%'heightGQ& 0.0exF'/%&widthGQ&0.3emF'/%&depthGFer/%*linebreakGQ%autoF'-F56-Q0&Diff erentialD;F'F8F;F>F@FBFDFFFHFJFMFioF+" }{TEXT 208 17 " and \+ " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin g:-mrow(Typesetting:-msub(Typesetting:-mi(\"b\", italic = \"true\", ma thvariant = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", ital ic = \"true\", mathvariant = \"italic\")), subscriptshift = \"0\"), Ty pesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mfrac(Typese tting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\")), Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\" c\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"& InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"n\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\"⁢\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mi(\"π\", italic = \"false\", mathvariant \+ = \"normal\")), Typesetting:-mi(\"\")), linethickness = \"1\", denomal ign = \"center\", numalign = \"center\", bevelled = \"false\")), Types etting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysli bGF'6%-I#miGF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"bF'/%'italicGQ%trueF'/%,m athvariantGQ'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#mo GF$6-Q\"=F'/F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGF M/%*symmetricGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceG Q,0.2777778emF'/%'rspaceGFfn-I&mfracGF$6(-F#6#-I#mnGF$6$Q\"2F'FI-F#6%F +-F#6'-F,6%Q\"cF'F7F:-FF6-Q1⁢F'FIFKFNFPFRFTFVFX/FenQ&0. 0emF'/FhnF]pF?Fio-F,6%Q%πF'/F8FMFIF+/%.linethicknessGQ\"1F'/%+denom alignGQ'centerF'/%)numalignGFhp/%)bevelledGFMF+" }{TEXT 208 1 " " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-msubsup(Typesetting:-mo(\"∫\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mn(\"0\", mathvariant = \"normal\"), Typesetting:-mi(\"L\", it alic = \"true\", mathvariant = \"italic\"), superscriptshift = \"2\", \+ subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"⁡ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi (\"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \+ \"normal\")), Typesetting:-mo(\"⁢\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mrow(Typesetting:-mi(\"sin\", italic = \"false\", mathvariant \+ = \"normal\"), Typesetting:-mo(\"⁡\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mfenced(Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow (Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"&p i;\", italic = \"false\", mathvariant = \"normal\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\" L\", italic = \"true\", mathvariant = \"italic\")), linethickness = \" 1\", denomalign = \"center\", numalign = \"center\", bevelled = \"fals e\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting :-mi(\"\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differen tialD;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6)-I(msubsupGF$6'-I# moGF$6-Q&∫F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separato rGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'ac centGF=/%'lspaceGQ&0.0emF'/%'rspaceGFL-I#mnGF$6$Q\"0F'F8-F,6%Q\"LF'/%' italicGQ%trueF'/F9Q'italicF'/%1superscriptshiftGQ\"2F'/%/subscriptshif tGFRF+-F#6'F+-F#6%-F,6%Q\"gF'FVFY-F56-Q0⁡F'F8F;F>F@FBFDF FFHFJFM-I(mfencedGF$6$-F#6#-F,6%Q\"xF'FVFYF8-F56-Q1⁢F'F 8F;F>F@FBFDFFFHFJFM-F#6%-F,6%Q$sinF'/FWF=F8Fao-Feo6$-F#6#-I&mfracGF$6( -F#6'-F,6%Q\"nF'FVFYF\\p-F,6%Q%πF'FdpF8F\\pFio-F#6#FS/%.linethickne ssGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF[r/%)bevelledGF=F8F+F+-I 'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.3emF'/%&depthGFer/%*linebr eakGQ%autoF'-F56-Q0ⅆF'F8F;F>F@FBFDFFFHFJFMFioF+" }} {PARA 0 "" 0 "" {TEXT 208 27 "Since the initial velocity " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Types etting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"g\", italic = \" true\", mathvariant = \"italic\"), Typesetting:-mo(\"⁡\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace \+ = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( \"x\", italic = \"true\", mathvariant = \"italic\")), mathvariant = \" normal\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesettin g:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I% mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6 &F+-F#6%-F,6%Q\"gF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$ 6-Q0⁡F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%) stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%'accentGFD /%'lspaceGQ&0.0emF'/%'rspaceGFS-I(mfencedGF$6$-F#6#-F,6%Q\"xF'F6F9F@-F =6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ,0.2777778emF'/FUF\\o-I#mnGF$6$Q\"0F'F@F+ " }{TEXT 208 10 ", we have " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\" b\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mrow(T ypesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic\")), s ubscriptshift = \"0\"), Typesetting:-mo(\"=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6%-I%msubGF$6%-F,6%Q\"bF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-F#6#-F,6%Q\"nF'F7F:/%/subscriptshiftGQ\"0F'-I#moGF$6-Q\"=F' /F;Q'normalF'/%&fenceGQ&falseF'/%*separatorGFM/%)stretchyGFM/%*symmetr icGFM/%(largeopGFM/%.movablelimitsGFM/%'accentGFM/%'lspaceGQ,0.2777778 emF'/%'rspaceGFfn-I#mnGF$6$FDFIF+" }{TEXT 208 1 "." }}{PARA 0 "" 0 "" {TEXT 208 18 "Now let's compute " }{XPPEDIT 18 0 "Typesetting:-mrow(Ty pesetting:-msub(Typesetting:-mi(\"a\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"n\", italic = \"tru e\", mathvariant = \"italic\")), subscriptshift = \"0\"));" "-I%mrowG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q \"aF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-F#6#-F/6%Q\"nF'F2F5/% /subscriptshiftGQ\"0F'" }{TEXT 208 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "assume(n, integer):\n" }{MPLTEXT 1 0 45 "a:=n->(2/L)*int(f(x)* sin(n*Pi*x/L), x=0..L);\n" }{MPLTEXT 1 0 10 "a[n]=a(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 18 "So the solution is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "u:=(x,t)->sum(a(n)*cos(sqrt(lambda[n])*t)*sin(n*Pi*x/ L), n=1..infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 6 "where " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(\"& lambda;\", italic = \"false\", mathvariant = \"normal\"), Typesetting: -mrow(Typesetting:-mi(\"n\", italic = \"true\", mathvariant = \"italic \")), subscriptshift = \"0\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6#-I%msubGF$6%-I#miGF$6%Q)λF'/%'italicGQ&false F'/%,mathvariantGQ'normalF'-F#6#-F/6%Q\"nF'/F3Q%trueF'/F6Q'italicF'/%/ subscriptshiftGQ\"0F'" }{TEXT 208 14 " is defined as" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lambda := n -> (c*n*Pi/L)^2;\n" }{MPLTEXT 1 0 20 "lambda[n]=lambda(n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 8 "And the " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"N\", italic = \" true\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6#-I#miGF$6%Q\"NF'/%'italicGQ%trueF'/%,mathvarian tGQ'italicF'" }{TEXT 208 57 "-th partial sum approximation of the solu tion is given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "u:=(x,t,N)->sum (a(n)*cos(sqrt(lambda(n))*t)*sin(n*Pi*x/L), n=1..N);\n" }{MPLTEXT 1 0 19 "U[N](x,t)=u(x,t,N);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 64 "We pl ot the (partial sum approximation to the) solution at time " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"t\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"0 \", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\" tF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'nor malF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%( largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'r spaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 208 2 ". " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,0,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 35 "Some more plots at different times " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"t\", italic = \"tru e\", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typese ttingGI(_syslibGF'6#-I#miGF$6%Q\"tF'/%'italicGQ%trueF'/%,mathvariantGQ 'italicF'" }{TEXT 208 1 ":" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot( u(x,1/2,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,1,20), x=0..L, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(u(x,3/2,20), x=0..L, thickness =2);" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 210 12 "An Animation" }} {EXCHG {PARA 0 "" 0 "" {TEXT 208 23 "And now a little movie:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "animate(u(x,t,2), x=0..L, t=0..20, frames =100, color=red, thickness=2); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 209 26 "Normal Modes and \+ Overtones" }}{EXCHG {PARA 0 "" 0 "" {TEXT 208 29 "The normal modes are given by" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "mode := (x,t,n) -> a(n )*cos(sqrt(lambda(n))*t)*sin(n*Pi*x/L);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 30 "Here is an animation of mode 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "animate(mode(x,t,1), x=0..L, t=0..20, frames=100, col or=red, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 23 "Now for some overtones." }}{PARA 0 "" 0 "" {TEXT 208 6 "Mode 2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate(mode(x,t,2), x=0..L, t=0..2, frames=20 , color=red, thickness=2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 208 18 "An d here is mode 3" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "animate(mode(x, t,3), x=0..L, t=0..2, frames=20, color=red, thickness=2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }