{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Empha sis" -1 263 "Times" 1 12 115 0 230 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 268 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Time s" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Outpu t" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 59 "A procedure for constructing solu tions to the wave equation" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter St one, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 2 7.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "load Fourier series and Fou rier transform procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m- file " }{TEXT 262 9 "fourier.m" }{TEXT -1 37 " contains the code for t he procedure " }{TEXT 0 13 "FourierSeries" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \"K:\\\\Maple/procdrs/fourier.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "A procedure for constructing s olutions to the wave equation: " }{TEXT 0 20 "convert(..,wave,x,t)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 19 "convert/wave: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 18 "Calling Sequence:\n" }} {PARA 0 "" 0 "" {TEXT 265 2 " " }{TEXT -1 27 " convert(SS, wave, x, t) " }{TEXT 266 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 " " {TEXT 23 10 " SS - " }{TEXT -1 52 " a finite trigonometric s ine series of the form " }}{PARA 257 "" 0 "" {TEXT -1 18 " \+ " }{XPPEDIT 18 0 "b[1]*sin(c[1]*x)+b[2]*sin(c[2]*x)+` . . . ` + \+ b[n]*sin(c[n]*x)" "6#,**&&%\"bG6#\"\"\"F(-%$sinG6#*&&%\"cG6#F(F(%\"xGF (F(F(*&&F&6#\"\"#F(-F*6#*&&F.6#F4F(F0F(F(F(%(~.~.~.~GF(*&&F&6#%\"nGF(- F*6#*&&F.6#F>F(F0F(F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 26 " where " }{XPPEDIT 18 0 "b[1],b[2],` . . . `,b[ n]" "6&&%\"bG6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "c[1],c[2],` . . . `,c[n]" "6&&%\"cG6#\"\"\"&F$6#\" \"#%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 27 " are finite real constants." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {TEXT 23 7 "x - " }{TEXT 268 12 "the variable" }{TEXT -1 32 " used \+ in the finite sine series " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 10 " t - " }{TEXT 267 104 "the 2n d \"time\" variable to be used in the description of the corresponding solution to the wave equation" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 12 "convert/wave" }{TEXT -1 57 " converts a finite trigonometric sine series of the form " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[1]*sin(c[1]*x)+b[2]*sin(c [2]*x)+` . . . ` + b[n]*sin(c[n]*x)" "6#,**&&%\"bG6#\"\"\"F(-%$sinG6#* &&%\"cG6#F(F(%\"xGF(F(F(*&&F&6#\"\"#F(-F*6#*&&F.6#F4F(F0F(F(F(%(~.~.~. ~GF(*&&F&6#%\"nGF(-F*6#*&&F.6#F>F(F0F(F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "to a finite sum " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[1]*sin(c[1]*x)*cos(c[1]*t)+b[2]*sin(c[2]*x)*cos( c[2]*t)+` . . . ` + b[n]*sin(c[n]*x)*cos(c[2]*t)" "6#,**(&%\"bG6#\"\" \"F(-%$sinG6#*&&%\"cG6#F(F(%\"xGF(F(-%$cosG6#*&&F.6#F(F(%\"tGF(F(F(*(& F&6#\"\"#F(-F*6#*&&F.6#F;F(F0F(F(-F26#*&&F.6#F;F(F7F(F(F(%(~.~.~.~GF(* (&F&6#%\"nGF(-F*6#*&&F.6#FJF(F0F(F(-F26#*&&F.6#F;F(F7F(F(F(" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "to provide a solution for the wave equation " }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 " diff(u,`$`(x,2)) = diff(u,`$`(t,2));" "6#/-%%diffG6$%\"uG-%\"$G6$%\"xG \"\"#-F%6$F'-F)6$%\"tGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 16 " How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedu re active open the subsection, place the cursor anywhere after the pro mpt [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "convert/wave: implementation" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 238 "A user can make his own conversions known to the convert funct ion by defining a Maple procedure in the following way. If the procedu re `convert/f` is defined, then the function call convert(a,f,x,y,...) will invoke `convert/f`(a,x,y,...);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 889 "`convert/wave` := proc(ff ,x,t)\n local i,wv,A,B,la,aa,bb,term;\n if type(ff,`+`) then\n \+ wv := 0; \n for i to nops(ff) do\n term := op(i,ff);\n \+ if patmatch(term,A::realcons*sin(B::realcons*x),'la') then\n \+ aa := subs(la,A);\n bb := subs(la,B);\n \+ if not type(aa,infinity) and not type(bb,infinity) then\n \+ wv := wv + term*cos(bb*t)\n else\n error \+ \"unrecognized conversion\"\n end if;\n else error \+ \"unrecognized conversion\"\n end if;\n end do;\n elif \+ patmatch(ff,A::realcons*sin(B::realcons*x),'la') then\n aa := sub s(la,A);\n bb := subs(la,B);\n if not type(aa,infinity) and \+ not type(bb,infinity) then\n wv := ff*cos(bb*t)\n else\n \+ error \"unrecognized conversion\"\n end if;\n else erro r \"unrecognized conversion\"\n end if;\n wv;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a=1" "6#/%\"aG\"\"\"" }{TEXT -1 23 " in the wave equation " }} {PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "a^2*diff(u,`$`(x,2) ) = diff(u,`$`(t,2));" "6#/*&%\"aG\"\"#-%%diffG6$%\"uG-%\"$G6$%\"xGF& \"\"\"-F(6$F*-F,6$%\"tGF&" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 52 "which describes the motion of a vibrating string. " }}{PARA 258 "" 0 "" {TEXT -1 80 "In the following examples we find a solution \+ subject to the boundary conditions " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "u(0,t)=u(Pi,t)" "6#/-%\"uG6$\"\"!%\"tG-F%6$%#PiGF(" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " }{TEXT 272 2 "t " }{TEXT -1 5 "with " }{XPPEDIT 18 0 "0 " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 21 "Initial deflection: " }{XPPEDIT 18 0 "f(x)=sin(x)^3 /2" "6#/-%\"fG6#%\"xG*&-%$sinG6#F'\"\"$\"\"#!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "f := x -> 1/2*sin(x)^3:\n'f(x)'=f(x);\nplot(f(x),x=0..Pi,thicknes s=2);\nFourierSeries(f(x),x=0..Pi,type=sin);\nconvert(%,wave,x,t);\nu \+ := unapply(%,x,t);\nplots[animate](u(x,t),x=0..Pi,t=0..2*Pi,thickness= 2,frames=50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*$)-% $sinGF&\"\"$\"\"\"#F.\"\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 463 81 81 {PLOTDATA 2 "6&-%'CURVESG6$7W7$$\"\"!F)F(7$$\"3%)eD2LzxZo!#>$\"3u`o#[k rF4$\"3 \"4)o`z9>TOF77$$\"3[99!=3o^i#F4$\"35h$y:!\\gQ()F77$$\"35!\\D0[nkH$F4$ \"3!)*[;OBIgp\"F-7$$\"37\"=Za&z%)=RF4$\"3;_4%3VQcy#F-7$$\"3edXa()oGjXF 4$\"3')Q)>e>i#yUF-7$$\"3W%3**Hbm(H_F4$\"3ZS?:%>9)HiF-7$$\"37PRr4)3T*eF 4$\"3-hRo/\">!)e)F-7$$\"3y\"[)yykYxlF4$\"3c7]j`NPU6F47$$\"3[s'ocGo$zrF 4$\"3Jz)RS%=OB9F47$$\"3UQ0;gr'p&yF4$\"3!4ueN]]$p*F4$\"3DypX]N*R^#F47$$\"3#*R7\\HeV)y*F4 $\"3?+7`$y7u&GF47$$\"3#G[))*)3W'\\5!#<$\"3m!z5IBZ8E$F47$$\"30'[@XS@'46 Ffp$\"3a#GN&*4i4f$F47$$\"3G9w$3G*Qz6Ffp$\"3SV!3J^E#\\RF47$$\"3>%3*3tc9 T7Ffp$\"3$4zxtES]B%F47$$\"3Ey0DBA!*38Ffp$\"3!HK*p\">id]%F47$$\"3qjE.#[ AMP\"Ffp$\"3c())y)** *\\F47$$\"3\")o0'))oxQg\"Ffp$\"3%H/EUT(z\"*\\F47$$\"3mwM!*4(4&Q;Ffp$\" 3a*eOqe.d'\\F47$$\"31,%Gg)plo;Ffp$\"3Rzws;[dG\\F47$$\"3CDL:iU!))p\"Ffp $\"37))))R$)QFy[F47$$\"3o(R1/.CRw\"Ffp$\"3*)=0=pMGEZF47$$\"32Id)H7*>J= Ffp$\"3!eV()zy_6^%F47$$\"3Gfb7wY,(*=Ffp$\"3/IO\"H]-*\\UF47$$\"3cM5'zg% pg>Ffp$\"3#)f;pwZfcRF47$$\"3**oJ8:.SJ?Ffp$\"3QporDv,%f$F47$$\"3)**p!zP D$\\4#Ffp$\"3xc)*)>srXC$F47$$\"3k#3/4JdGF47$$\"3wa k3@YBCAFfp$\"3?ciHJZ\"G]#F47$$\"39/b_s`7QC'F-7$$\"3'=BS\\0:[o#Ffp$\"3?%yqR%*3+H %F-7$$\"3[gN,?R*3v#Ffp$\"3A*p;?*orhFF-7$$\"3@/0LMNh6GFfp$\"3lgU77x(4q \"F-7$$\"3w#oUX107)GFfp$\"3i\"Qzv@kC`)F77$$\"3W46xe&[M%HFfp$\"34!)e*pr lR\"QF77$$\"3H.6)e58)4IFfp$\"3kXaX)*QLM6F77$$\"3Kt2RXCLtIFfp$\"31,+/;g d'e\"F07$$\"3!)***\\/l#fTJFfp$\"3-/gOD%\"uGf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),&*&-%$sinG6#9$\" \"\"-%$cosG6#9%F3#\"\"$\"\")*&#F3F:F3*&-F06#,$F2F9F3-F56#,$F7F9F3F3!\" \"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The following alternative solution due to d`Alembert gives essenti ally the same animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "f := x -> 1/2*sin(x)^3:\n'f(x)'=f( x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x)),x):\n 'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-2*Pi*floor(x/(2*Pi)+1/2));\nu := (x,t)-> (f_(x+t)+f_(x-t))/2:\nplots[animate](u(x,t),x=0..Pi,t=0..2 *Pi,thickness=2,frames=50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 21 "Initial deflec tion: " }{XPPEDIT 18 0 "f(x) = 1/2;" "6#/-%\"fG6#%\"xG*&\"\"\"F)\"\"# !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(2*x)*cos(x)=1/4" "6#/*&-%$s inG6#*&\"\"#\"\"\"%\"xGF*F*-%$cosG6#F+F**&F*F*\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(3*x)+1/4" "6#,&-%$sinG6#*&\"\"$\"\"\"%\"xGF)F) *&F)F)\"\"%!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG 6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "f := x -> 1/2*sin(2*x)*cos(x):\n'f (x)'=f(x);\nplot(f(x),x=0..Pi,thickness=2);\nFourierSeries(f(x),x=0..P i,type=sin,numterms=3);\nconvert(%,wave,x,t);\nu := unapply(%,x,t);\np lots[animate](u(x,t),x=0..Pi,t=0..2*Pi,thickness=2,frames=50);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,$*&-%$sinG6#,$F'\"\"#\" \"\"-%$cosGF&F/#F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 486 101 101 {PLOTDATA 2 "6&-%'CURVESG6$7gp7$$\"\"!F)F(7$$\"3Uzi`m*))QU$!#>$\"3E:6t b&3#>MF-7$$\"3%)eD2LzxZoF-$\"3PyO&4)>R5oF-7$$\"32b4U:a)o#)*F-$\"3[Jf`X $Qmr*F-7$$\"3)\\$px*G*f!G\"!#=$\"3a[R@%fsiD\"F=7$$\"3NA&zO3Jch\"F=$\"3 %RD&48m)pc\"F=7$$\"3+5@exGm]>F=$\"3i:IGI=\\l=F=7$$\"3Ci!4b$F=7$$\"3+@=F?n_'*[F=$\"3=rR\\+f%Gm$F=7$$\"3W%3**Hbm(H_F=$\"3+O o5u%['[PF=7$$\"3C5lN\"oP>c&F=$\"3]*o0]\"4&z!QF=7$$\"37PRr4)3T*eF=$\"3$ o^gL@56%QF=7$$\"3'Qd#)pA[\\1'F=$\"33n)*=0x1[QF=7$$\"3Y47DWwyNiF=$\"3=p KB`eC[QF=7$$\"32X)>:1FmS'F=$\"3f;1i*F=$\"3=EAwJ&*zBHF=7$$\"3#e'***\\tG2\\*F=$\"3czVZ?= adFF=7$$\"3#*R7\\HeV)y*F=$\"3m'GjC3,Pe#F=7$$\"3_.)of$QC95!#<$\"3M1_tAQ XpBF=7$$\"3#G[))*)3W'\\5Fet$\"3;GCTIjw\\@F=7$$\"3W%)\\vYFjz5Fet$\"3%RQ )zj8eh>F=7$$\"30'[@XS@'46Fet$\"3G!G]uayLx\"F=7$$\"3=]&zEM0X9\"Fet$\"3q Lgv$Hxnb\"F=7$$\"3G9w$3G*Qz6Fet$\"3aoS#G**y_M\"F=7$$\"3C\\L'pZn-@\"Fet $\"3GL4)psaW;\"F=7$$\"3>%3*3tc9T7Fet$\"32s4GbNu9**F-7$$\"3Ey0DBA!*38Fe t$\"3UJCyPp)[Z'F-7$$\"3qjE.#[AMP\"Fet$\"3;v6Iy:mqPF-7$$\"3*R:_MgU2W\"F et$\"3AUd#f()ywm\"F-7$$\"3!*[urYIlr9Fet$\"3!ytr\\d:\"\\(*!#?7$$\"3\"Qu #)**[jD]\"Fet$\"3X5+F-CoPYF_x7$$\"3?t9WMSB>:Fet$\"3k%G)o-Vy_EF_x7$$\"3 Q--!*yX!f`\"Fet$\"3Er@u0%*>;7F_x7$$\"3yJ*eL7vDb\"Fet$\"3pFA\"z./#>L!#@ 7$$\"3=hw\"ymX#p:Fet$\"3aUT\\X%QXS#!#B7$$\"3)\\6R$y;c'e\"Fet$\"3')G*Q! Q$[\\[#Fdy7$$\"3\")o0'))oxQg\"Fet$\"3yO%4uzyL4\"F_x7$$\"3&G-#Q*p$>@;Fe t$\"32M1B9M_MDF_x7$$\"3mwM!*4(4&Q;Fet$\"3w)e>zH2wc%F_x7$$\"31,%Gg)plo; Fet$\"3eo4\\6D]+&*F_x7$$\"3CDL:iU!))p\"Fet$\"3A=4&Q!*Qjh\"F-7$$\"3o(R1 /.CRw\"Fet$\"3mA%RdV/_h$F-7$$\"32Id)H7*>J=Fet$\"304Fp2\"zbS'F-7$$\"3Gf b7wY,(*=Fet$\"3'H2L[H\\!G(*F-7$$\"3/(HV?ka)G>Fet$\"3#Gy?U=..:\"F=7$$\" 3cM5'zg%pg>Fet$\"3'egi0s(GO8F=7$$\"3y,rahu/'*>Fet$\"3!f&4z%f&Q]:F=7$$ \"3**oJ8:.SJ?Fet$\"3)f@#y%*o!)pYEk;j?Fet$\"3%H'pfyZ;p>F=7$ $\"3)**p!zPD$\\4#Fet$\"3/#[1j;A%o@F=7$$\"3#)e[(>k\\)G@Fet$\"3g#*QNV!o' yBF=7$$\"3k#Fet$\"3S/=2B(\\ Jw#F=7$$\"3wak3@YBCAFet$\"3g=YJ=MNMHF=7$$\"39/bDFet$\"3 YnASb:X[QF=7$$\"3m**o^.n6ODFet$\"3!Q+3KeVy%QF=7$$\"3#Q'HHd#HIb#Fet$\"3 Kuw'oO(eSQF=7$$\"3!y0vg&og&e#Fet$\"3X];$Q7hw!QF=7$$\"3z^r&[X%==EFet$\" 3Uo.pdpe\\PF=7$$\"3%=p)*[v*\\^EFet$\"3C3#3@_]Tm$F=7$$\"3'=BS\\0:[o#Fet $\"3'RBtN(=e_NF=7$$\"3[gN,?R*3v#Fet$\"3]q*eV\\-gD$F=7$$\"3MK?=\\-+d(3$F=7$$\"3@/0LMNh6GFet$\"3ua'o,?P+!HF=7$$\"3E$fO%*H4k%G Fet$\"3=;bBWi%Hm#F=7$$\"3w#oUX107)GFet$\"38JQ3j;!RS#F=7$$\"36'*ol6oK7H Fet$\"3YJ8Xo!\\_:#F=7$$\"3W46xe&[M%HFet$\"3!*=v=w9A#*=F=7$$\"3Q1hKK3jw HFet$\"39&\\\\j#Q'yf\"F=7$$\"3H.6)e58)4IFet$\"3g%\\aK+)H\"H\"F=7$$\"34 QfjvFdTIFet$\"3AAUM&Hfd))*F-7$$\"3Kt2RXCLtIFet$\"3?sWW5'*)*)y'F-7$$\"3 M'Q?zaiu5$Fet$\"3hs\\S6YP3MF-7$$\"3!)***\\/l#fTJFet$\"3pawpOMzRJ!#E-%' COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6 \"Q!F\\il-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#%\"xG#\"\"\"\"\"%*&F(F)-F%6#,$F'\"\"$F)F)" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$cosG6#%\"t GF)#F)\"\"%*(F.F)-F&6#,$F(\"\"$F)-F+6#,$F-F4F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),&*&-%$si nG6#9$\"\"\"-%$cosG6#9%F3#F3\"\"%*(F8F3-F06#,$F2\"\"$F3-F56#,$F7F>F3F3 F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The following alternative solution due to d`Alembert gives essenti ally the same animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "f := x -> 1/2*sin(2*x)*cos(x):\n'f (x)'=f(x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x) ),x):\n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-2*Pi*floor(x/(2*Pi)+1/ 2));\nu := (x,t)-> (f_(x+t)+f_(x-t))/2:\nplots[animate](u(x,t),x=0..Pi ,t=0..2*Pi,thickness=2,frames=50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 21 "Initial defle ction: " }{XPPEDIT 18 0 "f(x) = cos(4*x)*sin(x)/2;" "6#/-%\"fG6#%\"xG *(-%$cosG6#*&\"\"%\"\"\"F'F.F.-%$sinG6#F'F.\"\"#!\"\"" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "f := x -> 1/2*cos(4*x)*sin(x):\n'f(x)'=f(x);\nplot(f(x),x=0.. Pi,thickness=2);\nFourierSeries(f(x),x=0..Pi,type=sin,numterms=7);\nco nvert(%,wave,x,t);\nu := unapply(%,x,t);\nplots[animate](u(x,t),x=0..P i,t=0..2*Pi,thickness=2,frames=50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/-%\"fG6#%\"xG,$*&-%$cosG6#,$F'\"\"%\"\"\"-%$sinGF&F/#F/\"\"#" }} {PARA 13 "" 1 "" {GLPLOT2D 486 101 101 {PLOTDATA 2 "6&-%'CURVESG6$7gq7 $$\"\"!F)F(7$$\"3%)eD2LzxZo!#>$\"3o>n$H$F-7$$\"32b4U:a)o#)*F-$\"3( RdO1%)>9`%F-7$$\"3)\\$px*G*f!G\"!#=$\"3Gi43.I#fc&F-7$$\"3NA&zO3Jch\"F8 $\"3$)Q\\T3W9@kF-7$$\"3+5@exGm]>F8$\"3=oi#ftm!))oF-7$$\"36Opjy\"*G>@F8 $\"374,TGQFfpF-7$$\"3CiF87$$\"3y\"[)yykYxlF8$!3- :&Gy=#zmEF87$$\"3[s'ocGo$zrF8$!3w#zm(GJ8qJF87$$\"3+1Y\"Hsn\"=vF8$!3%Q' pP&[tSQ$F87$$\"3UQ0;gr'p&yF8$!3PTAUpneONF87$$\"3JhPI]C1F!)F8$!3Cyyme&z ve$F87$$\"33$)pWSx:(>)F8$!3raO'p()z.i$F87$$\"3)Rf=bQ0AG)F8$!3Goyxs\"R( HOF87$$\"3'[?!fIIDn$)F8$!3rp%f*)R-Vj$F87$$\"3'o\"=mv1I_%)F8$!3$4e%\\$H -Sj$F87$$\"3vFMt?$[t`)F8$!3#3A-eFz(GOF87$$\"3.Wsn]mE,()F8$!3KXqmA/k/OF 87$$\"3?f5i!)\\=l))F8$!3ad\\*F8$!3_\"p/>'p3>MF87$$\"3#e'***\\tG2\\*F8$!3)pKp(pI*R A$F87$$\"3#*R7\\HeV)y*F8$!3#*yfYbf&y'HF87$$\"3_.)of$QC95!#<$!3]U^%H:@v e#F87$$\"3#G[))*)3W'\\5Feu$!3!Ri6Vp\"F87$$\"30'[@XS@'46Feu$!3sSa`Hf;67F87$$\"3=]&zEM0X9\"Feu$!3;9=))pI t)4'F-7$$\"3G9w$3G*Qz6Feu$\"3W&GdszN')Q#!#?7$$\"3C\\L'pZn-@\"Feu$\"3&G 4oqr-Q+'F-7$$\"3>%3*3tc9T7Feu$\"3oy-O$el/=\"F87$$\"37J)p\"[R-v7Feu$\"3 0qY&G!)[#3=F87$$\"3Ey0DBA!*38Feu$\"3_c4&4^oJT#F87$$\"3)4iTENi6M\"Feu$ \"33;*[APa_&HF87$$\"3qjE.#[AMP\"Feu$\"3'>n'>@wl_MF87$$\"3u3CuUD329Feu$ \"3$3q(=E\\l7RF87$$\"3*R:_MgU2W\"Feu$\"3p&[&y[C#>I%F87$$\"3!*[urYIlr9F eu$\"332vr&f-$*e%F87$$\"3\"Qu#)**[jD]\"Feu$\"3f[`Y\\-s.[F87$$\"3?t9WMS B>:Feu$\"3;\\Ae!HDv)[F87$$\"3Q--!*yX!f`\"Feu$\"3)eYL!\\zO[\\F87$$\"3'p cH6&)RUa\"Feu$\"39B#*p&ej+(\\F87$$\"3yJ*eL7vDb\"Feu$\"3CJVv$e(*e)\\F87 $$\"3e'H)e&R54c\"Feu$\"3c#zby5Ze*\\F87$$\"3=hw\"ymX#p:Feu$\"3e)\\Ot!y* )**\\F87$$\"3?)QyIn.zd\"Feu$\"3OaL(oK`y*\\F87$$\"3)\\6R$y;c'e\"Feu$\"3 U&eQSKT%*)\\F87$$\"3yT)*f$o>_f\"Feu$\"3OK$4)*)[nu\\F87$$\"3\")o0'))oxQ g\"Feu$\"3/xCe,qd`\\F87$$\"3&G-#Q*p$>@;Feu$\"3+v$*R[z_#*[F87$$\"3mwM!* 4(4&Q;Feu$\"3'p^6*pPn1[F87$$\"31,%Gg)plo;Feu$\"3qW_yn\\o*f%F87$$\"3CDL :iU!))p\"Feu$\"3G6X/!o\"4BVF87$$\"3Yh)zi9k8t\"Feu$\"3]\\=Q(o&=_RF87$$ \"3o(R1/.CRw\"Feu$\"3Q^U*)p5B9NF87$$\"3wjgpw:c(z\"Feu$\"3(o]JZEd9+$F87 $$\"32Id)H7*>J=Feu$\"3\\z_\"*4A0RCF87$$\"3oWcb**o5k=Feu$\"3SblTYj9`=F8 7$$\"3Gfb7wY,(*=Feu$\"3ZQD/C.qW7F87$$\"3/(HV?ka)G>Feu$\"3?0sy2&>!okF-7 $$\"3cM5'zg%pg>Feu$\"3eC/O#zz5=&Faw7$$\"3y,rahu/'*>Feu$!3Cxx$)Q\"=Q\"f F-7$$\"3**oJ8:.SJ?Feu$!3%*)zQH#)e;?\"F87$$\"3[M>YEk;j?Feu$!3\"))='HgWY 5k\\)G@Feu$!3:(3$R &4U_g#F87$$\"3k#Feu$!3%zU*eI WSJKF87$$\"3wak3@YBCAFeu$!3%)3fzxZmHMF87$$\"3)oreoxM5C#Feu$!3c]1vg7&*4 NF87$$\"3Yz4jK\\$yD#Feu$!3u!3%HK(*GqNF87$$\"3-UKS)3NYF#Feu$!3,!Re1zi2h $F87$$\"39/ber=OF87$$\"3(eL?h:e)QBFeu$!3#H()p#G$ \\we$F87$$\"3#*z_V$zlYN#Feu$!3O4GEL$o3a$F87$$\"3YB3&\\'Gr(Q#Feu$!3/I#* o'p*>&R$F87$$\"3cmjYO*f2U#Feu$!3!*Q6!oV21>$F87$$\"3)eq)=U!z`[#Feu$!3s] ^(Qo/Bl#F87$$\"3#Q'HHd#HIb#Feu$!31fAjEu**e>F87$$\"3z^r&[X%==EFeu$!3WcA 7/r$zC\"F87$$\"3'=BS\\0:[o#Feu$!3o(\\xQ_%y!f&F-7$$\"3[gN,?R*3v#Feu$\"3 :H))*eC(eB:Faw7$$\"3@/0LMNh6GFeu$\"3U!zlJ%>/ASF-7$$\"3E$fO%*H4k%GFeu$ \"3EkFT#Hx4`&F-7$$\"3w#oUX107)GFeu$\"3wW?Z#3I#*\\'F-7$$\"3?*y*4Qfw'*GF eu$\"3UjOd[)fxv'F-7$$\"36'*ol6oK7HFeu$\"3J#3*Q+L`5pF-7$$\"3+.S@&o()y#H Feu$\"3!4#\\J1!\\+'pF-7$$\"3W46xe&[M%HFeu$\"3^QLwuTc4pF-7$$\"3Q1hKK3jw HFeu$\"3!eZy(=W=(['F-7$$\"3H.6)e58)4IFeu$\"3a(*Q+8U7ycF-7$$\"34QfjvFdT IFeu$\"3%Gc;1c*Q)f%F-7$$\"3Kt2RXCLtIFeu$\"3Ww'>E(\\-%G$F-7$$\"3!)***\\ /l#fTJFeu$\"39F)[$=n*)p:!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABE LSG6$Q\"x6\"Q!Fg]m-%*THICKNESSG6#\"\"#-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFA ULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$sinG6#,$%\"xG\"\"$#!\"\"\"\" %*&#\"\"\"F,F/-F%6#,$F(\"\"&F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&-%$sinG6#,$%\"xG\"\"$\"\"\"-%$cosG6#,$%\"tGF*F+#!\"\"\"\"%*(#F+F3F+- F&6#,$F)\"\"&F+-F-6#,$F0F9F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" uGf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),&*&-%$sinG6#,$9$\"\"$\"\"\" -%$cosG6#,$9%F4F5#!\"\"\"\"%*(#F5F=F5-F06#,$F3\"\"&F5-F76#,$F:FCF5F5F) F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 " The following alternative solution due to d`Alembert gives essentially the same animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "f := x -> 1/2*cos(4*x)*sin(x):\n'f(x)'=f (x);\nf_odd := unapply(simplify(f(piecewise(x<0,-x,x))*signum(x)),x): \n'f_odd(x)'=f_odd(x);\nf_ := x -> f_odd(x-2*Pi*floor(x/(2*Pi)+1/2)); \nu := (x,t)-> (f_(x+t)+f_(x-t))/2:\nplots[animate](u(x,t),x=0..Pi,t=0 ..2*Pi,thickness=2,frames=50);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 21 "Initial deflec tion: " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x/Pi, 0 <= x and x < Pi/2], [1-x/Pi, Pi/2 <= x and x <= Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$* &F'\"\"\"%#PiG!\"\"31\"\"!F'2F'*&F.F-\"\"#F/7$,&F-F-*&F'F-F.F/F/31*&F. F-F5F/F'1F'F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "f := x -> piecewise(x$\"3!)*>)[9arz@F-7$$\"3)\\$px*G*f!G\"!#=$ \"3o$Rf<9ui2%F-7$$\"3+5@exGm]>F3$\"3Aq5Y&y_\"4iF-7$$\"3[99!=3o^i#F3$\" 3-$G:L`phN)F-7$$\"35!\\D0[nkH$F3$\"3+OY`)=)H\\5F37$$\"37\"=Za&z%)=RF3$ \"3-t*pm/3uC\"F37$$\"3edXa()oGjXF3$\"3[#H)zH$RDX\"F37$$\"3W%3**Hbm(H_F 3$\"3-MHD'R'ok;F37$$\"37PRr4)3T*eF3$\"3K@\\P/J:w=F37$$\"3y\"[)yykYxlF3 $\"3/g3C1En$4#F37$$\"3[s'ocGo$zrF3$\"3Y6F)=!RE&G#F37$$\"3UQ0;gr'p&yF3$ \"3*>[+DK]4]#F37$$\"3vFMt?$[t`)F3$\"3:MSGsB_*F3$\" 3gYa2s'Hi#HF37$$\"3#*R7\\HeV)y*F3$\"3L&p-[*ev:JF37$$\"3#G[))*)3W'\\5!# <$\"3^ET*\\!47TLF37$$\"30'[@XS@'46Fdp$\"3s@L!)HY.KNF37$$\"3G9w$3G*Qz6F dp$\"3oP!)\\xE6aPF37$$\"3>%3*3tc9T7Fdp$\"3#o!\\Qz$*o]RF37$$\"3Ey0DBA!* 38Fdp$\"3i@g3x^OmTF37$$\"3qjE.#[AMP\"Fdp$\"35l2QR&RJ=Fdp$\"3G\\b#e:76<%F37$$\"3Gfb7wY,(*=Fdp$\"3#p# \\`VZhhRF37$$\"3cM5'zg%pg>Fdp$\"3=5v)\\A:*ePF37$$\"3**oJ8:.SJ?Fdp$\"3! [Wi*o>&Q`$F37$$\"3)**p!zPD$\\4#Fdp$\"3;D7LBEiJLF37$$\"3k%F-7$$\" 3Kt2RXCLtIFdp$\"3'*)=r-\"**ys@F-7$$\"3!)***\\/l#fTJFdp$\"3=Wmfw?F%***! #F-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!F]\\l-%*THI CKNESSG6#\"\"#-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),@*(%#PiG !\"#-%$sinG6#9$\"\"\"-%$cosG6#9%F5\"\"%*&#F:\"\"*F5*(F/F0-F26#,$F4\"\" $F5-F76#,$F9FBF5F5!\"\"**#F:\"#DF5F/F0-F26#,$F4\"\"&F5-F76#,$F9FMF5F5* &#F:\"#\\F5*(F/F0-F26#,$F4\"\"(F5-F76#,$F9FXF5F5FF**#F:\"#\")F5F/F0-F2 6#,$F4F=F5-F76#,$F9F=F5F5*&#F:\"$@\"F5*(F/F0-F26#,$F4\"#6F5-F76#,$F9Ff oF5F5FF**#F:\"$p\"F5F/F0-F26#,$F4\"#8F5-F76#,$F9F`pF5F5*&#F:\"$D#F5*(F /F0-F26#,$F4\"#:F5-F76#,$F9F[qF5F5FF**#F:\"$*GF5F/F0-F26#,$F4\"#F5-F76#,$F9F`rF5F5FF**#F: \"$T%F5F/F0-F26#,$F4\"#@F5-F76#,$F9FjrF5F5*&#F:\"$H&F5*(F/F0-F26#,$F4 \"#BF5-F76#,$F9FesF5F5FF**#F:\"$D'F5F/F0-F26#,$F4FIF5-F76#,$F9FIF5F5*& #F:\"$H(F5*(F/F0-F26#,$F4\"#FF5-F76#,$F9FitF5F5FF**#F:\"$T)F5F/F0-F26# ,$F4\"#HF5-F76#,$F9FcuF5F5F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "The following alternative solution due to d`Alembert gives essentially the same animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "f := x -> p iecewise(x f_odd(x-2*Pi*floor(x/(2*Pi)+1/2));\nu := (x,t)-> (f_(x+t)+f_(x-t))/ 2:\nplots[animate](u(x,t),x=0..Pi,t=0..2*Pi,thickness=2,frames=50);" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 " " 0 "" {TEXT -1 21 "Initial deflection: " }{XPPEDIT 18 0 "f(x) = PIEC EWISE([sin(2*x)^2/2, 0 <= x and x < Pi/2],[0, Pi/2 <= x and x <= Pi]); " "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&-%$sinG6#*&\"\"#\"\"\"F'F2F1F1! \"\"31\"\"!F'2F'*&%#PiGF2F1F37$F631*&F9F2F1F3F'1F'F9" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> piecewise(xr\"!#>$\"3e8)Q1q<#feF07 $$\"3u4A!\\s;zc#F4$\"3-Qs$46!o<8F-7$$\"3Uzi`m*))QU$F4$\"3)pW!pX1%4M#F- 7$$\"3[>W!)\\M$e8&F4$\"3aw3v'>IoD&F-7$$\"3%)eD2LzxZoF4$\"3SB'3zbA*>$*F -7$$\"32b4U:a)o#)*F4$\"3r$=x6v8m!>F47$$\"3)\\$px*G*f!G\"!#=$\"3%p>5$pk x3KF47$$\"3NA&zO3Jch\"FS$\"3$fkPf^X8/&F47$$\"3+5@exGm]>FS$\"3`P#3g0C=B (F47$$\"3Ci#f)3!*pd:FS7$$\"35!\\D0[nkH$FS$\"3'\\OK k)*Rh(=FS7$$\"3hNj)zrdwg$FS$\"3W%pKVbM:=#FS7$$\"37\"=Za&z%)=RFS$\"3?cy &G:d=\\#FS7$$\"3Npe\\@u1TUFS$\"3kUYpu5D8GFS7$$\"3edXa()oGjXFS$\"3ema<] %[%HJFS7$$\"3+@=F?n_'*[FS$\"3)Gbo!z\\TXMFS7$$\"3W%3**Hbm(H_FS$\"3G)y*R :%3Yu$FS7$$\"3C5lN\"oP>c&FS$\"3oaBfbD(3-%FS7$$\"37PRr4)3T*eFS$\"3Inz0] mKqUFS7$$\"3Y47DWwyNiFS$\"3kVI)4-@V\\%FS7$$\"3y\"[)yykYxlFS$\"33(GhT?@ 6o%FS7$$\"3cw&GAQ<%yoFS$\"3b#et(o!e?\"[FS7$$\"3[s'ocGo$zrFS$\"3$RafUMI &4\\FS7$$\"3CR;H/!o([tFS$\"3wCY,H`7\\\\FS7$$\"3+1Y\"Hsn\"=vFS$\"3UbJ#e pzu(\\FS7$$\"3Q*3EAenGg(FS$\"39/UiY*)R()\\FS7$$\"3wsv`Tuc(o(FS$\"3#Rn# =3LY%*\\FS7$$\"39c!\\3InAx(FS$\"3#4iUCnk')*\\FS7$$\"3UQ0;gr'p&yFS$\"3' 3*oK<#)****\\FS7$$\"3J\\@B0[,UzFS$\"3gl]I#>]%)*\\FS7$$\"3JhPI]C1F!)FS$ \"3)\\TT*)*4,%*\\FS7$$\"3?s`P&45@6)FS$\"3S*e-Ux&o')\\FS7$$\"33$)pWSx:( >)FS$\"31(fEZ*H[w\\FS7$$\"3'[?!fIIDn$)FS$\"3'*[Ww8`\\Z\\FS7$$\"3vFMt?$ [t`)FS$\"3y,@k$4#=2\\FS7$$\"3?f5i!)\\=l))FS$\"3w*Gz&*em#)z%FS7$$\"3u\" p30k@I>*FS$\"3%3Nem+'))\\YFS7$$\"3#e'***\\tG2\\*FS$\"3I/zql$yI[%FS7$$ \"3#*R7\\HeV)y*FS$\"3#4*fWdC=)G%FS7$$\"3_.)of$QC95!#<$\"38r>'>1=P-%FS7 $$\"3#G[))*)3W'\\5F]y$\"3#p6*RWGvGPFS7$$\"3W%)\\vYFjz5F]y$\"3W7zwuYQfM FS7$$\"30'[@XS@'46F]y$\"3QG0Kb$Gi<$FS7$$\"3=]&zEM0X9\"F]y$\"3l\"4 \\$GFS7$$\"3G9w$3G*Qz6F]y$\"3/()>5+(zq[#FS7$$\"3C\\L'pZn-@\"F]y$\"3U<< fzZ=z@FS7$$\"3>%3*3tc9T7F]y$\"3kXvU2y2LF47$$\"3!*[urYIlr9F]y$ \"3$>cNf<[-%>F47$$\"3\"Qu#)**[jD]\"F]y$\"3=&fJ&H9y`#*F-7$$\"3?t9WMSB>: F]y$\"3e)e@)*G<&)H&F-7$$\"3Q--!*yX!f`\"F]y$\"3U#y*4>$=4V#F-7$$\"3'pcH6 &)RUa\"F]y$\"3TzSnrp;49F-7$$\"3yJ*eL7vDb\"F]y$\"3kx!)*)*31tj'F07$$\"3e 'H)e&R54c\"F]y$\"3I'HrP-yV&>F07$$\"3=hw\"ymX#p:F]y$\"3EAF<4624[!#B7$$ \"3\")o0'))oxQg\"F]yF(7$$\"3mwM!*4(4&Q;F]yF(7$$\"3CDL:iU!))p\"F]yF(7$$ \"3o(R1/.CRw\"F]yF(7$$\"32Id)H7*>J=F]yF(7$$\"3Gfb7wY,(*=F]yF(7$$\"3cM5 'zg%pg>F]yF(7$$\"3**oJ8:.SJ?F]yF(7$$\"3)**p!zPD$\\4#F]yF(7$$\"3k%\"uGf*6$% \"xG%\"tG6\"6$%)operatorG%&arrowGF),@*(%#PiG!\"\"-%$sinG6#9$\"\"\"-%$c osG6#9%F5#\"\")\"#:**#\"\"#\"\"$F5F/F0-F26#,$F4F?F5-F76#,$F9F?F5F5**#F ;\"#@F5F/F0-F26#,$F4F@F5-F76#,$F9F@F5F5*&#F;\"#XF5*(F/F0-F26#,$F4\"\"& F5-F76#,$F9FWF5F5F0*&#F?FF5*(F/F0-F26#,$F4\"#8F5-F76#,$F9FbrF5F5F0*&#F?\"$:$F5*(F/F0-F26 #,$F4\"#9F5-F76#,$F9F]sF5F5F0*&#F;\"%NJF5*(F/F0-F26#,$F4F F5-F76#,$F9FhuF5F5F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The following alternative solution due to d`Ale mbert gives essentially the same animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "f := x -> piecewi se(x f_odd(x-2*Pi*floor(x/(2*Pi)+1/2));\nu := (x,t)-> (f_(x+t)+f_(x-t))/2 :\nplots[animate](u(x,t),x=0..Pi,t=0..2*Pi,thickness=2,frames=50);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 32 "In the following questions, let " }{XPPEDIT 18 0 "a=1" "6#/%\"a G\"\"\"" }{TEXT -1 23 " in the wave equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2" "6#*$%\"aG\"\"#" }{TEXT -1 1 " " } {XPPEDIT 18 0 "diff(u,`$`(x,2)) = diff(u,`$`(t,2));" "6#/-%%diffG6$%\" uG-%\"$G6$%\"xG\"\"#-F%6$F'-F)6$%\"tGF," }{TEXT -1 4 ". " }}{PARA 258 "" 0 "" {TEXT -1 51 "Find a solution subject to the boundary condi tions " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(0,t)=u(Pi ,t)" "6#/-%\"uG6$\"\"!%\"tG-F%6$%#PiGF(" }{XPPEDIT 18 0 "``=0" "6#/%!G \"\"!" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " }{TEXT 269 1 "t" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "t>0" "6#2\"\"!%\"tG" } {TEXT -1 45 " (that is, assume that the string has length " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 12 " units) and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "diff(u,t)" "6#-%%diffG6$%\"uG%\"tG" } {TEXT -1 1 " " }{TEXT 271 1 "|" }{XPPEDIT 18 0 "``[t=0]=``" "6#/&%!G6# /%\"tG\"\"!F%" }{XPPEDIT 18 0 "u[t](x,0)=0" "6#/-&%\"uG6#%\"tG6$%\"xG \"\"!F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 55 "(that is, ass ume that the string is initially at rest)." }}{PARA 0 "" 0 "" {TEXT -1 17 "Use the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 61 " given in the particular question for the initial deflect ion." }}{PARA 0 "" 0 "" {TEXT -1 50 "Construct an animation to illustr ate the solution." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 36 " : Where the Fourier sine series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6 #%\"xG" }{TEXT -1 62 " is infinite, take a suitable truncated series t o approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 22 "Initial deflection: " }{XPPEDIT 18 0 "f (x) = cos(6*x)*sin(x)/2;" "6#/-%\"fG6#%\"xG*(-%$cosG6#*&\"\"'\"\"\"F'F .F.-%$sinG6#F'F.\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = sin(7 *x)/4-sin(5*x)/4;" "6#/%!G,&*&-%$sinG6#*&\"\"(\"\"\"%\"xGF,F,\"\"%!\" \"F,*&-F(6#*&\"\"&F,F-F,F,F.F/F/" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 48 "________________________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 48 "________________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflection: " }{XPPEDIT 18 0 "f(x) = sin(x)^5/2;" "6#/-%\"fG6#%\"xG*&-%$sinG6#F' \"\"&\"\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "5*sin(x)/16 -5*sin( 3*x)/32+sin(5*x)/32" "6#,(*(\"\"&\"\"\"-%$sinG6#%\"xGF&\"#;!\"\"F&*(F% F&-F(6#*&\"\"$F&F*F&F&\"#KF,F,*&-F(6#*&F%F&F*F&F&F2F,F&" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "___ _____________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "__ ______________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " } }{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflection: " }{XPPEDIT 18 0 " f(x) = PIECEWISE([3/(2*Pi), 0 <= x and x < Pi/3],[1/2, Pi/3 <= x and x < 2*Pi/3],[3/2-3*x/(2*Pi), 2*Pi/3 <= x and x <= Pi]);" "6#/-%\"fG6#% \"xG-%*PIECEWISEG6%7$*&\"\"$\"\"\"*&\"\"#F.%#PiGF.!\"\"31\"\"!F'2F'*&F 1F.F-F27$*&F.F.F0F231*&F1F.F-F2F'2F'*(F0F.F1F.F-F27$,&*&F-F.F0F2F.*(F- F.F'F.*&F0F.F1F.F2F231*(F0F.F1F.F-F2F'1F'F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "f := x -> piecewise(x $\"3')*HK<7t&pKF-7$$\"3)\\$px*G*f!G\"!#=$\"3_!4RE@6W6'F-7$$\"3+5@exGm] >F3$\"3n0;>y\"HPJ*F-7$$\"3[99!=3o^i#F3$\"3Y#H(**HaU`7F37$$\"35!\\D0[nk H$F3$\"31a>!GGZRd\"F37$$\"37\"=Za&z%)=RF3$\"3Qf\\+q?6r=F37$$\"3edXa()o GjXF3$\"3sQup%**3)y@F37$$\"3W%3**Hbm(H_F3$\"3-,%zVfHq\\#F37$$\"37PRr4) 3T*eF3$\"3+#QillHU\"GF37$$\"3y\"[)yykYxlF3$\"3M!Hh$4*309$F37$$\"3[s'oc Go$zrF3$\"3*F3$\"3!*pJ63XM*Q%F37$$\"3#*R7\\H eV)y*F3$\"3)H//A%QjtYF37$$\"3_.)of$QC95!#<$\"3z;w%)*fdE%[F37$$\"3#G[)) *)3W'\\5Fdp$\"3++++++++]F37$$\"3W%)\\vYFjz5FdpFjp7$$\"30'[@XS@'46FdpFj p7$$\"3G9w$3G*Qz6FdpFjp7$$\"3>%3*3tc9T7FdpFjp7$$\"3Ey0DBA!*38FdpFjp7$$ \"3qjE.#[AMP\"FdpFjp7$$\"3*R:_MgU2W\"FdpFjp7$$\"3\"Qu#)**[jD]\"FdpFjp7 $$\"3=hw\"ymX#p:FdpFjp7$$\"3mwM!*4(4&Q;FdpFjp7$$\"3CDL:iU!))p\"FdpFjp7 $$\"3o(R1/.CRw\"FdpFjp7$$\"32Id)H7*>J=FdpFjp7$$\"3Gfb7wY,(*=FdpFjp7$$ \"3cM5'zg%pg>FdpFjp7$$\"3**oJ8:.SJ?FdpFjp7$$\"3[M>YEk;j?FdpFjp7$$\"3)* *p!zPD$\\4#Fdp$\"3uPo*\\$RV(*\\F37$$\"3#)e[(>k\\)G@Fdp$\"3^'o39p\"\\N[ F37$$\"3k1!Q %F37$$\"39/b *=LJF37$$\"3#Q'HHd#HIb#Fdp$\"3WNyI:K=5GF37$$\"3z^r&[X%==EFdp$\"3a\\P\\ i+4*\\#F37$$\"3'=BS\\0:[o#Fdp$\"3!\\qh:Q_4=#F37$$\"3[gN,?R*3v#Fdp$\"3[ F?DT!\\a'=F37$$\"3@/0LMNh6GFdp$\"3)ex;%QU`v:F37$$\"3w#oUX107)GFdp$\"3S f))\\d\"fKC\"F37$$\"3W46xe&[M%HFdp$\"3QA)e/k$og%*F-7$$\"3H.6)e58)4IFdp $\"3c&ov[#*4?H'F-7$$\"3Kt2RXCLtIFdp$\"3a%z1a'[=fKF-7$$\"3!)***\\/l#fTJ Fdp$\"3i'\\*[639*\\\"!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%*THICKNESSG6 #\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fiz-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG " 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 49 "_______________________________________ _________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "_______________________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q4 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflec tion: " }{XPPEDIT 18 0 "f(x) = PIECEWISE([sqrt(3)*x/(3*Pi), 0 <= x an d x < Pi/3],[sqrt(3)/6*(1-x/Pi), Pi/3 <= x and x <= Pi]);" "6#/-%\"fG6 #%\"xG-%*PIECEWISEG6$7$*(-%%sqrtG6#\"\"$\"\"\"F'F1*&F0F1%#PiGF1!\"\"31 \"\"!F'2F'*&F3F1F0F47$*(-F.6#F0F1\"\"'F4,&F1F1*&F'F1F3F4F4F131*&F3F1F0 F4F'1F'F3" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "f := x -> piecewise(x$\"3iQ:J\"Gf%e7F-7$$\"3)\\$px*G*f !G\"!#=$\"3_'\\NI(zV`BF-7$$\"3+5@exGm]>F3$\"3cja7Kg&[e$F-7$$\"3[99!=3o ^i#F3$\"3HFPXHnVC[F-7$$\"35!\\D0[nkH$F3$\"3w'e1h\"f7egF-7$$\"37\"=Za&z %)=RF3$\"39T@M:P\">?(F-7$$\"3edXa()oGjXF3$\"3%yiv1tRiQ)F-7$$\"3W%3**Hb m(H_F3$\"39$RH**Qr5h*F-7$$\"37PRr4)3T*eF3$\"3!H\"=&)*\\(>$3\"F37$$\"3y \"[)yykYxlF3$\"3$f53GW#y37F37$$\"3[s'ocGo$zrF3$\"3o<8#*GxR>8F37$$\"3UQ 0;gr'p&yF3$\"3JMb#>MCRW\"F37$$\"3vFMt?$[t`)F3$\"3?*F3$\"37HM)))[f%*o\"F37$$\"3#*R7\\HeV)y*F3$\"3=*\\$f/D)))z\"F37 $$\"3_.)of$QC95!#<$\"3EsMCz(RR'=F37$$\"3#G[))*)3W'\\5Fdp$\"3a19+>GDA>F 37$$\"3W%)\\vYFjz5Fdp$\"3Ug()R'z'p%*=F37$$\"30'[@XS@'46Fdp$\"3g9hzt29n =F37$$\"3G9w$3G*Qz6Fdp$\"37*G([jB..=F37$$\"3>%3*3tc9T7Fdp$\"3)H*=hdbGY J=Fdp$\"3W+I*HO'4/7F37$$\"3Gfb7 wY,(*=Fdp$\"31SFIp'>O9\"F37$$\"3cM5'zg%pg>Fdp$\"3!3w.&e`5&3\"F37$$\"3* *oJ8:.SJ?Fdp$\"3>y$yF>N,-\"F37$$\"3)**p!zPD$\\4#Fdp$\"3z\\E4#4mvh*F-7$ $\"3k\"yF-7$$\"3#*z_V$zlYN#Fdp$\"3DYaWw_#4 B(F-7$$\"3cmjYO*f2U#Fdp$\"3!HY#fG%)fBmF-7$$\"3)eq)=U!z`[#Fdp$\"3&*3^dD a#)HgF-7$$\"3#Q'HHd#HIb#Fdp$\"3n@QV>,?3aF-7$$\"3z^r&[X%==EFdp$\"3T:'*p n5]4[F-7$$\"3'=BS\\0:[o#Fdp$\"3_`8G:[C(>%F-7$$\"3[gN,?R*3v#Fdp$\"3e@Yv `$e+f$F-7$$\"3@/0LMNh6GFdp$\"3t#*zbbq6KIF-7$$\"3w#oUX107)GFdp$\"3)zW\" =kLl#R#F-7$$\"3W46xe&[M%HFdp$\"3(>E7bT42#=F-7$$\"3H.6)e58)4IFdp$\"3>*e BY(y*3@\"F-7$$\"3Kt2RXCLtIFdp$\"3uny$z>/BF'!#?7$$\"3!)***\\/l#fTJFdp$ \"3E;())o$y4&)G!#F-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x 6\"Q!F^\\l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 49 "_________________________________________ _______ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "_______________________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflec tion: " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x/Pi, 0 <= x and x < Pi/4], [1/2-x/Pi, Pi/4 <= x and x < 3*Pi/4],[x/Pi-1, 3*Pi/4 <= x and x <= Pi] );" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$*&F'\"\"\"%#PiG!\"\"31\"\"!F'2F' *&F.F-\"\"%F/7$,&*&F-F-\"\"#F/F-*&F'F-F.F/F/31*&F.F-F5F/F'2F'*(\"\"$F- F.F-F5F/7$,&*&F'F-F.F/F-F-F/31*(F@F-F.F-F5F/F'1F'F." }{TEXT -1 3 " . \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "f := x -> piecewise(x$\"3!)*>)[9arz@F-7$$\"3)\\$px*G*f!G\"!#=$\"3o$Rf<9ui2%F-7$ $\"3+5@exGm]>F3$\"3Aq5Y&y_\"4iF-7$$\"3[99!=3o^i#F3$\"3-$G:L`phN)F-7$$ \"35!\\D0[nkH$F3$\"3+OY`)=)H\\5F37$$\"37\"=Za&z%)=RF3$\"3-t*pm/3uC\"F3 7$$\"3edXa()oGjXF3$\"3[#H)zH$RDX\"F37$$\"3W%3**Hbm(H_F3$\"3-MHD'R'ok;F 37$$\"37PRr4)3T*eF3$\"3K@\\P/J:w=F37$$\"3y\"[)yykYxlF3$\"3/g3C1En$4#F3 7$$\"3[s'ocGo$zrF3$\"3Y6F)=!RE&G#F37$$\"3+1Y\"Hsn\"=vF3$\"3+(f\">7r5$R #F37$$\"3UQ0;gr'p&yF3$\"3+=&*\\x'\\!*\\#F37$$\"33$)pWSx:(>)F3$\"3#>u2E lj2R#F37$$\"3vFMt?$[t`)F3$\"3'e'frFwZ#G#F37$$\"3u\"p30k@I>*F3$\"3S`X#z KqP2#F37$$\"3#*R7\\HeV)y*F3$\"3o/t>0TC%)=F37$$\"3#G[))*)3W'\\5!#<$\"3] te+&4z)e;F37$$\"30'[@XS@'46F^q$\"3Gym>q`'zY\"F37$$\"3G9w$3G*Qz6F^q$\"3 Ki>]At)eC\"F37$$\"3>%3*3tc9T7F^q$\"3;$4:1i5$\\5F37$$\"3Ey0DBA!*38F^q$ \"3\"RyR\"H#[jL)F-7$$\"3qjE.#[AMP\"F^q$\"31\\B>1Yg#G'F-7$$\"3*R:_MgU2W \"F^q$\"3gd1](eQ(RTF-7$$\"3\"Qu#)**[jD]\"F^q$\"3!GNn>l=><#F-7$$\"3=hw \"ymX#p:F^q$\"3:Vn`D3!f$\\!#@7$$\"3mwM!*4(4&Q;F^q$!3ca6/FRQb@F-7$$\"3C DL:iU!))p\"F^q$!3_ZjXL\">Y2%F-7$$\"3o(R1/.CRw\"F^q$!3I#Q^btXu9'F-7$$\" 32Id)H7*>J=F^q$!3=2XuT%y))G)F-7$$\"3Gfb7wY,(*=F^q$!32t]Yc_QQ5F37$$\"3c M5'zg%pg>F^q$!3#)*[7]x%3T7F37$$\"3**oJ8:.SJ?F^q$!3?bv.J![hY\"F37$$\"3) **p!zPD$\\4#F^q$!3%[xomPx$o;F37$$\"3k\"o.I%)=F37$$ \"3wak3@YBCAF^q$!3!>wS3pe*z?F37$$\"39/b%F-7$$\"3Kt2RXCLtIF^q$!3'*)=r-\"* *ys@F-7$$\"3!)***\\/l#fTJF^q$!3=Wmfw?F%***!#F-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fh\\l-%*THICKNESSG6#\"\"#-%%VIEWG6$;F ($\"+aEfTJ!\"*%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 49 "_____________ ___________________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "____________________ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflection: " }{XPPEDIT 18 0 "f(x) = PIECEWISE( [x/Pi, 0 <= x and x < Pi/4],[1/2-x/Pi, Pi/4 <= x and x < Pi/2],[0, Pi/ 2 <= x and x <= Pi]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$*&F'\"\"\"%#P iG!\"\"31\"\"!F'2F'*&F.F-\"\"%F/7$,&*&F-F-\"\"#F/F-*&F'F-F.F/F/31*&F.F -F5F/F'2F'*&F.F-F9F/7$F231*&F.F-F9F/F'1F'F." }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := x -> piecewise(x$\"3!)*>)[ 9arz@F-7$$\"3)\\$px*G*f!G\"!#=$\"3o$Rf<9ui2%F-7$$\"3+5@exGm]>F3$\"3Aq5 Y&y_\"4iF-7$$\"3[99!=3o^i#F3$\"3-$G:L`phN)F-7$$\"35!\\D0[nkH$F3$\"3+OY `)=)H\\5F37$$\"37\"=Za&z%)=RF3$\"3-t*pm/3uC\"F37$$\"3edXa()oGjXF3$\"3[ #H)zH$RDX\"F37$$\"3W%3**Hbm(H_F3$\"3-MHD'R'ok;F37$$\"37PRr4)3T*eF3$\"3 K@\\P/J:w=F37$$\"3y\"[)yykYxlF3$\"3/g3C1En$4#F37$$\"3[s'ocGo$zrF3$\"3Y 6F)=!RE&G#F37$$\"3+1Y\"Hsn\"=vF3$\"3+(f\">7r5$R#F37$$\"3UQ0;gr'p&yF3$ \"3+=&*\\x'\\!*\\#F37$$\"33$)pWSx:(>)F3$\"3#>u2Elj2R#F37$$\"3vFMt?$[t` )F3$\"3'e'frFwZ#G#F37$$\"3u\"p30k@I>*F3$\"3S`X#zKqP2#F37$$\"3#*R7\\HeV )y*F3$\"3o/t>0TC%)=F37$$\"3#G[))*)3W'\\5!#<$\"3]te+&4z)e;F37$$\"30'[@X S@'46F^q$\"3Gym>q`'zY\"F37$$\"3G9w$3G*Qz6F^q$\"3Ki>]At)eC\"F37$$\"3>%3 *3tc9T7F^q$\"3;$4:1i5$\\5F37$$\"3Ey0DBA!*38F^q$\"3\"RyR\"H#[jL)F-7$$\" 3qjE.#[AMP\"F^q$\"31\\B>1Yg#G'F-7$$\"3*R:_MgU2W\"F^q$\"3gd1](eQ(RTF-7$ $\"3\"Qu#)**[jD]\"F^q$\"3!GNn>l=><#F-7$$\"3Q--!*yX!f`\"F^q$\"36g86I)Q1 6\"F-7$$\"3=hw\"ymX#p:F^q$\"3:Vn`D3!f$\\!#@7$$\"3\")o0'))oxQg\"F^qF(7$ $\"3mwM!*4(4&Q;F^qF(7$$\"3CDL:iU!))p\"F^qF(7$$\"3o(R1/.CRw\"F^qF(7$$\" 32Id)H7*>J=F^qF(7$$\"3Gfb7wY,(*=F^qF(7$$\"3cM5'zg%pg>F^qF(7$$\"3**oJ8: .SJ?F^qF(7$$\"3)**p!zPD$\\4#F^qF(7$$\"3k " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "_______________________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflec tion: " }{XPPEDIT 18 0 "f(x) = PIECEWISE([sin(4*x)^3, 0 <= x and x < \+ Pi/2],[0, Pi/2 <= x and x <= Pi])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$* $-%$sinG6#*&\"\"%\"\"\"F'F2\"\"$31\"\"!F'2F'*&%#PiGF2\"\"#!\"\"7$F631* &F9F2F:F;F'1F'F9" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> piecewise(xr\"!#>$\"3[2P l*GVN?$!#@7$$\"3Uzi`m*))QU$F-$\"3mX-V$Qd[a#!#?7$$\"3[>W!)\\M$e8&F-$\"3 I'p*3*)4f)[)F67$$\"3%)eD2LzxZoF-$\"3u`W:)4H#z>F-7$$\"32b4U:a)o#)*F-$\" 3y[sQGrb>cF-7$$\"3)\\$px*G*f!G\"!#=$\"3u#zb:$=Vx6FI7$$\"3NA&zO3Jch\"FI $\"3I(p>$>S$Q=#FI7$$\"3+5@exGm]>FI$\"3Ei)e*45B\"[$FI7$$\"3Ci3'R] 5%QFI$\"3G56g+kG#)**FI7$$\"37\"=Za&z%)=RFI$\"3kp!>h'3%)****FI7$$\"3/`$ f>#GS**RFI$\"3]&R.4w@u)**FI7$$\"3'\\_r%)od*zSFI$\"3)zS&=N_'R%**FI7$$\" 3W(p$)\\b70;%FI$\"3w>%pjn'yp)*FI7$$\"3Npe\\@u1TUFI$\"3%H)p*z#QUl(*FI7$ $\"3u8-_ar<-WFI$\"3e(*ff#*oPp%*FI7$$\"3edXa()oGjXFI$\"3W3vyS-Mk!*FI7$$ \"3+@=F?n_'*[FI$\"3#3'GZ!)[[LzFI7$$\"3W%3**Hbm(H_FI$\"3W)eozklJ_'FI7$$ \"3C5lN\"oP>c&FI$\"3s9uH`fT**\\FI7$$\"37PRr4)3T*eFI$\"36.P([uv,_$FI7$$ \"3Y47DWwyNiFI$\"37?j&zmYF>#FI7$$\"3y\"[)yykYxlFI$\"3k/t#ze%>n6FI7$$\" 3cw&GAQ<%yoFI$\"3]Ku4vGU/bF-7$$\"3[s'ocGo$zrFI$\"3KF-ek')\\%*=F-7$$\"3 CR;H/!o([tFI$\"3ofLor'Qe3)F67$$\"3+1Y\"Hsn\"=vFI$\"3MSz;PP!>S#F67$$\"3 wsv`Tuc(o(FI$\"37&[\"y'o*)H%HF07$$\"3UQ0;gr'p&yFI$!3-U\\*z]3Jq\"!#E7$$ \"3JhPI]C1F!)FI$!3z5dz11W5LF07$$\"33$)pWSx:(>)FI$!3)>&3_r,MiDF67$$\"3' [?!fIIDn$)FI$!3wW&fvK]LZ)F67$$\"3vFMt?$[t`)FI$!3Qxjj&)RKn>F-7$$\"3?f5i !)\\=l))FI$!31HMwnR*\\4'F-7$$\"3u\"p30k@I>*FI$!3&3pyl+1%H8FI7$$\"3#e'* **\\tG2\\*FI$!3u]&)eQ&zxD#FI7$$\"3#*R7\\HeV)y*FI$!32B`1m=08MFI7$$\"3_. )of$QC95!#<$!3SMbYDH#H)\\FI7$$\"3#G[))*)3W'\\5Fhy$!3@?TY%ei\\g'FI7$$\" 3W%)\\vYFjz5Fhy$!3aT!fLAQW(yFI7$$\"30'[@XS@'46Fhy$!3IY3huF%G#*)FI7$$\" 3B=0gtL1F6Fhy$!3CfRu+A%**Q*FI7$$\"3=]&zEM0X9\"Fhy$!3zsY84v,K(*FI7$$\"3 ;m!>sKEK:\"Fhy$!3c1r\\%f*R_)*FI7$$\"37#ee;\"Fhy$!3_%R@KJbv$**FI7$$ \"35)4)H'Ho1<\"Fhy$!3+w/r$Rhn)**FI7$$\"3G9w$3G*Qz6Fhy$!3_alrg$*f****FI 7$$\"3'z/p)H)3r=\"Fhy$!39cKX)oC0)**FI7$$\"3'=[+*y$G[>\"Fhy$!3b'yT)**=* H$**FI7$$\"3c:>$z#za-7Fhy$!3w*RQ%exJd)*FI7$$\"3C\\L'pZn-@\"Fhy$!3kn`' \\'f+a(*FI7$$\"3g;i-vlqD7Fhy$!3ad5B5BRn%*FI7$$\"3>%3*3tc9T7Fhy$!3Q01&* HHs!3*FI7$$\"37J)p\"[R-v7Fhy$!3e&Rd$Q/`MzFI7$$\"3Ey0DBA!*38Fhy$!3'y_>, F_%*\\'FI7$$\"3)4iTENi6M\"Fhy$!3(p3C*QQ<>]FI7$$\"3qjE.#[AMP\"Fhy$!3y6% *Qe)z!zNFI7$$\"3u3CuUD329Fhy$!3IApf;)e\"fAFI7$$\"3*R:_MgU2W\"Fhy$!3\"R 5&*4VG\"G7FI7$$\"3!*[urYIlr9Fhy$!3Y*\\SFahGw&F-7$$\"3\"Qu#)**[jD]\"Fhy $!3Yw;sPLfe>F-7$$\"3?t9WMSB>:Fhy$!3cQ]:)4y')e)F67$$\"3Q--!*yX!f`\"Fhy$ !3G07BX8D#p#F67$$\"3yJ*eL7vDb\"Fhy$!3EDJ=FhyF(7$$\"3Gfb7 wY,(*=FhyF(7$$\"3cM5'zg%pg>FhyF(7$$\"3**oJ8:.SJ?FhyF(7$$\"3)**p!zPD$\\ 4#FhyF(7$$\"3k " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "________________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }}{PARA 0 "" 0 "" {TEXT -1 22 " Initial deflection: \+ " }{XPPEDIT 18 0 "f(x) = PIECEWISE([3*x/(2*Pi), 0 <= x and x < Pi/6],[ 1/4, Pi/6 <= x and x < Pi/3],[3*x/(2*Pi)-1/4, Pi/3 <= x and x < Pi/2], [5/4-3*x/(2*Pi), Pi/2 <= x and x < 2*Pi/3],[1/4, 2*Pi/3 <= x and x < 5 *Pi/6],[3/2-3*x/(2*Pi), 5*Pi/6 <= x and x <= Pi]);" "6#/-%\"fG6#%\"xG- %*PIECEWISEG6(7$*(\"\"$\"\"\"F'F.*&\"\"#F.%#PiGF.!\"\"31\"\"!F'2F'*&F1 F.\"\"'F27$*&F.F.\"\"%F231*&F1F.F8F2F'2F'*&F1F.F-F27$,&*(F-F.F'F.*&F0F .F1F.F2F.*&F.F.F;F2F231*&F1F.F-F2F'2F'*&F1F.F0F27$,&*&\"\"&F.F;F2F.*(F -F.F'F.*&F0F.F1F.F2F231*&F1F.F0F2F'2F'*(F0F.F1F.F-F27$*&F.F.F;F231*(F0 F.F1F.F-F2F'2F'*(FNF.F1F.F8F27$,&*&F-F.F0F2F.*(F-F.F'F.*&F0F.F1F.F2F23 1*(FNF.F1F.F8F2F'1F'F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "f:=x->piecewise(x$\"3')*HK <7t&pKF-7$$\"3)\\$px*G*f!G\"!#=$\"3_!4RE@6W6'F-7$$\"3+5@exGm]>F3$\"3n0 ;>y\"HPJ*F-7$$\"3[99!=3o^i#F3$\"3Y#H(**HaU`7F37$$\"35!\\D0[nkH$F3$\"31 a>!GGZRd\"F37$$\"37\"=Za&z%)=RF3$\"3Qf\\+q?6r=F37$$\"3edXa()oGjXF3$\"3 sQup%**3)y@F37$$\"3+@=F?n_'*[F3$\"3t>%QXH>zL#F37$$\"3W%3**Hbm(H_F3$\"3 -,%zVfHq\\#F37$$\"3C5lN\"oP>c&F3$\"3++++++++DF37$$\"37PRr4)3T*eF3Ffn7$ $\"3y\"[)yykYxlF3Ffn7$$\"3[s'ocGo$zrF3Ffn7$$\"3UQ0;gr'p&yF3Ffn7$$\"3vF Mt?$[t`)F3Ffn7$$\"3u\"p30k@I>*F3Ffn7$$\"3#*R7\\HeV)y*F3Ffn7$$\"3_.)of$ QC95!#\"\\d8o6DF37$$\"3W%)\\vYFjz5F`p $\"3s'3)4^m'[l#F37$$\"30'[@XS@'46F`p$\"3I#)\\qW>0)z#F37$$\"3G9w$3G*Qz6 F`p$\"3CcqC;!p68$F37$$\"3>%3*3tc9T7F`p$\"3_gt2pS.EMF37$$\"3Ey0DBA!*38F `p$\"39K!HcwZ&\\PF37$$\"3qjE.#[AMP\"F`p$\"3OZ624$4w0%F37$$\"3*R:_MgU2W \"F`p$\"33,\\(=@R!zVF37$$\"3\"Qu#)**[jD]\"F`p$\"33(*[?-A@uYF37$$\"3Q-- !*yX!f`\"F`p$\"3r&H$[vTSL[F37$$\"3=hw\"ymX#p:F`p$\"3K%ph([hf#*\\F37$$ \"3\")o0'))oxQg\"F`p$\"3!o1\"4J\"[?%[F37$$\"3mwM!*4(4&Q;F`p$\"3#o#Q%4T #pwYF37$$\"3CDL:iU!))p\"F`p$\"3'ya\")*Hr!))Q%F37$$\"3o(R1/.CRw\"F`p$\" 3A$Hn'RJ)y2%F37$$\"32Id)H7*>J=F`p$\"3#RKQPBomv$F37$$\"3Gfb7wY,(*=F`p$ \"3R!R-`6ACW$F37$$\"3cM5'zg%pg>F`p$\"3sk7[PGPQJF37$$\"3**oJ8:.SJ?F`p$ \"3?nOW`zx+GF37$$\"3[M>YEk;j?F`p$\"3[_-AWf5\\EF37$$\"3)**p!zPD$\\4#F`p Ffn7$$\"3#)e[(>k\\)G@F`pFfn7$$\"3k " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 48 "_______________________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }