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438 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 439 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 440 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 441 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 442 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 443 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 444 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 445 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 262 41 "Study of Membrane Oscillation \+ Using Maple" }}{PARA 257 "" 0 "" {TEXT 260 44 "D. Frenkel , L. Golebio wski and R. Portugal " }}{PARA 442 "" 0 "" {TEXT 270 21 "CBPF and PUC- Rio/1996" }}{PARA 258 "" 0 "" {TEXT 263 17 "Circular Membrane" }} {PARA 443 "" 0 "" {TEXT 273 53 "The following four sections contain va rious examples." }}{SECT 0 {PARA 259 "" 0 "" {TEXT 256 28 "Resolution \+ of Wave Equation " }}{EXCHG {PARA 260 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 0 54 "protect('r,theta,t, m,n'); # protect reserved variables" }}{PARA 262 "> " 0 "" {MPLTEXT 1 0 22 "alias(u=u(r,theta,t)):" }}{PARA 263 "> " 0 "" {MPLTEXT 1 0 79 "w ave_eq := linalg[laplacian](u,[r,theta,z],coords=cylindrical)=Diff(u,t ,t)/c^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(wave_eqG/*&%\"rG!\"\",( -%%diffG6$%\"uGF'\"\"\"*&F'F.-F+6$F--%\"$G6$F'\"\"#F.F.*&F'F(-F+6$F--F 36$%&thetaGF5F.F.F.*&-%%DiffG6$F--F36$%\"tGF5F.%\"cG!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Method of separation of variables: " }{XPPEDIT 18 0 "u(r,theta,t)=R(r)*Theta(th eta)*T(t)" "6#/-%\"uG6%%\"rG%&thetaG%\"tG*(-%\"RG6#F'\"\"\"-%&ThetaG6# F(F.-%\"TG6#F)F." }{TEXT 261 1 "." }}}{EXCHG {PARA 265 "> " 0 "" {MPLTEXT 1 0 87 "Eq1 := expand(value(subs(u=R(r)*Theta(theta)*T(t), wa ve_eq)/(R(r)*Theta(theta)*T(t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$Eq1G/,(*(-%\"RG6#%\"rG!\"\"F+F,-%%diffG6$F(F+\"\"\"F0*&F(F,-F.6$F(- %\"$G6$F+\"\"#F0F0*(-%&ThetaG6#%&thetaGF,F+!\"#-F.6$F9-F56$F " 0 "" {MPLTEXT 1 0 44 "Eqt := expand(T(t)*c^2*( rhs(Eq1)=lambda ));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EqtG/-%%diffG6$-%\"TG6#%\"tG-%\"$G6 $F,\"\"#*(F)\"\"\")%\"cGF0F2%'lambdaGF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Equation for variable " } {TEXT 275 5 "theta" }{TEXT -1 1 ":" }}}{EXCHG {PARA 268 "> " 0 "" {MPLTEXT 1 0 65 "Eqtheta := diff(Theta(theta),theta,theta) = lambda1*T heta(theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(EqthetaG/-%%diffG6$ -%&ThetaG6#%&thetaG-%\"$G6$F,\"\"#*&%(lambda1G\"\"\"F)F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Equation \+ for variable " }{TEXT 276 1 "r" }{TEXT -1 1 ":" }}}{EXCHG {PARA 269 "> " 0 "" {MPLTEXT 1 0 46 "Eqr := simplify(subs(Eqt, Eqtheta, R(r)*Eq1)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EqrG/*&,(*&-%%diffG6$-%\"RG6#% \"rGF/\"\"\"F/F0F0*&-F*6$F,-%\"$G6$F/\"\"#F0)F/F7F0F0*&%(lambda1GF0F,F 0F0F0F/!\"#*&F,F0%'lambdaGF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "1. " }{XPPEDIT 18 0 "u(a,theta,t)=0" "6 #/-%\"uG6%%\"aG%&thetaG%\"tG\"\"!" }{TEXT -1 21 " => constant \+ " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 9 " negative" }} {PARA 0 "" 0 "" {TEXT -1 19 "2. periodicity in " }{XPPEDIT 18 0 "thet a" "6#%&thetaG" }{TEXT -1 19 " => constant " }{XPPEDIT 18 0 "lam bda1" "6#%(lambda1G" }{TEXT -1 17 " negative integer" }}}{EXCHG {PARA 271 "> " 0 "" {MPLTEXT 1 0 15 "lambda := -k^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG,$*$)%\"kG\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Dependence of " }{XPPEDIT 18 0 "k;" "6#%\"kG" } {TEXT -1 4 " on " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 26 " will be introduced below. " }}}{EXCHG {PARA 272 "> " 0 "" {MPLTEXT 1 0 16 "lambda1 := -m^2;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda1G,$*$)%\"mG\"\"#\"\"\"!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "General solution for " }{XPPEDIT 18 0 "R(r)" "6#-%\"RG6#%\"rG" } {TEXT -1 1 ":" }}}{EXCHG {PARA 274 "> " 0 "" {MPLTEXT 1 0 29 "alias(J= BesselJ, Y=BesselY): " }}{PARA 275 "> " 0 "" {MPLTEXT 1 0 57 "sol_R := dsolve(Eqr,R(r)):\nsol_R := subs(k=k(m,n),sol_R);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&sol_RG/-%\"RG6#%\"rG,&*&%$_C1G\"\"\"-%\"JG6$%\"mG* &-%\"kG6$F1%\"nGF-F)F-F-F-*&%$_C2GF--%\"YGF0F-F-" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "Y(m, k(m,n)*r)" "6#-%\"YG6$%\"mG*&-%\"kG6$F&%\"nG\"\"\" %\"rGF," }{TEXT -1 20 " is not analytic at " }{XPPEDIT 18 0 "r=0" "6#/ %\"rG\"\"!" }{TEXT -1 1 ":" }}}{EXCHG {PARA 277 "> " 0 "" {MPLTEXT 1 0 42 "assign(eval(subs(Y=0,_C1=1,_C2=1,sol_R)));" }}{PARA 278 "> " 0 " " {MPLTEXT 1 0 5 "R(r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"JG6$%\" mG*&-%\"kG6$F&%\"nG\"\"\"%\"rGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "General solution for " }{XPPEDIT 18 0 "Theta(theta)" "6#-%&ThetaG6#%&thetaG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 279 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(Eqtheta,Theta(theta));" }} {PARA 280 "> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&ThetaG6#%&thetaG,&*&%$_C1G\"\"\"-%$sinG6#*&%\"mGF+F 'F+F+F+*&%$_C2GF+-%$cosGF.F+F+" }}}{EXCHG {PARA 281 "> " 0 "" {MPLTEXT 1 0 98 "subs_set := \{eval(subs(cos=1,sin=0,Theta(theta))) = \+ C1,eval(subs(cos=0,sin=1,Theta(theta))) = C2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)subs_setG<$/%$_C2G%#C1G/%$_C1G%#C2G" }}}{EXCHG {PARA 282 "> " 0 "" {MPLTEXT 1 0 45 "Theta(theta) := subs(subs_set, Th eta(theta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&ThetaG6#%&thetaG,& *&%#C2G\"\"\"-%$sinG6#*&%\"mGF+F'F+F+F+*&%#C1GF+-%$cosGF.F+F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Ge neral solution for T(t):" }}}{EXCHG {PARA 283 "> " 0 "" {MPLTEXT 1 0 57 "sol_T := dsolve(Eqt,T(t)):\nsol_T := subs(k=k(m,n),sol_T);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol_TG/-%\"TG6#%\"tG,&*&%$_C1G\"\" \"-%$sinG6#*(-%\"kG6$%\"mG%\"nGF-%\"cGF-F)F-F-F-*&%$_C2GF--%$cosGF0F-F -" }}}{EXCHG {PARA 284 "> " 0 "" {MPLTEXT 1 0 41 "C1 := eval(subs(sin= 0,cos=1,rhs(sol_T)));" }}{PARA 285 "> " 0 "" {MPLTEXT 1 0 41 "C2 := ev al(subs(cos=0,sin=1,rhs(sol_T)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#C1G%$_C2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G%$_C1G" }}} {EXCHG {PARA 286 "> " 0 "" {MPLTEXT 1 0 47 "expand(subs(C1=-I*C2,C2=I, convert(sol_T,exp)));" }}{PARA 287 "> " 0 "" {MPLTEXT 1 0 10 "assign(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"TG6#%\"tG-%$expG6#**^#\"\" \"F--%\"kG6$%\"mG%\"nGF-%\"cGF-F'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Unassign variables that are not going to be used anymore:" }}}{EXCHG {PARA 288 "> " 0 "" {MPLTEXT 1 0 73 "unassign('C1,C2,sol_T,sol_R,wave_eq,Eq1,Eqr,Eqtheta,Eqt,lambda,l ambda1');" }}}}{SECT 0 {PARA 289 "" 0 "" {TEXT 257 12 "Normal Modes" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Sequence of normal modes:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#**^#\"\"\"F(-%\"kG6$%\"mG%\"nGF(%\"cGF(%\"tGF( " }}}{EXCHG {PARA 290 "> " 0 "" {MPLTEXT 1 0 27 "omega := (m,n) -> k(m ,n)*c;" }}{PARA 291 "> " 0 "" {MPLTEXT 1 0 17 "protect('omega');" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaGf*6$%\"mG%\"nG6\"6$%)operator G%&arrowGF)*&-%\"kG6$9$9%\"\"\"%\"cGF3F)F)F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 277 20 "Boundary conditions:" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "R(a)=0" "6#/-%\"RG6#%\"aG\"\"!" } {TEXT -1 8 " => " }{XPPEDIT 18 0 "J(m,k(m,n)*a)=0" "6#/-%\"JG6$%\" mG*&-%\"kG6$F'%\"nG\"\"\"%\"aGF-\"\"!" }{TEXT -1 9 " => " } {XPPEDIT 18 0 "a*k(m,n)" "6#*&%\"aG\"\"\"-%\"kG6$%\"mG%\"nGF%" }{TEXT -1 34 " root of Bessel Function of order " }{XPPEDIT 18 0 "m" "6#%\"mG " }{TEXT -1 1 ":" }}}{EXCHG {PARA 292 "> " 0 "" {MPLTEXT 1 0 34 "k := \+ (m,n) -> BesselJZeros(m,n)/a;" }}{PARA 293 "> " 0 "" {MPLTEXT 1 0 13 " protect('k');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kGf*6$%\"mG%\"nG6 \"6$%)operatorG%&arrowGF)*&-%-BesselJZerosG6$9$9%\"\"\"%\"aG!\"\"F)F)F )" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Example of a root:" }}}{EXCHG {PARA 294 "> " 0 "" {MPLTEXT 1 0 22 "alias(j=BesselJZeros):" }}{PARA 295 "> " 0 "" {MPLTEXT 1 0 14 "evalf( j(2,3));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;\"!\")" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 14 "N ormal Modes: " }{TEXT -1 3 "If " }{XPPEDIT 18 0 "m<>0" "6#0%\"mG\"\"! " }{TEXT -1 41 ", there is a degeneresce or fourth order:" }}}{EXCHG {PARA 296 "> " 0 "" {MPLTEXT 1 0 78 "normal_mode := unapply(R(r)*evalc (Re(T(t)))*subs(C1=1,C2=0,Theta(theta)),m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,normal_modeGf*6$%\"mG%\"nG6\"6$%)operatorG%&arrowGF) *(-%\"JG6$9$*&-%\"kG6$F19%\"\"\"%\"rGF7F7-%$cosG6#*(F3F7%\"cGF7%\"tGF7 F7-F:6#*&F1F7%&thetaGF7F7F)F)F)" }}}{EXCHG {PARA 297 "> " 0 "" {MPLTEXT 1 0 79 "normal_mode1 := unapply(subs(C1=1,C2=0,R(r)*evalc(Im( T(t)))*Theta(theta)),m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-norma l_mode1Gf*6$%\"mG%\"nG6\"6$%)operatorG%&arrowGF)*(-%\"JG6$9$*&-%\"kG6$ F19%\"\"\"%\"rGF7F7-%$sinG6#*(F3F7%\"cGF7%\"tGF7F7-%$cosG6#*&F1F7%&the taGF7F7F)F)F)" }}}{EXCHG {PARA 298 "> " 0 "" {MPLTEXT 1 0 79 "normal_m ode2 := unapply(subs(C1=0,C2=1,R(r)*evalc(Re(T(t)))*Theta(theta)),m,n) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-normal_mode2Gf*6$%\"mG%\"nG6\" 6$%)operatorG%&arrowGF)*(-%\"JG6$9$*&-%\"kG6$F19%\"\"\"%\"rGF7F7-%$cos G6#*(F3F7%\"cGF7%\"tGF7F7-%$sinG6#*&F1F7%&thetaGF7F7F)F)F)" }}}{EXCHG {PARA 299 "> " 0 "" {MPLTEXT 1 0 79 "normal_mode3 := unapply(subs(C1=0 ,C2=1,R(r)*evalc(Im(T(t)))*Theta(theta)),m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-normal_mode3Gf*6$%\"mG%\"nG6\"6$%)operatorG%&arrowGF )*(-%\"JG6$9$*&-%\"kG6$F19%\"\"\"%\"rGF7F7-%$sinG6#*(F3F7%\"cGF7%\"tGF 7F7-F:6#*&F1F7%&thetaGF7F7F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 68 " => Axially symmetric nor mal modes - degeneresce or second order" }}}{PARA 300 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 301 "" 0 "" {TEXT 258 25 "Animation of Normal Modes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Procedure for animation: " }}}{EXCHG {PARA 302 "> " 0 "" {MPLTEXT 1 0 44 "animate_mode := proc( m::nonnegint,n::posint)" }}{PARA 303 "> " 0 "" {MPLTEXT 1 0 19 "local \+ FRAM,Options;" }}{PARA 304 "> " 0 "" {MPLTEXT 1 0 21 "global a,c,r,the ta,t;" }}{PARA 305 "> " 0 "" {MPLTEXT 1 0 35 "if not type([a,c],list(' numeric')) " }}{PARA 306 "> " 0 "" {MPLTEXT 1 0 59 " then ERROR(`Th e constants a and c must be numerical.`) " }}{PARA 307 "> " 0 "" {MPLTEXT 1 0 25 "elif not assigned(omega) " }}{PARA 308 "> " 0 "" {MPLTEXT 1 0 54 " then ERROR(`The function omega must be defined.`) " }}{PARA 309 "> " 0 "" {MPLTEXT 1 0 30 "elif not assigned(normal_mod e)" }}{PARA 310 "> " 0 "" {MPLTEXT 1 0 59 " then ERROR(`The functio n normal_mode must be defined.`)" }}{PARA 311 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 312 "> " 0 "" {MPLTEXT 1 0 10 "if nargs=1" }} {PARA 313 "> " 0 "" {MPLTEXT 1 0 52 " then Options:='frames'=8,'sty le=patch'; FRAM:=8;" }}{PARA 314 "> " 0 "" {MPLTEXT 1 0 33 " else O ptions:=args[3..nargs];" }}{PARA 315 "> " 0 "" {MPLTEXT 1 0 35 " \+ if has(\{Options\},'frames')" }}{PARA 316 "> " 0 "" {MPLTEXT 1 0 47 " then FRAM:=subs(\{Options\},'frames');" }}{PARA 317 "> \+ " 0 "" {MPLTEXT 1 0 52 " else Options:=Options,'frames'=8; F RAM:=8" }}{PARA 318 "> " 0 "" {MPLTEXT 1 0 16 " end if;" }} {PARA 319 "> " 0 "" {MPLTEXT 1 0 38 " if not has(\{Options\},' style')" }}{PARA 320 "> " 0 "" {MPLTEXT 1 0 46 " then Option s:=Options,'style=patch'" }}{PARA 321 "> " 0 "" {MPLTEXT 1 0 15 " \+ end if" }}{PARA 322 "> " 0 "" {MPLTEXT 1 0 8 "end if; " }}{PARA 323 "> " 0 "" {MPLTEXT 1 0 137 "plots[animate3d]([r,theta,normal_mode( m,n)],r=0..a,theta=0..2*Pi,t=0..evalf(2*Pi*(FRAM-1)/(omega(m,n)*FRAM)) ,coords=cylindrical,Options) " }}{PARA 324 "> " 0 "" {MPLTEXT 1 0 9 "e nd proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-animate_modeGf*6$'%\"mG %*nonnegintG'%\"nG%'posintG6$%%FRAMG%(OptionsG6\"F0C%@(4-%%typeG6$7$% \"aG%\"cG-%%listG6#.%(numericG-%&ERRORG6#%IThe~constants~a~and~c~must~ be~numerical.G4-%)assignedG6#%&omegaG-F@6#%DThe~function~omega~must~be ~defined.G4-FE6#%,normal_modeG-F@6#%JThe~function~normal_mode~must~be~ defined.G@%/9#\"\"\"C$>8%6$/.%'framesG\"\")./%&styleG%&patchG>8$FgnC%> FX&9\"6#;\"\"$FT@%-%$hasG6$<#FXFen>F]o-%%subsGFhoC$>FX6$FXFZ>F]oFgn@$4 -Fgo6$Fio.Fjn>FX6$FXFhn-&%&plotsG6#%*animate3dG6(7%%\"rG%&thetaG-FN6$9 $9%/F_q;\"\"!F8/F`q;Fgq,$%#PiG\"\"#/%\"tG;Fgq-%&evalfG6#,$**F[rFU,&F]o FUFU!\"\"FU-FGFbqFfrF]oFfrF\\r/%'coordsG%,cylindricalGFXF06'F8F9F_qF`q F^rF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Numerical values to constants:" }}}{EXCHG {PARA 325 "> " 0 "" {MPLTEXT 1 0 24 "a:=1; # membrane radius " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"\"" }}}{EXCHG {PARA 325 "> " 0 "" {MPLTEXT 1 0 21 "c:=1; # wave velocity" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG \"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 4 "Mode" }{TEXT -1 7 " <0,1>" }}}{EXCHG {PARA 326 "> " 0 " " {MPLTEXT 1 0 18 "animate_mode(0,1);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6$-%(ANIMATEG6*7#-%%MESHG6#7;7;7%$\"\"!F.F-$ \"\"\"F.F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,F,7;7%$\"3Wmmmmm mmT!#>F-$\"+8^\"\\(**!#57%$\"3]<6(G%4pCSF5$\"3#)or#zo7%y5F5F67%$\"3@i^ 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axes=none, scalin g=constrained, title=sprintf( \"Nodal lines of mode m = %d and n = %d \", m, n ) )\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,nodal_li nesGf*6$'%\"mG%*nonnegintG'%\"nG%'posintG6%%#ciG%#slG%\"pG6\"F1C&>&8$6 #9%-&%&plotsG6#%*polarplotG6$%\"aG/%*thicknessG\"\"#?(8&\"\"\"FD,&F7FD FD!\"\"%%trueG>&F56#FC-F96#*(F>FD-%\"jG6$\"\"!FCFD-FO6$FQF7FF?(FCFDFD9 $FG>&8%FJ-F96#7%%\"rG,$*(,&FCFAFDFFFD%#PiGFDFUFF#FDFA/Ffn;,$F>FFF>-&F: 6#%(displayG6&<$-%$seqG6$FW/FC;FDFU-Ffo6$FI/FC;FDF7/%%axesG%%noneG/%(s calingG%,constrainedG/%&titleG-%(sprintfG6%QFNodal~lines~of~mode~m~=~% d~and~n~=~%dF1FUF7F16$F>FfnF1" }}}{EXCHG {PARA 445 "" 0 "" {TEXT -1 0 "" }}{PARA 336 "" 0 "" {TEXT 271 26 "Nodal lines of mode <2,3>:" }}} {EXCHG {PARA 337 "> " 0 "" {MPLTEXT 1 0 17 "nodal_lines(2,3);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'CURVESG6$7S7$ $\"3iua'=\"y1rq!#=$!3tva'=\"y1rqF*7$$\"3J-\"H9]4Gw'F*$!3U.\"H9]4Gw'F*7 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+S0*)3fFh[l$\"+]J0.CFh[l7$$\"+\"4GJ`&Fh[l$\"+B%zR<$Fh[l7$$\"+VFno]Fh[l $\"+a?vsQFh[l7$$\"+\\\">d`%Fh[l$\"+Or=&[%Fh[l7$$\"+$)*)HeQFh[l$\"+SJoz ]Fh[l7$$\"+LB[$=$Fh[l$\"+hgmFbFh[l7$$\"+([)f1CFh[l$\"+nuW2fFh[l7$$\"+0 r-k;Fh[l$\"+vd(z:'Fh[l7$$\"+YJqQ#)F`\\l$\"+dhTDjFh[l7$$\"+0wD]>F][o$\" +rU\")yjFh[l7$$!+5e=9#)F`\\l$\"+1]tDjFh[l7$$!+H5#)H;Fh[l$\"+ds6nhFh[l7 $$!+zfzYCFh[l$\"+`<\"4*eFh[l7$$!+Au_\">$Fh[l$\"+Z\\-BbFh[l7$$!+f$fq*QF h[l$\"+Tz+]]Fh[l7$$!+CI`GXFh[l$\"+)fUC\\%Fh[l7$$!+:1SR]Fh[l$\"+Ijw5RFh [l7$$!+^YALbFh[l$\"+&36Q<$Fh[l7$$!+sU[%)eFh[l$\"+#y8AY#Fh[l7$$!+'\\v&e hFh[l$\"+eh!=m\"Fh[l7$$!+qC\\>jFh[l$\"+>$)R\"o)F`\\l7$Facn$!+u\\k;EF^r Fiz-%(SCALINGG6#%,CONSTRAINEDG-%&TITLEG6#QDNodal~lines~of~mode~m~=~2~a nd~n~=~36\"-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAULTGFdco" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" }}}}}{SECT 0 {PARA 338 "" 0 "" {TEXT 259 33 " Animation with Initial Conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 18 "General solution: " }{TEXT -1 33 "superp osition of all normal modes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 " \+ Unassign the reserved variables of this problem\n a - radius of the m embrane\n c - the constant of wave equation\n u0 - the initial displ acement of the membrane\n v0 - the initial velocity of the membrane\n AB_simp - simplifying function" }}}{EXCHG {PARA 439 "> " 0 "" {MPLTEXT 1 0 30 "unassign('u0,v0,a,c,AB_simp');" }}}{EXCHG {PARA 429 " > " 0 "" {MPLTEXT 1 0 11 "alias(u=u):" }}{PARA 430 "> " 0 "" {MPLTEXT 1 0 14 "u := unapply( " }}{PARA 431 "> " 0 "" {MPLTEXT 1 0 70 "2*Sum(f actor(A1(0,n)*normal_mode(0,n) + A2(0,n)*normal_mode1(0,n)) + " }} {PARA 432 "> " 0 "" {MPLTEXT 1 0 68 " Sum(factor(A1(m,n)*normal_mode( m,n) + A2(m,n)*normal_mode1(m,n) +" }}{PARA 433 "> " 0 "" {MPLTEXT 1 0 68 " B1(m,n)*normal_mode2(m,n) + B2(m,n)*normal_mode3(m, n))," }}{PARA 434 "> " 0 "" {MPLTEXT 1 0 36 "m=1..p1),n=1..p2), r,thet a,t,p1,p2);" }}{PARA 435 "> " 0 "" {MPLTEXT 1 0 13 "protect('u');" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"uGf*6'%\"rG%&thetaG%\"tG%#p1G%#p2G 6\"6$%)operatorG%&arrowGF,,$-%$SumG6$,&*&-%\"JG6$\"\"!*(-%\"jG6$F9%\"n G\"\"\"%\"aG!\"\"9$F?F?,&*&-%#A1GF=F?-%$cosG6#**F;F?F@FA%\"cGF?9&F?F?F ?*&-%#A2GF=F?-%$sinGFIF?F?F?F?-F26$*&-F76$%\"mG*(-F<6$FWF>F?F@FAFBF?F? ,**(-FH6#*&FWF?9%F?F?-FH6#**FYF?F@FAFKF?FLF?F?-FFFZF?F?*(-FOFZF?-FQF\\ oF?FgnF?F?*(-%#B1GFZF?F[oF?-FQFhnF?F?*(-%#B2GFZF?FaoF?FeoF?F?F?/FW;F?9 'F?/F>;F?9(\"\"#F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 13 "Coefficients " }{XPPEDIT 18 0 "A1(m,n)" " 6#-%#A1G6$%\"mG%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "A2(m,n)" "6#-%# A2G6$%\"mG%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B1(m,n)" "6#-%#B1G6$ %\"mG%\"nG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "B2(m,n)" "6#-%#B2G6$% \"mG%\"nG" }{TEXT -1 14 " are constants" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Case m=0 (Axially symmetric): " }}}{EXCHG {PARA 436 "> " 0 "" {MPLTEXT 1 0 25 "value(u(r,theta,t,0,p2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$sumG6$*&-%\"JG6$\"\"!*(-%\" jG6$F+%\"nG\"\"\"%\"aG!\"\"%\"rGF1F1,&*&-%#A1GF/F1-%$cosG6#**F-F1F2F3% \"cGF1%\"tGF1F1F1*&-%#A2GF/F1-%$sinGF;F1F1F1/F0;F1%#p2G\"\"#" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Generic b oundary conditions:" }}{PARA 425 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "u[0](r,theta) = u(r,theta,0)*` `" "6#/-&%\"uG 6#\"\"!6$%\"rG%&thetaG*&-F&6%F*F+F(\"\"\"%8~~~~~~~~~~~~~~~~~~~~~~~GF/ " }}{PARA 426 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "v[0](r,theta) = \+ Diff(u(r,theta,t),t)*` at t =`*0" "6#/-&%\"vG6#\"\"!6$%\"rG%&theta G*(-%%DiffG6$-%\"uG6%F*F+%\"tGF3\"\"\"%,~~~at~~~t~=GF4F(F4" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[0](r,theta)" "6#-&%\"uG6#\"\" !6$%\"rG%&thetaG" }{TEXT -1 24 " is the initial position" }}{PARA 0 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "v[0](r,theta)" "6#-&%\"vG6#\"\"! 6$%\"rG%&thetaG" }{TEXT -1 25 " is the initial velocity " }}{PARA 427 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A1(m,n)=int(int(u0(r,theta)*cos (m*theta)*r*J(m,k(m,n)*r),theta=0..2*Pi),r=0..a)/(s*a^2*(J(m+1,k(m,n)* a))^2)" "6#/-%#A1G6$%\"mG%\"nG*&-%$intG6$-F+6$**-%#u0G6$%\"rG%&thetaG \"\"\"-%$cosG6#*&F'F5F4F5F5F3F5-%\"JG6$F'*&-%\"kG6$F'F(F5F3F5F5/F4;\" \"!*&\"\"#F5%#PiGF5/F3;FC%\"aGF5*(%\"sGF5*$FIFEF5-F;6$,&F'F5F5F5*&-F?6 $F'F(F5FIF5FE!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 424 "" 0 " " {XPPEDIT 18 0 "A2(m,n)=int(int(v0(r,theta)*cos(m*theta)*r*J(m,k(m,n) *r),theta=0..2*Pi),r=0..a)/(s*omega(m,n)*a^2*(J(m+1,k(m,n)*a))^2)" "6# /-%#A2G6$%\"mG%\"nG*&-%$intG6$-F+6$**-%#v0G6$%\"rG%&thetaG\"\"\"-%$cos G6#*&F'F5F4F5F5F3F5-%\"JG6$F'*&-%\"kG6$F'F(F5F3F5F5/F4;\"\"!*&\"\"#F5% #PiGF5/F3;FC%\"aGF5**%\"sGF5-%&omegaG6$F'F(F5FIFE-F;6$,&F'F5F5F5*&-F?6 $F'F(F5FIF5FE!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "s=2" "6#/%\"sG\"\"#" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "m=0" "6#/%\"mG\"\"!" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "s=1" "6#/%\"sG\"\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "m<>0" "6#0%\"mG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B1(m,n)" "6#-%#B1G6$%\"mG%\"nG" }{TEXT -1 3 " e \+ " }{XPPEDIT 18 0 "B2(m,n)" "6#-%#B2G6$%\"mG%\"nG" }{TEXT -1 5 " are:" }}{PARA 423 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B1(m,n)=int(int(u0( r,theta)*sin(m*theta)*r*J(m,k(m,n)*r),theta=0..2*Pi),r=0..a)/(a^2*J(m+ 1,k(m,n)*a)^2)" "6#/-%#B1G6$%\"mG%\"nG*&-%$intG6$-F+6$**-%#u0G6$%\"rG% &thetaG\"\"\"-%$sinG6#*&F'F5F4F5F5F3F5-%\"JG6$F'*&-%\"kG6$F'F(F5F3F5F5 /F4;\"\"!*&\"\"#F5%#PiGF5/F3;FC%\"aGF5*&FIFE-F;6$,&F'F5F5F5*&-F?6$F'F( F5FIF5FE!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 428 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "B2(m,n)=int(int(v0(r,theta)*sin(m*theta )*r*J(m,k(m,n)*r),theta=0..2*Pi),r=0..a)/(omega(m,n)*a^2*(J(m+1,k(m,n) *a))^2)" "6#/-%#B2G6$%\"mG%\"nG*&-%$intG6$-F+6$**-%#v0G6$%\"rG%&thetaG \"\"\"-%$sinG6#*&F'F5F4F5F5F3F5-%\"JG6$F'*&-%\"kG6$F'F(F5F3F5F5/F4;\" \"!*&\"\"#F5%#PiGF5/F3;FC%\"aGF5*(-%&omegaG6$F'F(F5*$FIFEF5-F;6$,&F'F5 F5F5*&-F?6$F'F(F5FIF5FE!\"\"" }}{EXCHG {PARA 339 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Procedures to find " }{XPPEDIT 18 0 " A1(m,n)" "6#-%#A1G6$%\"mG%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "A2(m, n)" "6#-%#A2G6$%\"mG%\"nG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "B1(m,n)" " 6#-%#B1G6$%\"mG%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B2(m,n)" "6# -%#B2G6$%\"mG%\"nG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 340 "> " 0 "" {MPLTEXT 1 0 83 "A1:=proc(m,n) global u0,v0,a,c; `A1/A1uv`(m,n,eval(u0 ),a,c,eval(AB_simp)) end proc;" }}{PARA 341 "> " 0 "" {MPLTEXT 1 0 37 "`A1/A1uv` := proc(m,n,u0,a,c,AB_simp)" }}{PARA 342 "> " 0 "" {MPLTEXT 1 0 16 "local s,r,theta;" }}{PARA 343 "> " 0 "" {MPLTEXT 1 0 17 "options remember;" }}{PARA 344 "> " 0 "" {MPLTEXT 1 0 34 "if m=0 t hen s:=2 else s:=1 end if;" }}{PARA 345 "> " 0 "" {MPLTEXT 1 0 46 "AB_ simp(int(r*J(m,j(m,n)/a*r)*int(u0(r,theta)*" }}{PARA 346 "> " 0 "" {MPLTEXT 1 0 64 "cos(m*theta),theta=0..2*Pi),r=0..a)/(s*Pi*a^2*J(m+1,j (m,n))^2));" }}{PARA 347 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 348 "> " 0 "" {MPLTEXT 1 0 83 "A2:=proc(m,n) global u0,v0,a,c; `A2/A2u v`(m,n,eval(v0),a,c,eval(AB_simp)) end proc;" }}{PARA 349 "> " 0 "" {MPLTEXT 1 0 37 "`A2/A2uv` := proc(m,n,v0,a,c,AB_simp)" }}{PARA 350 "> " 0 "" {MPLTEXT 1 0 16 "local s,r,theta;" }}{PARA 351 "> " 0 "" {MPLTEXT 1 0 17 "options remember;" }}{PARA 352 "> " 0 "" {MPLTEXT 1 0 34 "if m=0 then s:=2 else s:=1 end if;" }}{PARA 353 "> " 0 "" {MPLTEXT 1 0 80 "AB_simp(int(r*J(m,k(m,n)*r)*int(v0(r,theta)*cos(m*the ta),theta=0..2*Pi),r=0..a)/" }}{PARA 354 "> " 0 "" {MPLTEXT 1 0 43 "(s *omega(m,n)*Pi*a^2*(J(m+1,k(m,n)*a))^2));" }}{PARA 355 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 356 "> " 0 "" {MPLTEXT 1 0 83 "B1:= proc(m,n) global u0,v0,a,c; `B1/B1uv`(m,n,eval(u0),a,c,eval(AB_simp)) \+ end proc;" }}{PARA 357 "> " 0 "" {MPLTEXT 1 0 52 "`B1/B1uv` := proc(m, n,u0,a,c,AB_simp)\nlocal r,theta;" }}{PARA 358 "> " 0 "" {MPLTEXT 1 0 17 "options remember;" }}{PARA 359 "> " 0 "" {MPLTEXT 1 0 46 "AB_simp( int(r*J(m,j(m,n)/a*r)*int(u0(r,theta)*" }}{PARA 360 "> " 0 "" {MPLTEXT 1 0 62 "sin(m*theta),theta=0..2*Pi),r=0..a)/(Pi*a^2*J(m+1,j(m ,n))^2));" }}{PARA 361 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 362 "> " 0 "" {MPLTEXT 1 0 83 "B2:=proc(m,n) global u0,v0,a,c; `B2/B2u v`(m,n,eval(v0),a,c,eval(AB_simp)) end proc;" }}{PARA 363 "> " 0 "" {MPLTEXT 1 0 37 "`B2/B2uv` := proc(m,n,v0,a,c,AB_simp)" }}{PARA 364 "> " 0 "" {MPLTEXT 1 0 14 "local r,theta;" }}{PARA 365 "> " 0 "" {MPLTEXT 1 0 17 "options remember;" }}{PARA 366 "> " 0 "" {MPLTEXT 1 0 80 "AB_simp(int(r*J(m,k(m,n)*r)*int(v0(r,theta)*sin(m*theta),theta=0 ..2*Pi),r=0..a)/" }}{PARA 367 "> " 0 "" {MPLTEXT 1 0 41 "(omega(m,n)*P i*a^2*(J(m+1,k(m,n)*a))^2));" }}{PARA 368 "> " 0 "" {MPLTEXT 1 0 9 "en d proc;" }}{PARA 369 "> " 0 "" {MPLTEXT 1 0 23 "protect('A1,A2,B1,B2') ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1Gf*6$%\"mG%\"nG6\"F)F)-%(A1/ A1uvG6(9$9%-%%evalG6#%#u0G%\"aG%\"cG-F06#%(AB_simpGF)6&F2%#v0GF3F4F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(A1/A1uvGf*6(%\"mG%\"nG%#u0G%\"aG% \"cG%(AB_simpG6%%\"sG%\"rG%&thetaG6#%)rememberG6\"C$@%/9$\"\"!>8$\"\"# >F:\"\"\"-9)6#*,-%$intG6$*(8%F=-%\"JG6$F7*(-%\"jG6$F79%F=9'!\"\"FFF=F= -FC6$*&-9&6$FF8&F=-%$cosG6#*&F7F=FWF=F=/FW;F8,$%#PiGF;F=/FF;F8FOF=F:FP FinFPFO!\"#-FH6$,&F7F=F=F=FKF\\oF3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2Gf*6$%\"mG%\"nG6\"F)F)-%(A2/A2uvG6(9$9%-%%evalG6#%#v0G%\"aG %\"cG-F06#%(AB_simpGF)6&%#u0GF2F3F4F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(A2/A2uvGf*6(%\"mG%\"nG%#v0G%\"aG%\"cG%(AB_simpG6%%\"sG%\"rG%&the taG6#%)rememberG6\"C$@%/9$\"\"!>8$\"\"#>F:\"\"\"-9)6#*.-%$intG6$*(8%F= -%\"JG6$F7*&-%\"kG6$F79%F=FFF=F=-FC6$*&-9&6$FF8&F=-%$cosG6#*&F7F=FUF=F =/FU;F8,$%#PiGF;F=/FF;F89'F=F:!\"\"-%&omegaGFMF[oFgnF[oFjn!\"#-FH6$,&F 7F=F=F=*&FKF=FjnF=F^oF3F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#B1Gf *6$%\"mG%\"nG6\"F)F)-%(B1/B1uvG6(9$9%-%%evalG6#%#u0G%\"aG%\"cG-F06#%(A B_simpGF)6&F2%#v0GF3F4F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(B1/B1uv Gf*6(%\"mG%\"nG%#u0G%\"aG%\"cG%(AB_simpG6$%\"rG%&thetaG6#%)rememberG6 \"-9)6#**-%$intG6$*(8$\"\"\"-%\"JG6$9$*(-%\"jG6$F@9%F<9'!\"\"F;F6$,&F@F%#B2Gf*6$%\"mG%\"nG6\"F)F)-%(B2/B2uvG6(9$9%-%%evalG6#%#v0G%\"aG%\"cG -F06#%(AB_simpGF)6&%#u0GF2F3F4F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% (B2/B2uvGf*6(%\"mG%\"nG%#v0G%\"aG%\"cG%(AB_simpG6$%\"rG%&thetaG6#%)rem emberG6\"-9)6#*,-%$intG6$*(8$\"\"\"-%\"JG6$9$*&-%\"kG6$F@9%F6$,&F@F " 0 "" {MPLTEXT 1 0 28 "AB_simp := x -> simplify(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(AB_simpG%)simplifyG" }}} {EXCHG {PARA 371 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 " Procedure to animate with initial conditions:" }}}{EXCHG {PARA 372 "> \+ " 0 "" {MPLTEXT 1 0 59 "animate_ic := proc(nt::numeric, p1::nonnegint, p2::posint) " }}{PARA 373 "> " 0 "" {MPLTEXT 1 0 20 "local FRAM,Optio ns; " }}{PARA 374 "> " 0 "" {MPLTEXT 1 0 22 "global a,c,r,theta,t; " } }{PARA 375 "> " 0 "" {MPLTEXT 1 0 11 "if nargs<3 " }}{PARA 376 "> " 0 "" {MPLTEXT 1 0 179 "then ERROR(\"This procedure has at least three ar guments. The first one is the number of dominant oscillations. The sec ond and the third are truncation values of the double sum.\") " }} {PARA 377 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 378 "> " 0 "" {MPLTEXT 1 0 32 "if not type([a,c],list(numeric))" }}{PARA 379 "> " 0 "" {MPLTEXT 1 0 58 " then ERROR(\"The constants a and c must be nu meric.\") " }}{PARA 380 "> " 0 "" {MPLTEXT 1 0 24 "elif not assigned(o mega)" }}{PARA 381 "> " 0 "" {MPLTEXT 1 0 55 " then ERROR(\"The fu nction omega must be defined.\") " }}{PARA 382 "> " 0 "" {MPLTEXT 1 0 20 "elif not assigned(u)" }}{PARA 383 "> " 0 "" {MPLTEXT 1 0 59 " t hen ERROR(\"The function normal_mode must be defined.\")" }}{PARA 384 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 385 "> " 0 "" {MPLTEXT 1 0 11 "if nargs=3 " }}{PARA 386 "> " 0 "" {MPLTEXT 1 0 19 "then FRAM := 8 *nt; " }}{PARA 387 "> " 0 "" {MPLTEXT 1 0 48 " Options := 'frames' = FRAM,'style = patch';" }}{PARA 388 "> " 0 "" {MPLTEXT 1 0 33 "else \+ Options := args[4 .. nargs];" }}{PARA 389 "> " 0 "" {MPLTEXT 1 0 32 " \+ if has(\{Options\}, 'frames')" }}{PARA 390 "> " 0 "" {MPLTEXT 1 0 43 " then FRAM := subs(\{Options\},'frames');" }}{PARA 391 "> " 0 "" {MPLTEXT 1 0 23 " else FRAM := 8*nt;" }}{PARA 392 "> " 0 "" {MPLTEXT 1 0 37 " Options := 'frames' = FRAM;" }}{PARA 393 "> " 0 "" {MPLTEXT 1 0 12 " end if;" }}{PARA 394 "> " 0 "" {MPLTEXT 1 0 8 "end if; " }}{PARA 395 "> " 0 "" {MPLTEXT 1 0 27 "if not type(FR AM,'posint') " }}{PARA 437 "> " 0 "" {MPLTEXT 1 0 110 "then ERROR(\"Th e first argument is a integer number representing the number of oscill ations of the membrane.\") " }}{PARA 438 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }}{PARA 395 "> " 0 "" {MPLTEXT 1 0 30 "if not has(\{Options\},'s tyle') " }}{PARA 396 "> " 0 "" {MPLTEXT 1 0 37 " then Options:=Option s,'style=patch'" }}{PARA 397 "> " 0 "" {MPLTEXT 1 0 7 "end if;" }} {PARA 398 "> " 0 "" {MPLTEXT 1 0 132 "plots[animate3d]([r,theta,value( u(r,theta,t,p1,p2))],r=0..a,theta=0..2*Pi,t=0..evalf(2*Pi*nt/omega(0,1 )),coords=cylindrical,Options)" }}{PARA 399 "> " 0 "" {MPLTEXT 1 0 9 " end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+animate_icGf*6%'%#ntG% (numericG'%#p1G%*nonnegintG'%#p2G%'posintG6$%%FRAMG%(OptionsG6\"F3C(@$ 29#\"\"$-%&ERRORG6#Q_uThis~procedure~has~at~least~three~arguments.~The ~first~one~is~the~number~of~dominant~oscillations.~The~second~and~the~ third~are~truncation~values~of~the~double~sum.F3@(4-%%typeG6$7$%\"aG% \"cG-%%listG6#F)-F:6#QGThe~constants~a~and~c~must~be~numeric.F34-%)ass ignedG6#%&omegaG-F:6#QDThe~function~omega~must~be~defined.F34-FM6#%\"u G-F:6#QJThe~function~normal_mode~must~be~defined.F3@%/F7F8C$>8$,$9$\" \")>8%6$/.%'framesGFhn./%&styleG%&patchGC$>F]o&9\"6#;\"\"%F7@%-%$hasG6 $<#F]oF`o>Fhn-%%subsGF`pC$>FhnFin>F]oF_o@$4-F@6$Fhn.F/-F:6#QjpThe~firs t~argument~is~a~integer~number~representing~the~number~of~oscillations ~of~the~membrane.F3@$4-F_p6$Fap.Fdo>F]o6$F]oFbo-&%&plotsG6#%*animate3d G6(7%%\"rG%&thetaG-%&valueG6#-FV6'F^rF_r%\"tG9%9&/F^r;\"\"!FC/F_r;Fjr, $%#PiG\"\"#/Fer;Fjr-%&evalfG6#,$*(F^s\"\"\"FjnFgs-FO6$FjrFgs!\"\"F_s/% 'coordsG%,cylindricalGF]oF36'FCFDF^rF_rFerF3" }}}{EXCHG {PARA 400 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT 267 52 "First example: Ax ially symmetric initial conditions." }}}{EXCHG {PARA 401 "> " 0 "" {MPLTEXT 1 0 21 "u0 := (r,theta) -> 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G\"\"\"" }}}{EXCHG {PARA 402 "> " 0 "" {MPLTEXT 1 0 21 "v0 \+ := (r,theta) -> 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G\"\"!" }}} {EXCHG {PARA 403 "> " 0 "" {MPLTEXT 1 0 15 "a := 5; c := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let's see the coefficients: " }}}{EXCHG {PARA 404 "> " 0 "" {MPLTEXT 1 0 15 "unprotect('m');" }}{PARA 405 "> " 0 "" {MPLTEXT 1 0 18 "assume(m,integer);" }}{PARA 406 "> " 0 "" {MPLTEXT 1 0 13 "protect('m');" }}}{EXCHG {PARA 407 "> " 0 "" {MPLTEXT 1 0 8 "A1( 0,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&-%\"jG6$\"\"!%\"n GF$-%\"JG6$F$F&F$!\"\"" }}}{EXCHG {PARA 408 "> " 0 "" {MPLTEXT 1 0 8 " A1(m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 409 "> " 0 "" {MPLTEXT 1 0 8 "A2(0,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 410 "> " 0 "" {MPLTEXT 1 0 8 "A2(m,n);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 411 "> " 0 "" {MPLTEXT 1 0 8 "B1(m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 412 "> " 0 "" {MPLTEXT 1 0 8 "B2(m,n);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 270 "" 0 "" {TEXT 269 62 "Fourier approximation of the initial p osition with five terms:" }}}{EXCHG {PARA 440 "> " 0 "" {MPLTEXT 1 0 89 "plots[cylinderplot]([r,theta,u(r,theta,0,0,5)],r=0..a,theta=0..2*P i,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%%MESHG6#7;7;7%$\"\"!F*F)$\"+1IX28!\"*F(F(F(F(F(F(F(F( F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(7;7%$\"3ULLLLLL$3#!#=F)$\"+T\\9s7F-7%$ \"3.fbVraM7?F2$\"37WejRM1#R&!#>F37%$\"3C\"e<7f>U!=F2$\"3ulmmmmmT5F2F37 %$\"3ZF\"RJZ\"F2$\"3&H(>ZF\"RJZ\"F2F37%$\"3^ommmmmT5F2$\"3ezv@\"f>U !=F2F37%$\"3MvejRM1#R&F:$\"3ZebVraM7?F2F37%$\"3pA7SjyrlL!#LF0F37%$!3!3 \"ejRM1#R&F:$\"3')fbVraM7?F2F37%$!3'Hmmmmm;/\"F2$\"3\"He<7f>U!=F2F37%$ !3tq>ZF\"RJZ\"F2$\"3nx>ZF\"RJZ\"F2F37%$!3\"zd<7f>U!=F2$\"3HrmmmmmT5F2F 37%$!3OdbVraM7?F2$\"3'z!fjRM1#R&F:F37%$!3ULLLLLL$3#F2$\"3PXC!osN9t'FRF 37%$!3(4cN9ZXB,#F2$!3?ydjRM1#R&F:F37%$!3d%e<7f>U!=F2$!3xfmmmmmT5F2F37% $!3ZF\"RJZ\"F2$!3&z'>ZF\"RJZ\"F2F37%$!31ummmmmT5F2$!3Dwv@\"f>U!=F2 F37%$!3qNfjRM1#R&F:$!3\"obN9ZXB,#F2F37%$!3;>lFj1cM'*FRFgoF37%$\"3W]djR M1#R&F:$!3`hbVraM7?F2F37%$\"3FdmmmmmT5F2$!3'fe<7f>U!=F2F37%$\"3tl>ZF\" RJZ\"F2$!3R#)>ZF\"RJZ\"F2F37%$\"3euv@\"f>U!=F2$!37xmmmmmT5F2F37%$\"3qb bVraM7?F2$!3[sfjRM1#R&F:F37%F0$!33*[g`9(GY8!#KF37;7%$\"3&ommmmmm;%F2F) $\"+(4)*y<\"F-7%$\"31=6(G%4pCSF2$\"3#)or#zo7%y5F2Fhs7%$\"3[i^V#=R%3OF2 $\"3[JLLLLL$3#F2Fhs7%$\"3N]R%\\Dyi%HF2$\"3\"f%R%\\Dyi%HF2Fhs7%$\"3.PLL LLL$3#F2$\"3:f^V#=R%3OF2Fhs7%$\"31vr#zo7%y5F2$\"3&p6rG%4pCSF2Fhs7%FioF fsFhs7%$!3;ir#zo7%y5F2$\"3s>6(G%4pCSF2Fhs7%$!3#fKLLLLL3#F2$\"3\"e;NC=R %3OF2Fhs7%$!3ZTR%\\Dyi%HF2$\"3MbR%\\Dyi%HF2Fhs7%$!3#e:NC=R%3OF2$\"3eUL LLLL$3#F2Fhs7%$!3t96(G%4pCSF2$\"3f\"=Fzo7%y5F2Fhs7%$!3&ommmmmm;%F2$\"3 3*[g`9(GY8FcsFhs7%$!3%>7rG%4pCSF2$!3kbr#zo7%y5F2Fhs7%$!3:p^V#=R%3OF2$! 3a>LLLLL$3#F2Fhs7%$!3MgR%\\Dyi%HF2$!3#f$R%\\Dyi%HF2Fhs7%$!38[LLLLL$3#F 2$!3\\_^V#=R%3OF2Fhs7%$!39(=Fzo7%y5F2$!3i86(G%4pCSF2Fhs7%$!3$QIbE87p#> FcsF^wFhs7%$\"33]r#zo7%y5F2$!31B6(G%4pCSF2Fhs7%$\"3b9LLLLL$3#F2$!3#>Fzo7%y5F2Fhs7%Ffs$!3;y4s!HuDp #FcsFhs7;7%$\"3+++++++]iF2F)$\"+1$3c0\"F-7%$\"3\"on1VTOq.'F2$\"3B`2*=. >wh\"F2F^[l7%$\"3WF2$\"3')=fT#Q<%>WF2F^[l7%$\"3b0+++++DJF2$\"3tQFlt(eET&F2F^[l7%$\"3n i2*=.>wh\"F2$\"3fumI9k.PgF2F^[l7%$\"3rm.-f`r45FcsF\\[lF^[l7%$!3CV2*=.> wh\"F2$\"39!o1VTOq.'F2F^[l7%$!3!*))*********\\7$F2$\"3hZFlt(eET&F2F^[l 7%$!3?7fT#Q<%>WF2$\"3uKfT#Q<%>WF2F^[l7%$!3=LFlt(eET&F2$\"3)Q,++++]7$F2 F^[l7%$!3PsmI9k.PgF2$\"3Qs2*=.>wh\"F2F^[l7%$!3+++++++]iF2$\"3TL2/=2V>? 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