{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 18 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 13 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } 0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 263 "" 0 "" {TEXT -1 18 "J\374rgen Krummenauer" } }{PARA 264 "" 0 "" {TEXT -1 21 "Steinstra\337er Allee 38" }}{PARA 265 "" 0 "" {TEXT -1 12 "52428 J\374lich" }}{PARA 0 "" 0 "" {TEXT -1 8 "E- Mail: " }{TEXT 280 24 "krummenauer@fh-aachen.de" }{MPLTEXT 1 0 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Die eindimensionale sinus-f\366r mige Welle" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 163 "Man definiert zwar eine Welle als eine St\366rung, die sich im Raum ausbreitet, im Unterricht beschr\344nkt man sich doch h\344ufig \+ auf eine sinus-f\366rmige St\366rung der Form " }{XPPEDIT 256 0 "y(x,t ) = A*sin(k*x-omega*t);" "6#/-%\"yG6$%\"xG%\"tG*&%\"AG\"\"\"-%$sinG6#, &*&%\"kGF+F'F+F+*&%&omegaGF+F(F+!\"\"F+" }{TEXT -1 2 ". " }{MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f1:=(A,x,t,omega,k)->A*sin(k*x-omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6'%\"AG%\"xG%\"tG%&omegaG%\"kG6\"6$%)ope ratorG%&arrowGF,*&9$\"\"\"-%$sinG6#,&*&9(F29%F2F2*&9'F29&F2!\"\"F2F,F, F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "animate(f1(1,x,t,1,1 ),x=0..4*Pi,t=0..10,frames=30,numpoints=200,color=red,title=`nach rech ts laufende Welle`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Das die A usbreitungsrichtung durch das Vorzeichen von " }{XPPEDIT 18 0 "omega*t ;" "6#*&%&omegaG\"\"\"%\"tGF%" }{TEXT -1 29 " festgelegt wird, wird du rch " }{XPPEDIT 18 0 "omega = -1;" "6#/%&omegaG,$\"\"\"!\"\"" }{TEXT -1 14 " verdeutlicht." }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "animate(f1(2,x,t,-1,1),x=0..4*Pi,t =0..10,frames=30,numpoints=200,color=blue,title=`nach links laufende W elle`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Nun l\344\337t sich au ch die \334berlagerung zeigen (im Beispiel stehende Welle)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fr:=f1(1,x, t,1,1)+f1(1,x,t,-1,1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#frG,&-%$sinG6#,&%\"xG\"\"\"%\"tG!\"\"F+-F '6#,&F*F+F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "animate( fr,x=0..4*Pi,t=0..10,frames=30,numpoints=200,title=`stehende Welle`); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Interferenz (konstruktiv und \+ destruktiv) l\344\337t sich besser an einzelnen Peaks erkennen." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "peak1:=(x,t)->Heaviside(x-t)*sin(x- t)*Heaviside(t+Pi-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&peak1GR6$% \"xG%\"tG6\"6$%)operatorG%&arrowGF)*(-%*HeavisideG6#,&9$\"\"\"9%!\"\"F 3-%$sinGF0F3-F/6#,(F4F3%#PiGF3F2F5F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 109 "animate(peak1(x,t),x=0..8*Pi,t=0..8*Pi,frames=40,n umpoints=100,color=red,title=`nach rechts laufender Peak`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "peak2:=(x,t)->Heaviside(x+t- 7*Pi)*sin(-t-x)*Heaviside(-t+8*Pi-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&peak2GR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)*(-%*HeavisideG6#,( 9$\"\"\"9%F3%#PiG!\"(F3-%$sinG6#,&F4!\"\"F2F;F3-F/6#,(F4F;F5\"\")F2F;F 3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "animate(peak2(x ,t),x=0..8*Pi,t=0..8*Pi,frames=40,numpoints=100,color=blue,title=`nach links laufender Peak`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "animate(peak2(x,t)+peak1(x,t),x=0..8*Pi,t=0..8*Pi,frames=40,numpo ints=100,color=green,title=`konstruktive Interferenz`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "animate(peak2(x,t)-peak1(x,t),x=0. .8*Pi,t=0..8*Pi,frames=40,numpoints=100,color=green,title=`destruktive Interferenz`);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "Die eindimensionale Wellengleichu ng" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Ei ne Welle ist nicht notwendigerweise sinusf\366rmig. Zupft man an einer eingespannten Saite der L\344nge " }{TEXT 257 1 "L" }{TEXT -1 109 ", \+ so erzeugt man bestimmt keine Sinusst\366rung. Die allgemeine Form ein er Welle ergibt sich aus der L\366sung der " }{TEXT 258 15 "Wellenglei chung" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "wel lengleichung:=diff(welle(x,t),t$2)=c^2*diff(welle(x,t),x$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0wellengleichungG/-%%diffG6$-%&welleG6$%\" xG%\"tG-%\"$G6$F-\"\"#*&)%\"cGF1\"\"\"-F'6$F)-F/6$F,F1\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Die Anfangswert- bzw. Randwertbewe gungen lauten f\374r die eingespannte Saite:" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{XPPEDIT 18 0 "welle(x = 0,t) = 0;" "6#/-%&welleG6$/%\"xG \"\"!%\"tGF)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " } {XPPEDIT 18 0 "welle(x = L,t) = 0;" "6#/-%&welleG6$/%\"xG%\"LG%\"tG\" \"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "3. " }{XPPEDIT 18 0 "welle(x,t = 0) = f(x);" "6#/-%&welleG6$%\"xG/%\"tG\"\"!-%\"fG6#F'" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "4. " }{XPPEDIT 18 0 "diff( welle(x,t),t)(t = 0) = 0;" "6#/--%%diffG6$-%&welleG6$%\"xG%\"tGF,6#/F, \"\"!F/" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Die Forderungen 1) und 2) sind die Bedingungen f\374r eine stehende Welle" }}{PARA 0 "" 0 "" {TEXT -1 48 "Die Forderung 3) definiert die Anfangsauslenkung " }}{PARA 0 "" 0 "" {TEXT -1 67 "Die Forderung 4) legt die Anfangsgesc hwindigkeit der St\366rung fest. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Die \+ L\366sung erfolgt durch den Produktansatz: " }}{PARA 256 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "welle(x,t): =ort(x)*zeit(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&welleG6$%\"xG% \"tG*&-%$ortG6#F'\"\"\"-%%zeitG6#F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Einsetzen und umordnen \374berf\374hrt die Gleichung in" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "wellengleichung/(ort(x)*zeit(t));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/*&-%%diffG6$-%%zeitG6#%\"tG-%\"$G6$F+\"\"#\"\"\"F(! \"\"*&*&)%\"cGF/F0-F&6$-%$ortG6#%\"xG-F-6$F;F/\"\"\"F0F8F1" }}}{PARA 0 "" 0 "" {TEXT -1 325 "Die linke Seite der Gleichung ist unabh\344ngi g vom Ort, also bzgl. des Orts konstant. Das gleiche gilt f\374r die r echte eite der Gleichung bzgl. der Zeit. Damit geht die partielle Dgl \+ \374ber in zwei gew\366hnliche Differentialgleichungen die sehr an die Schwingungsgleichung erinnern. Diese Gleichungen k\366nnen elementar \+ gel\366st werden." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "dglzeit:=diff(zeit(t),t$2)=-omega^2 *zeit(t);dglort:=diff(ort(x),x$2)=-omega^2*ort(x)/c^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dglzeitG/-%%diffG6$-%%zeitG6#%\"tG-%\"$G6$F,\" \"#,$*&)%&omegaGF0\"\"\"F)\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dglortG/-%%diffG6$-%$ortG6#%\"xG-%\"$G6$F,\"\"#,$*&*&)%&omegaGF0 \"\"\"F)\"\"\"F6*$)%\"cG\"\"#F6!\"\"!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Nun zu den Bedingungen 1 ) und 2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "initort:=ort(0)=0,ort(L )=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(initortG6$/-%$ortG6#\"\"!F* /-F(6#%\"LGF*" }}}{EXCHG {PARA 257 "" 1 "" {TEXT -1 98 "Mit den oben f ormulierten Anfangswertbedingungen findet Maple nur die triviale L\366 sung f\374r den Ort." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "tri viale_loesung:=dsolve(\{dglort,initort\},ort(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%1triviale_loesungG/-%$ortG6#%\"xG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Folglich mu\337 die Konstante " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 25 " geeignet gew\344hlt werden." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "omega:=n*Pi*c/L;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&*(%\"nG\"\"\"%#PiGF(%\"cGF( \"\"\"%\"LG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Dies f\374hrt zur allgemeinen L\366sung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "loesung_zeit:=dsolve(dglzeit,zeit(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-loesung_zeitG/-%%zeitG6#%\"tG,&*&%$_C1G\"\"\"-%$sinG 6#*&**%\"nGF-%#PiGF-%\"cGF-F)F-\"\"\"%\"LG!\"\"F-F-*&%$_C2GF--%$cosGF0 F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "_C1:=A1;_C2:=A2;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_C1G%#A1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_C2G%#A2G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "loesung_ort:=dsolve(dglort,o rt(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,loesung_ortG/-%$ortG6#% \"xG,&*&%$_C3G\"\"\"-%$sinG6#*&*(%#PiGF-%\"nGF-F)F-\"\"\"%\"LG!\"\"F-F -*&%$_C4GF--%$cosGF0F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "_C3:=A3;_C4:=A4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_C3G%#A3G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$_C4G%#A4G" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ort:=unapply(rhs( loesung_ort),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ortGR6#%\"xG6\" 6$%)operatorG%&arrowGF(,&*&%#A3G\"\"\"-%$sinG6#*&*(%#PiGF/%\"nGF/9$F/ \"\"\"%\"LG!\"\"F/F/*&%#A4GF/-%$cosGF2F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "zeit:=unapply(rhs(loesung_zeit),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%zeitGR6#%\"tG6\"6$%)operatorG%&arrowGF(,& *&%#A1G\"\"\"-%$sinG6#*&**%\"nGF/%#PiGF/%\"cGF/9$F/\"\"\"%\"LG!\"\"F/F /*&%#A2GF/-%$cosGF2F/F/F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 " Nun zur Auswertung der Anfangsbedingung:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "g1:=ort(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g 1G/%#A4G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "init:=solv e(g1,\{A3,A4\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%initG<$/%#A4G\" \"!/%#A3GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "A31:=eval(A3 ,init):A41:=eval(A4,init):A3:=A31;A4:=A41;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A4G\"\"! " }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 14 "Die Bedingung " }{XPPEDIT 18 0 "ort(L) = 0;" "6#/-%$ortG6#%\"LG \"\"!" }{TEXT -1 25 " ist f\374r alle A3 erf\374llt." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ort =ort(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$ortG*&%#A3G\"\"\"-%$sin G6#*&*(%#PiGF'%\"nGF'%\"xGF'\"\"\"%\"LG!\"\"F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Dann b esitzt die L\366sung der Wellengleichung folgende Form" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "welle:=(x,t)->ort(x)*zeit(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&welleGR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)*& -%$ortG6#9$\"\"\"-%%zeitG6#9%F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "welle=expand(we lle(x,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&welleG,&**%#A3G\"\"\" -%$sinG6#*&*(%#PiGF(%\"nGF(%\"xGF(\"\"\"%\"LG!\"\"F(%#A1GF(-F*6#*&**F/ F1F.F1%\"cGF(%\"tGF(F1F2F3F(F(**F'F1F)F1%#A2GF(-%$cosGF6F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Zu r Bedingung 3)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g2:=f(x)=welle(x, 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G/-%\"fG6#%\"xG*(%#A3G\"\" \"-%$sinG6#*&*(%#PiGF,%\"nGF,F)F,\"\"\"%\"LG!\"\"F,%#A2GF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "und zu Bedingung 4)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "v:=diff(welle(x,t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG*(%#A3G\"\"\"-%$sinG6#*&*(%#PiGF'%\"nGF'%\"xGF'\"\"\"%\"LG !\"\"F',&*&*,%#A1GF'-%$cosG6#*&**F.F0F-F0%\"cGF'%\"tGF'F0F1F2F'F.F0F-F 0F " 0 "" {MPLTEXT 1 0 16 "v:=unapply(v,t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"vGR6#%\"tG6\"6$%)operatorG%&arrowGF(*(%#A3G\"\"\" -%$sinG6#*&*(%#PiGF.%\"nGF.%\"xGF.\"\"\"%\"LG!\"\"F.,&*&*,%#A1GF.-%$co sG6#*&**F5F7F4F7%\"cGF.9$F.F7F8F9F.F5F7F4F7FCF7F7F8F9F.*&*,%#A2GF.-F0F @F.F5F7F4F7FCF7F7F8F9!\"\"F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g3:=v0(x)=v(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #g3G/-%#v0G6#%\"xG*&*.%#A3G\"\"\"-%$sinG6#*&*(%#PiGF-%\"nGF-F)F-\"\"\" %\"LG!\"\"F-%#A1GF-F4F5F3F5%\"cGF-F5F6F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Untersuchung der Gleichung 2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "g2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG* (%#A3G\"\"\"-%$sinG6#*&*(%#PiGF*%\"nGF*F'F*\"\"\"%\"LG!\"\"F*%#A2GF*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Die Gleichung ist f\374r belieb ige n erf\374llt. Dann mu\337 auch gelten:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "hilf:=op(1,rhs(g2) )*op(3,rhs(g2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hilfG*&%#A3G\" \"\"%#A2GF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f(x):=sum(rh s(g2),n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"fG6#%\"x G-%$sumG6$*(%#A3G\"\"\"-%$sinG6#*&*(%#PiGF-%\"nGF-F'F-\"\"\"%\"LG!\"\" F-%#A2GF-/F4;F-%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Deu tet man das Produkt " }{TEXT 259 4 "A2A3" }{TEXT -1 136 " als neue Kon stante, so beschreibt obige Gleichung die Fourierdarstellung einer pun ktsymmetrischen periodischen Funktion der Periode 2L." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 29 "Untersuchung der Gleichung 3)" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 3 "g3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#v0G6#%\"x G*&*.%#A3G\"\"\"-%$sinG6#*&*(%#PiGF+%\"nGF+F'F+\"\"\"%\"LG!\"\"F+%#A1G F+F2F3F1F3%\"cGF+F3F4F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Analog e \334berlegungen f\374hren zu:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " v0(x):=sum(rhs(g3),n=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >-%#v0G6#%\"xG-%$sumG6$*&*.%#A3G\"\"\"-%$sinG6#*&*(%#PiGF.%\"nGF.F'F. \"\"\"%\"LG!\"\"F.%#A1GF.F5F6F4F6%\"cGF.F6F7F8/F5;F.%)infinityG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A1:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 195 "Dies ents pricht bis auf Konstanten ebenfalls der Fourierdarstellung einer punkt symmetrischen periodischen Funktion der Periode 2L. Da v0(x)=0 gew \344hlt wurde, verschwindet diese. Es sei also A=0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Wer den wir konkret:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "zupfen:=x->piec ewise(xl,y*(L-x)/(L-l));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'zupfenGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecewiseG6&29$%\"lG*& *&%\"yG\"\"\"F0F5\"\"\"F1!\"\"2F1F0*&*&F4F6,&%\"LGF5F0!\"\"F5F6,&F " 0 "" {MPLTEXT 1 0 17 "zupfen=zupfe n(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%'zupfenG-%*PIECEWISEG6$7$*& *&%\"yG\"\"\"%\"xGF,\"\"\"%\"lG!\"\"2F-F/7$*&*&F+F.,&%\"LGF,F-!\"\"F,F .,&F6F,F/F7F02F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "schla gen:=x->piecewise(x<3/4,0,x<5/4,-y,x>5/4,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)schlagenGR6#%\"xG6\"6$%)operatorG%&arrowGF(-%*piecew iseG6(29$#\"\"$\"\"%\"\"!2F0#\"\"&F3,$%\"yG!\"\"2F6F0F4F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "schlagen=schlagen(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)schlagenG-%*PIECEWISEG6%7$\"\"!2%\"xG#\"\"$\"\"%7$,$%\"yG!\" \"2F+#\"\"&F.7$F)2F4F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Die folgende Grafik zeigt die Anfangsausl enkung" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "parameter:=\{L=2,l=0.2,y=0.1,c=5\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "plot(\{subs(parameter,zupfen(x)),subs(parameter,schlagen(x)) \},x=0..2,color=[red,blue],title=`Rot = schlagen, Blau = zupfen`);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Besimmung der Fouruerkoeffizienten" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "a_zupfen:=n->2/L*int(zupfen(x)*sin(n*Pi*x/L),x=0..L); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)a_zupfenGR6#%\"nG6\"6$%)operato rG%&arrowGF(,$*&-%$intG6$*&-%'zupfenG6#%\"xG\"\"\"-%$sinG6#*&*(9$F6%#P iGF6F5F6\"\"\"%\"LG!\"\"F6/F5;\"\"!F?F>F?F@\"\"#F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a_schlagen:=n->2/L*int(schlagen(x)* sin(n*Pi*x/L),x=0..L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+a_schlage nGR6#%\"nG6\"6$%)operatorG%&arrowGF(,$*&-%$intG6$*&-%)schlagenG6#%\"xG \"\"\"-%$sinG6#*&*(9$F6%#PiGF6F5F6\"\"\"%\"LG!\"\"F6/F5;\"\"!F?F>F?F@ \"\"#F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Es werden nur die \+ ersten 25 Glieder der Fourierreihe ber\374cksichtigt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "loesung_zupfen:=sum(a_zupfen(n)*welle(x,t)/hilf,n=1.. 25):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "loesung_zupfen:=una pply(loesung_zupfen,(x,t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "loesung_schlagen:=sum(a_schlagen(n)*welle(x,t)/hilf,n=1..25):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "loesung_schlagen:=unapply(lo esung_schlagen,(x,t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "p 1:=plot(subs(parameter,loesung_zupfen(x,0)),x=0..2,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "p2:=plot(subs(parameter,loes ung_schlagen(x,0)),x=0..2,color=blue):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Dann hat die gen\344herte L\366sung der Wellengleichung \+ zu Zeit t=0 folgendes Aussehen:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "display(\{p1,p2\},title=`gen\344herte L\366sung zur Zeit t=0 Zupfen = Rot, schlagen = Blau`);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Mit den obe n definierten Parametern kann diese L\366sung als Animation dargestell t werden." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "loesung_zupfen(x,t):=s ubs(parameter,loesung_zupfen(x,t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "animate(loesung_zupfen(x,t),x=0..2,t=0..1,frames=100, numpoints=200,color=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "loesung_schlagen(x,t):=subs(parameter,loesung_schlagen(x,t)):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "animate(loesung_schlagen(x,t),x=0.. 2,t=0..1,frames=100,numpoints=200,color=black);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Sinu s-f\366rmige zweidimensinale Wellen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Die folgende Animation zeigt die A usbreitung einer ebenen Welle" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ebene_welle1:=(x,t)->sin(k*x-omega*t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-ebene_welle1GR6$%\"xG%\"tG6\"6$%)op eratorG%&arrowGF)-%$sinG6#,&*&%\"kG\"\"\"9$F3F3*&%&omegaGF39%F3!\"\"F) F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ebene_welle2:=(x,t) ->sin(k*x+omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-ebene_welle2 GR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%$sinG6#,&*&%\"kG\"\"\"9$F3F3 *&%&omegaGF39%F3F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "parameter:=\{k=1/2,omega=1\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*p arameterG<$/%\"kG#\"\"\"\"\"#/%&omegaGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "animate3d(subs(parameter,ebene_welle1(x,t)),x=-4*Pi. .4*Pi,y=-4*Pi..4*Pi,t=-8*Pi..8*Pi,title=`Die ebene Welle` );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "animate3d(subs(parameter,eb ene_welle1(x,t)+ebene_welle2(x,t)),x=-4*Pi..4*Pi,y=-4*Pi..4*Pi,t=-8*Pi ..8*Pi,title=`Die stehende ebene Welle` );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "\304ndert sich die Gr\366\337e der Phasenfl\344che, so mu ss die Amplitude der Welle ortsabh\344ngig sein." }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 76 "Die Kreiswelle ist ein Beispiel f\374r eine Welle \+ mit ortsabh\344ngiger Amplitude." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "kreis:=(x,y,t)->1/sqrt((x^2+y^2))*sin(sqr t(x^2+y^2)+t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kreisGR6%%\"xG%\" yG%\"tG6\"6$%)operatorG%&arrowGF**&-%$sinG6#,&-%%sqrtG6#,&*$)9$\"\"#\" \"\"\"\"\"*$)9%F:F;F " 0 " " {MPLTEXT 1 0 55 "animate3d(kreis(x,y,t),x=-24..24,y=-24..24,t=0..12* Pi);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Die zweidimensionale We llengleichung" }}{PARA 3 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "Die allgemeine Darstellung einer zweidimensionalen We lle folgt aus der L\366sung der zweidimensionalen Wellengleichung." }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Die zweidimensionale Wellengleich ung lautet." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "wellengleich ung2d:=diff(welle(x,y,t),t$2)=c^2*(diff(welle(x,y,t),x$2)+diff(welle(x ,y,t),y$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2wellengleichung2dG/ -%%diffG6$-%&welleG6%%\"xG%\"yG%\"tG-%\"$G6$F.\"\"#*&)%\"cGF2\"\"\",&- F'6$F)-F06$F,F2\"\"\"-F'6$F)-F06$F-F2F " 0 "" {MPLTEXT 1 0 38 "well e(x,y,t):=ortx(x)*orty(y)*zeit(t);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Einsetzen und umordnen \374berf\374hrt di e Gleichung in" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "wellengleichung2d :=expand(wellengleichung2d/(ortx(x)*orty(y)*zeit(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%&welleG6%%\"xG%\"yG%\"tG*(-%%ortxG6#F'\"\"\"-%% ortyG6#F(F.-%%zeitG6#F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2wellen gleichung2dG/*&-%%diffG6$-%%zeitG6#%\"tG-%\"$G6$F-\"\"#\"\"\"F*!\"\",& *&*&)%\"cGF1F2-F(6$-%%ortxG6#%\"xG-F/6$F>F1\"\"\"F2F;F3FA*&*&F7F2-F(6$ -%%ortyG6#%\"yG-F/6$FIF1FAF2FFF3FA" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "Die linke Seite der Gleichung ist unabh\344ngig vom Ort, also \+ bzgl. des Orts konstant. Das gleiche gilt f\374r die rechte Seite der \+ Gleichung bzgl. der Zeit. Die Gleichung bzgl. der Zeit kann elementar gel\366st werden." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "dglzeit:=diff (zeit(t),t$2)=-omega^2*zeit(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%( dglzeitG/-%%diffG6$-%%zeitG6#%\"tG-%\"$G6$F,\"\"#,$*&)%&omegaGF0\"\"\" F)\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "loesung_ze it:=dsolve(dglzeit,zeit(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-loe sung_zeitG/-%%zeitG6#%\"tG,&*&%$_C1G\"\"\"-%$cosG6#*&%&omegaGF-F)F-F-F -*&%$_C2GF--%$sinGF0F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Umben ennung der Integrationskonstanten:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "_C1:=A:_C2:=B:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "zeit: =t->A*cos(omega*t)+B*sin(omega*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%zeitGR6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"AG\"\"\"-%$cosG6#*&%& omegaGF/9$F/F/F/*&%\"BGF/-%$sinGF2F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "zeit=zeit(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%%zeitG,&*&%\"AG\"\"\"-%$cosG6#*&%&omegaGF(%\"tGF(F(F(*&%\"BGF(-%$si nGF+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Die Dgl. f\374r den O rt lautet dann:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "dglortxy:=expand(rhs(wellengleichung2d)/c^2)=-omega^2 /c^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dglortxyG/,&*&-%%diffG6$-% %ortxG6#%\"xG-%\"$G6$F.\"\"#\"\"\"F+!\"\"\"\"\"*&-F)6$-%%ortyG6#%\"yG- F06$F " 0 "" {MPLTEXT 1 0 35 "dglortx:=op(1,lhs(dglortxy))=-kx^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(dglortxG/*&-%%diffG6$-%%ortxG6#%\"xG-%\"$G6$F -\"\"#\"\"\"F*!\"\",$*$)%#kxGF1F2!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dglorty:=op(2,lhs(dglortxy))=-ky^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dglortyG/*&-%%diffG6$-%%ortyG6#%\"yG-%\"$G6$F-\" \"#\"\"\"F*!\"\",$*$)%#kyGF1F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Diese besitzen die L\366sungen " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "loesung_ortx:=dsolve(dglortx,ortx(x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-loesung_ortxG/-%%ortxG6#%\"x G,&*&%$_C3G\"\"\"-%$sinG6#*&%#kxGF-F)F-F-F-*&%$_C4GF--%$cosGF0F-F-" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ortx:=x->C*cos(kx*x)+D*sin( kx*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ortxGR6#%\"xG6\"6$%)opera torG%&arrowGF(,&*&%\"CG\"\"\"-%$cosG6#*&%#kxGF/9$F/F/F/*&%\"DGF/-%$sin GF2F/F/F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "loesung_or ty:=dsolve(dglorty,orty(y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-loe sung_ortyG/-%%ortyG6#%\"yG,&*&%$_C3G\"\"\"-%$sinG6#*&%#kyGF-F)F-F-F-*& %$_C4GF--%$cosGF0F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "or ty:=y->E*cos(ky*y)+F*sin(ky*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% ortyGR6#%\"yG6\"6$%)operatorG%&arrowGF(,&*&%\"EG\"\"\"-%$cosG6#*&%#kyG F/9$F/F/F/*&%\"FGF/-%$sinGF2F/F/F(F(F(" }}}{EXCHG {PARA 261 "" 1 "" {TEXT -1 34 "Damit lautet die gefundene L\366sung:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "welle:=(x,y,t)->zeit(t)*ortx(x)*orty(y):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "welle=welle(x,y,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&welleG*(,&*&%\"AG\"\"\"-%$cosG6#*&% &omegaGF)%\"tGF)F)F)*&%\"BGF)-%$sinGF,F)F)F),&*&%\"CGF)-F+6#*&%#kxGF)% \"xGF)F)F)*&%\"DGF)-F3F8F)F)F),&*&%\"EGF)-F+6#*&%#kyGF)%\"yGF)F)F)*&% \"FGF)-F3FCF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Nur zur Anfa ngsbedingungen:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g1:=welle(0,y,t) =0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G/*(,&*&%\"AG\"\"\"-%$cosG 6#*&%&omegaGF*%\"tGF*F*F**&%\"BGF*-%$sinGF-F*F*F*%\"CGF*,&*&%\"EGF*-F, 6#*&%#kyGF*%\"yGF*F*F**&%\"FGF*-F4F:F*F*F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Diese Gleichung mu\337 f\374r alle y,t erf\374llt se in => " }{TEXT 269 3 "C=0" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g2:=welle(x,0,t)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#g2G/*(,&*&%\"AG\"\"\"-%$cosG6#*&%&omegaGF*%\"tGF*F*F**&%\"BGF*-%$ sinGF-F*F*F*,&*&%\"CGF*-F,6#*&%#kxGF*%\"xGF*F*F**&%\"DGF*-F4F9F*F*F*% \"EGF*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Diese Gleichung mu \337 f\374r alle x,t erf\374llt sein => " }{TEXT 270 3 "E=0" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "C:=0:E:=0:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Die L\366sungsfunktion lautet nun " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "welle=welle(x,y,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&welleG*,,&*&%\"AG\"\"\"-%$cosG6#*&%&omega GF)%\"tGF)F)F)*&%\"BGF)-%$sinGF,F)F)F)%\"DGF)-F36#*&%#kxGF)%\"xGF)F)% \"FGF)-F36#*&%#kyGF)%\"yGF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 " Die Bedingungen " }{TEXT 271 36 "welle(x=a,y,t) = welle(x,y=b,t) = 0 \+ " }{TEXT -1 26 "werden durch die Wahl von " }{TEXT 272 0 "" }{TEXT 274 0 "" }{TEXT 275 0 "" }{XPPEDIT 261 0 "k[x];" "6#&%\"kG6#%\"xG" } {TEXT -1 5 " und " }{XPPEDIT 262 0 "k[y];" "6#&%\"kG6#%\"yG" }{TEXT -1 9 " erf\374llt." }{TEXT 273 1 " " }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "kx:=n*Pi/a;ky:=m*Pi/b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#kxG*&*&%\"nG\"\"\"%#PiGF(\"\"\"%\"aG!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#kyG*&*&%\"mG\"\"\"%#PiGF(\"\"\"%\" bG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Hierbei sind " }{TEXT 277 3 "n,m" }{TEXT -1 30 " beliebige nat\374rliche Zahlen." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Dann wird " }{XPPEDIT 263 0 "omega ;" "6#%&omegaG" }{TEXT -1 15 " bestimmt durch" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "g_omega:=kx^2+ky^2=omega^2/c^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(g_omegaG/,&*&*&)%\"nG\"\"#\"\"\")%#PiGF+F,F,*$) %\"aG\"\"#F,!\"\"\"\"\"*&*&)%\"mGF+F,F-F,F,*$)%\"bG\"\"#F,F3F4*&*$)%&o megaGF+F,F,*$)%\"cG\"\"#F,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "omega:=solve(g_omega,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&omegaG6$*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF/F0)%\"bGF/F0\"\" \"*()%\"mGF/F0F1F0)%\"aGF/F0F5F0%\"cGF5F0*&F4\"\"\"F:\"\"\"!\"\",$F&! \"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "Da " }{XPPEDIT 256 0 "omega;" "6#%&omegaG" }{TEXT -1 19 " \+ negativ sein soll " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "omega :=-abs(omega[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG,$-%$abs G6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF2F3)%\"bGF2F3\"\"\"*()%\" mGF2F3F4F3)%\"aGF2F3F8F3%\"cGF8F3*&F7\"\"\"F=\"\"\"!\"\"!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Jetzt lautet die L\366sungsfunktio n f\374r ein spezielles Paar von nat\374rlichen Zahlen " }{TEXT 278 5 "(n,m)" }{TEXT -1 1 ":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "welle=welle(x,y,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%&welleG*,,&*&%\"AG\"\"\"-%$cosG6#*&-%$absG6#*&*&-%%sqrtG6#,&*( )%\"nG\"\"#\"\"\")%#PiGF:F;)%\"bGF:F;F)*()%\"mGF:F;F " 0 "" {MPLTEXT 1 0 70 "loesung_allgemein:=Sum(Sum(w elle(x,y,t),n=1..infinity),m=1..infinity);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%2loesung_allgemeinG-%$SumG6$-F&6$*,,&*&%\"AG\"\"\"-%$ cosG6#*&-%$absG6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF?F@)%\"bGF? F@F.*()%\"mGF?F@FAF@)%\"aGF?F@F.F@%\"cGF.F@*&FD\"\"\"FI\"\"\"!\"\"F.% \"tGF.F.F.*&%\"BGF.-%$sinGF1F.!\"\"F.%\"DGF.-FS6#*&*(F>F.FBF.%\"xGF.F@ FIFNF.%\"FGF.-FS6#*&*(FGF.FBF@%\"yGF.F@FDFNF./F>;F.%)infinityG/FGF\\o " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Nun zu r Anfangsgeschwindigkeit:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "v:=diff(loesung_allgemein,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"vG-%$SumG6$-F&6$*,,&*(%\"AG\"\"\"-%$sinG6#*&-%$absG 6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF?F@)%\"bGF?F@F.*()%\"mGF?F @FAF@)%\"aGF?F@F.F@%\"cGF.F@*&FD\"\"\"FI\"\"\"!\"\"F.%\"tGF.F.F3F@!\" \"*(%\"BGF.-%$cosGF1F.F3F@FPF.%\"DGF.-F06#*&*(F>F.FBF.%\"xGF.F@FIFNF.% \"FGF.-F06#*&*(FGF.FBF@%\"yGF.F@FDFNF./F>;F.%)infinityG/FGF\\o" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "v:=unapply(v,x,y,t);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"vGR6%%\"xG%\"yG%\"tG6\"6$%)operato rG%&arrowGF*-%$SumG6$-F/6$*,,&*(%\"AG\"\"\"-%$sinG6#*&-%$absG6#*&*&-%% sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGFHFI)%\"bGFHFIF7*()%\"mGFHFIFJFI)%\" aGFHFIF7FI%\"cGF7FI*&FM\"\"\"FR\"\"\"!\"\"F79&F7F7F " 0 "" {MPLTEXT 1 0 21 "g2:=v0(x,y)=v(x,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G/-%#v0G6$%\"xG%\"yG-%$SumG6$-F,6$,$*.%\"BG\"\"\"- %$absG6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF@FA)%\"bGF@FAF3*()% \"mGF@FAFBFA)%\"aGF@FAF3FA%\"cGF3FA*&FE\"\"\"FJ\"\"\"!\"\"F3%\"DGF3-%$ sinG6#*&*(F?F3FCF3F)F3FAFJFOF3%\"FGF3-FR6#*&*(FHF3FCFAF*F3FAFEFOF3!\" \"/F?;F3%)infinityG/FHFgn" }}}{EXCHG {PARA 260 "" 1 "" {TEXT -1 52 "Au ch hier erkennt man die Fourierreihe der Funktion " }{XPPEDIT 260 0 "v [0](x,y);" "6#-&%\"vG6#\"\"!6$%\"xG%\"yG" }{TEXT -1 19 ". (Es soll gel ten " }{XPPEDIT 256 0 "v[0](x,y) = 0;" "6#/-&%\"vG6#\"\"!6$%\"xG%\"yG F(" }{TEXT -1 4 " => " }{TEXT 266 4 "B=0)" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Dies f\374hrt zur allgemeinen L\366sungsf unktion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG\" \"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "loesung_allgemein;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$-F$6$*.%\"AG\"\"\"-%$cosG6# *&-%$absG6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#PiGF;F<)%\"bGF;FF*%\"xGF*FF<% \"yGF*F " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "welle=welle(x,y,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&welleG*. %\"AG\"\"\"-%$cosG6#*&-%$absG6#*&*&-%%sqrtG6#,&*()%\"nG\"\"#\"\"\")%#P iGF8F9)%\"bGF8F9F'*()%\"mGF8F9F:F9)%\"aGF8F9F'F9%\"cGF'F9*&F=\"\"\"FB \"\"\"!\"\"F'%\"tGF'F'%\"DGF'-%$sinG6#*&*(F7F'F;F'%\"xGF'F9FBFGF'%\"FG F'-FK6#*&*(F@F'F;F9%\"yGF'F9F=FGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Nun zur Anfangsauslenkung f\374r dieses spezielle Wertepaar:" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 24 "g1:=f(x,y)=we lle(x,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g1G/-%\"fG6$%\"xG%\" yG*,%\"AG\"\"\"%\"DGF--%$sinG6#*&*(%\"nGF-%#PiGF-F)F-\"\"\"%\"aG!\"\"F -%\"FGF--F06#*&*(%\"mGF-F5F6F*F-F6%\"bGF8F-" }}}{EXCHG {PARA 12 "" 1 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 12 "Das Produkt " }{TEXT 267 3 "ADF" }{TEXT -1 67 " sind auch hier die Fourierkoeffizienten der periodischen Funktion " }{TEXT 268 6 "f(x,y)" }{TEXT -1 22 " (Periode 2a bzw. 2b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 " Ein konkretes Beispiel mit " }{TEXT 276 6 "a=b=1:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 78 "f:=(x,y)->Heaviside(x-2/5)*Heaviside(3/5-x)*Heavisi de(y-2/5)*Heaviside(3/5-y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF)**-%*HeavisideG6#,&9$\"\"\"#!\"#\"\"&F3F3-F/6#,&#\"\"$F6F3F2!\" \"F3-F/6#,&9%F3F4F3F3-F/6#,&F:F3F@F " 0 "" {MPLTEXT 1 0 40 "plot3d(f(x,y),x=0..1,y=0..1,axes=boxed);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Die Fourierkoeffizienten werden bestimmt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "a:=(n,m)->4*int (int(f(x,y)*sin(n*Pi*x),x=0..1)*sin(m*Pi*y),y=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aGR6$%\"nG%\"mG6\"6$%)operatorG%&arrowGF),$-%$in tG6$*&-F/6$*&-%\"fG6$%\"xG%\"yG\"\"\"-%$sinG6#*(9$F:%#PiGF:F8F:F:/F8; \"\"!F:F:-F<6#*(9%F:F@\"\"\"F9F:F:/F9FB\"\"%F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Die angen\344herte Funktion lautet dann " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "f1:=(x, y)->sum(sum(a(j,i)*sin(i*Pi*x),i=1..15)*sin(j*Pi*y),j=1..15);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1GR6$%\"xG%\"yG6\"6$%)operatorG%&a rrowGF)-%$sumG6$*&-F.6$*&-%\"aG6$%\"jG%\"iG\"\"\"-%$sinG6#*(F8F9%#PiGF 99$F9F9/F8;F9\"#:F9-F;6#*(F7F9F>\"\"\"9%F9F9/F7FAF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Damit geht die allgemeine L\366sung \374b er in:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "loesung:=(x,y,t)- >sum(sum(a(n,m)*welle(x,y,t)/(A*D*F),n=1..15),m=1..15);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(loesungGR6%%\"xG%\"yG%\"tG6\"6$%)operatorG%&a rrowGF*-%$sumG6$-F/6$*&*&-%\"aG6$%\"nG%\"mG\"\"\"-%&welleG6%9$9%9&F:\" \"\"*(%\"AG\"\"\"%\"DG\"\"\"%\"FG\"\"\"!\"\"/F8;F:\"#:/F9FKF*F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Und die L\366sung f\374r das spezi elle Randwertproblem erh\344lt man" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "parameter:=\{a= 1,b=1,c=0.25\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*parameterG<%/%\" aG\"\"\"/%\"bGF(/%\"cG$\"#D!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(subs(parameter,loesung(x,y,0)),x=0..1,y=0..1,title=`St \366rung zur Zeit t=0`,axes=boxed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Oder als Anim ation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "animate3d(subs(parameter,l oesung(x,y,t)),x=0..1,y=0..1,t=0..4*Pi,frames=20,axes=boxed);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "W\344hlt man als Anfangsbedingung eine Funktion, die mit ihrer Fourierreihe \374bereistimmt, so bildet \+ sich auf der Membran eine stehende Welle aus." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "stehende_welle:=(x,y)->sin(n *Pi*x)*sin(m*Pi*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/stehende_wel leGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&-%$sinG6#*(%\"nG\"\"\"%#Pi GF39$F3F3-F/6#*(%\"mGF3F4\"\"\"9%F3F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "loesung_stehende_welle:=(x,y,t)->subs(a=1,b=1,zeit (t))*stehende_welle(x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%7loesun g_stehende_welleGR6%%\"xG%\"yG%\"tG6\"6$%)operatorG%&arrowGF**&-%%subs G6%/%\"aG\"\"\"/%\"bGF4-%%zeitG6#9&F4-%/stehende_welleG6$9$9%F4F*F*F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "loesung_stehende_welle=loesung_stehende_welle(x,y,t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%7loesung_stehende_welleG**%\"AG \"\"\"-%$cosG6#*&-%$absG6#*&-%%sqrtG6#,&*&)%#PiG\"\"#\"\"\")%\"nGF7F8F '*&)%\"mGF7F8F5F8F'F8%\"cGF'F'%\"tGF'F'-%$sinG6#*(F:F'F6F'%\"xGF'F'-FA 6#*(F=F'F6F8%\"yGF'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "pa rameter1:=\{c=0.25,m=4,n=3\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+pa rameter1G<%/%\"mG\"\"%/%\"nG\"\"$/%\"cG$\"#D!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "animate3d(subs(parameter1,loesung_stehende_w elle(x,y,t)),x=0..1,y=0..1,t=0..4*Pi,frames=20,axes=boxed);" }}} {EXCHG {PARA 262 "" 1 "" {TEXT -1 0 "" }}}}}{MARK "6" 0 }{VIEWOPTS 1 1 0 1 1 1803 }