{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 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 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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 } 1 1 0 0 0 0 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 Output" -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 "Map le 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 "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 255 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 "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "c International Thomson Pu blishing Bonn 1995 filename: paket1" }}{PARA 0 "" 0 "" {TEXT -1 108 "Autor: Komma \+ Datum: 20.8.94 " }}{PARA 0 "" 0 "" {TEXT -1 19 "Thema: Wellenpakete" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 257 125 "In Maple 6 nicht stabil: Die Reihenfolg e der ops und das R\374cksetzen der Annahmen sind eine reines Lotterie spiel insbes. mit %" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Linearkombinat ionen von L\366sungen einer linearen DG sind wieder L\366sungen, d.h. \+ L\366sungen k\366nnen superponiert werden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Eine L\366sung ist die ebene We lle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "restart;with(inttrans);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7/%)addtableG%(fourierG%+fouriercosG%+fouriersinG%'hank elG%(hilbertG%+invfourierG%+invhilbertG%+invlaplaceG%*invmellinG%(lapl aceG%'mellinG%*savetableG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "yk:=(x,t)->exp(I*(k*x-omega*t));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#ykGR6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%$expG6#*&^#\"\"\"F2,& *&%\"kGF29$F2F2*&%&omegaGF29%F2!\"\"F2F)F)F)" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 25 "die z.B. mit dem Gewicht " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "wk:=exp(-((k-k0)^2)/(2*sk^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wkG-%$expG6#,$*&*$),&%\"kG\"\"\"%#k0G!\"\"\"\"#F.F.* $)%#skGF1F.F0#F0F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "auftritt. I m Falle des kontinuierlichen Spektrums mu\337 also " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "z:=(x,t)->wk*yk(x,t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "z(x,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* &-%$expG6#,$*&*$),&%\"kG\"\"\"%#k0G!\"\"\"\"#F-F-*$)%#skGF0F-F/#F/F0F- -F%6#*&^#F-F-,&*&F,F-%\"xGF-F-*&%&omegaGF-%\"tGF-F/F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "\374ber alle Wellenzahlen integriert wer den. Wenn wir den Zusammenhang von omega und k noch offen halten wolle n, k\366nnen wir das zun\344chst f\374r t=0 tun:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "assume(sk>0): # sk koennte komplex sein" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(z(x,0),k=-infinity..infi nity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%$expG6#*(^##\"\"\"\"\"#F *%\"xGF*,&%#k0GF+*(^#F*F*F,F*)%$sk|irGF+F*F*F*F*-%%sqrtG6#F+F*F2F*-F46 #%#PiGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Diese Integration ist aber nichts andere s als eine Fouriertransformation (hier invers wegen +kx und geeign. no rmiert)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "omega:='omega': " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "pak0:=invfourier(wk,k,x )*2*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pak0G*.-%$expG6#,$*&*$)% #k0G\"\"#\"\"\"F/*$)%$sk|irGF.F/!\"\"#F3F.F/-%%sqrtG6#%#PiGF/-F66#F.F/ F2F/-F'6#*(^#F/F/F-F/%\"xGF/F/-F'6#,$*&,&*$F,F/F3*&)F?F.F/)F2\"\"%F/F/ F/*$F1F/F3F4F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 347 "Es wird also wie bei der Schwingu ng eine Gau\337verteilung in eine Gau\337verteilung transformiert und \+ das Produkt der Varianzen ist wieder ~1: ein gut lokalisiertes Wellenp aket kann nur durch ein breites Spektrum der Wellenzahlen aufgebaut we rden. Die Frage ist nun, wie sich dieses Paket im Laufe der Zeit entwi ckelt. Im Falle der Wellengleichung gilt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "omega:=k*c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ome gaG*&%\"kG\"\"\"%\"cGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " pakw:=int(z(x,t),k=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pakwG**-%$expG6#*&^##\"\"\"\"\"#F,,,*&%#k0GF,%\"xGF,F-**F-F,F 0F,%\"cGF,%\"tGF,!\"\"*(^#F,F,)F1F-F,)%$sk|irGF-F,F,*,^#!\"#F,F1F,F9F, F3F,F4F,F,**F7F,)F3F-F,)F4F-F,F9F,F,F,F,-%%sqrtG6#F-F,F:F,-FB6#%#PiGF, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**-%$expG6#*(^##\"\"\"\"\"#F*,&%\"xGF**&%\"c GF*%\"tGF*!\"\"F*,(*(^#F*F*F-F*)%$sk|irGF+F*F***^#F1F*F5F*F/F*F0F*F**& F+F*%#k0GF*F*F*F*-%%sqrtG6#F+F*F6F*-F<6#%#PiGF*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Der Ex ponent l\344\337t sich vereinfachen zu: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "collect(op(op(1,%)),sk); # nicht stabil (+/-), bitte vorher die Ausgabe lesen, bzw. auf \304nderung achten." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&**^##\"\"\"\"\"#F',&%\"xGF'*&%\"cGF'%\"tGF'!\" \"F',&*&^#F'F'F*F'F'*(^#F.F'F,F'F-F'F'F')%$sk|irGF(F'F'*(F1F'F)F'%#k0G F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Das ist ein Gau\337-Paket , das sich mit der Geschwindigkeit c bewegt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pakw:=exp(subs(sk=1/sx,%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pakwG-%$expG6#,&*&*(^##\"\"\"\"\"#F-,&%\"xGF-*&%\"cG F-%\"tGF-!\"\"F-,&*&^#F-F-F0F-F-*(^#F4F-F2F-F3F-F-F-F-*$)%#sxGF.F-F4F- *(F7F-F/F-%#k0GF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "repa kw:=evalc(Re(pakw)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#rep akw;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k0:=2: c:=3: sx:=4: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "W arning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "animate(repakw,x=-20..20,t=-10..10,frames=8 0,numpoints=200);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Abe r es lohnt sich wieder, auf Vollbild zu stellen und mit den Parametern zu spielen." }}{PARA 0 "" 0 "" {TEXT -1 51 "(Man kann auch noch eine \+ Blende reinprogrammieren)." }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "I m Falle der sgl gilt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "sk :='sk': k0:='k0':b:='b':k:='k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "omega:=b*k^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*&% \"bG\"\"\")%\"kG\"\"#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " #omega:=taylor(omega,k=k0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Der \+ Exponent des Integranden sieht also so aus:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "expo:=op(combine(wk*yk(x,t),power));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%expoG,&*&*$),&%\"kG\"\"\"%#k0G!\"\"\"\"#F+F+* $)%#skGF.F+F-#F-F.*&^#F+F+,&*&F*F+%\"xGF+F+*(%\"bGF+)F*F.F+%\"tGF+F-F+ F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "F\374r t=0 gilt wieder das \+ Gleiche wie oben. Wie entwickelt sich das Paket?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "assume(sk>0,k0>0,t>0):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "pak:=int(z(x,t),k=-infinity..infinity);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$pakG-%*PIECEWISEG6$7$*&**-%$expG6#* &*&^##!\"\"\"\"#\"\"\",(*()%$k0|irGF3F4%\"bGF4%#t|irGF4F3*(F3F4F8F4%\" xGF4F2*(^#F2F4)F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 24 "In Maple 6 ein Ergebnis!" }{TEXT -1 227 " So ist kein Ergebni s zu bekommen. Wir k\366nnen aber versuchen, den Exponenten zu einem v ollst\344ndigen Quadrat zu erg\344nzen -- so wie man das von Hand auch machen w\374rde, weil dann die Fehlerfunktion ins Spiel gebracht werd en kann." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "with(student): \+ b:='b':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "expo;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*$),&%\"kG\"\"\"%$k0|irG!\"\"\"\"#F)F)*$ )%$sk|irGF,F)F+#F+F,*&^#F)F),&*&F(F)%\"xGF)F)*(%\"bGF))F(F,F)%#t|irGF) F+F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "expoc:=completesq uare(expo,k); # Reihenfolge fuer op-Befehl (s.u.) beachten!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&expocG,&*&*&,&\"\"\"F)**^#\"\"#F)%\"bGF)% #t|irGF))%$sk|irGF,F)F)F)),&%\"kGF)*&,&%$k0|irGF)*(^#F)F)%\"xGF)F/F)F) F)F(!\"\"F:F,F)F)*$F/F)F:#F:F,*&#F)F,F)*&,(**F+F))F6F,F)F-F)F.F)F)*(^# !\"#F)F6F)F9F)F)*&)F9F,F)F/F)F)F)F(F:F)F:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "op(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&*&,&\" \"\"F'**^#\"\"#F'%\"bGF'%#t|irGF')%$sk|irGF*F'F'F'),&%\"kGF'*&,&%$k0|i rGF'*(^#F'F'%\"xGF'F-F'F'F'F&!\"\"F8F*F'F'*$F-F'F8#F8F*,$*&,(**F)F')F4 F*F'F+F'F,F'F'*(^#!\"#F'F4F'F7F'F'*&)F7F*F'F-F'F'F'F&F8F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Der erste Summand ist nun ein vollst\344ndiges Quadrat in k un d von der Form a*(k-etwas)^2. Also setzen wir an:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "assume(a>0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "paks:=int(exp(-a*(k-etwas)^2+op(2,expoc)),k=-infinity ..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%paksG*&*&-%$expG6#, $*&,(**^#\"\"#\"\"\")%$k0|irGF/F0%\"bGF0%#t|irGF0F0*(^#!\"#F0F2F0%\"xG F0F0*&)F8F/F0)%$sk|irGF/F0F0F0,&F0F0**F.F0F3F0F4F0F;F0F0!\"\"#F?F/F0-% %sqrtG6#%#PiGF0F0*$-FB6#%#a|irGF0F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "# bis hier her l\344uft es, kopie von paks:" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 87 "paks := exp(-1/2*(2*I*k0^2*b* t-2*I*k0*x+x^2*sk^2)/(1+2*I*b*t*sk^2))*sqrt(Pi)/(sqrt(a));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "\"Etwas\" ist verschwunden und wir ben\366tigen noch a." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "op (op(1,\{op(expoc)\})); # nicht stabil: op(1,\{\}) oder op(2,\{\})" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&#!\"\"\"\"#,&\"\"\"F'**^#F%F'%\"bGF'%# t|irGF')%$sk|irGF%F'F'*&F'F'*$F,F'F$*$),&%\"kGF'*&,&%$k0|irGF'*(^#F'F' %\"xGF'F,F'F'F'F&F$F$F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "as:=op(1,\{op(expoc)\})/op(4,op(1,\{op(expoc)\})); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#asG,$*&,&\"\"\"F(**^#\"\"#F(%\"bGF(%#t|irGF() %$sk|irGF+F(F(F(*$F.F(!\"\"#F1F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "# kopie" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 35 "a s := -1/2*(1+2*I*b*t*sk^2)/(sk^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Bis man die Klammern in op(()) ri chtig gesetzt hat, hat man's auch von Hand nochmal geschrieben. Oder m it der Maus geholt..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "pa ks:=subs(a=as, sk=s,paks);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%paksG *&*&-%$expG6#,$*&,(**^#\"\"#\"\"\")%$k0|irGF/F0%\"bGF0%#t|irGF0F0*(^#! \"#F0F2F0%\"xGF0F0*&)F8F/F0)%\"sGF/F0F0F0,&F0F0**F.F0F3F0F4F0F;F0F0!\" \"#F?F/F0-%%sqrtG6#%#PiGF0F0*$-FB6#,$*&F=F0*$F;F0F?F@F0F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "apaks:=simplify(evalc(abs(paks))^2) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&apaksG,$*&**-%$expG6#,$*&*&)% \"sG\"\"#\"\"\"),&*(%$k0|irGF1%\"bGF1%#t|irGF1F0%\"xG!\"\"F0F1F1,&F1F1 **\"\"%F1)F6F0F1)F7F0F1)F/F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "t:='t': k0:='k0':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# Annahmen vo n Hand (C&P) entfernen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "apaks := 2*exp(-s^2*(2*k0*b*t-x)^2/(1+4*b^2*t^2*s^4))*Pi*csgn(conjuga te(s)^2)*s^2/(sqrt(1+4*b^2*t^2*s^4));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&apaksG,$*&**-%$expG6#,$*&*&)%\"sG\"\"#\"\"\"),&*(%#k0GF1%\"bGF1% \"tGF1F0%\"xG!\"\"F0F1F1,&F1F1**\"\"%F1)F6F0F1)F7F0F1)F/F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "with(plots): # es dauert wie der ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "s:=2: b:=3: k0:= 4:t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "animate(apaks, x=-40..40,t=-1..1,frames=25,numpoints=200);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Die folgenden Befehle sollen die Animation des \+ Realteils beschleunigen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 " repaks:=simplify(evalc(Re(paks))); # auch diese \"simple\" Umformung b enoetigt ihre Zeit" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'repaksG,$*&*( -%%sqrtG6#%#PiG\"\"\"-%$expG6#,$*&*$),&*&%$k0|irGF,%#t|irGF,\"\"'%\"xG !\"\"\"\"#F,F,,&F,F,*&\"$w&F,)F7F;F,F,F:!\"#F,,&*&-%$cosG6#*&,(*&F7F,) F9F;F,!#[*(\"\"$F,)F6F;F,F7F,F,*&F6F,F9F,F:F,F " 0 "" {MPLTEXT 1 0 31 "epaks:=simplify(evalf(repaks ));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&epaksG,$*&*&-%$expG6#,$*&*$) ,&*&%$k0|irG\"\"\"%#t|irGF2$\"\"'\"\"!*&$F2F6F2%\"xGF2!\"\"\"\"#F2F2,& $F2F6F2*&$\"$w&F6F2)F3F;F2F2F:$!\"#F6F2,&*&-%$cosG6#*&,(*&F3F2)F9F;F2$ !#[F6*($\"\"$F6F2)F1F;F2F3F2F2*($F2F6F2F1F2F9F2F:F2F " 0 "" {MPLTEXT 1 0 20 "# Anna hmen entfernen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "epaks := 3.544907702*exp(-2.*(6.*k0*t-1.*x)^2/(1.+576.*t^2))*(cos((-48.*t*x^2+ 3.*k0^2*t-1.*k0*x)/(1.+576.*t^2))*sqrt(sqrt(1.+576.*t^2)-1.)+sin((-48. *t*x^2+3.*k0^2*t-1.*k0*x)/(1.+576.*t^2))*sqrt(sqrt(1.+576.*t^2)+1.))/( sqrt(1.+576.*t^2));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&epaksG,$*&*& -%$expG6#,$*&*$),&%\"tG$\"#C\"\"!*&$\"\"\"F3F6%\"xGF6!\"\"\"\"#F6F6,&$ F6F3F6*&$\"$w&F3F6)F0F9F6F6F8$!\"#F3F6,&*&-%$cosG6#*&,(*&F0F6)F7F9F6$! #[F3*&$\"#[F3F6F0F6F6*&$\"\"%F3F6F7F6F8F6F:F8F6-%%sqrtG6#,&*$-FT6#F:F6 F6$F6F3F8F6F6*&-%$sinGFFF6-FT6#,&FWF6F;F6F6F6F6F6*$-FT6#F:F6F8$\"+-x! \\a$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Und jetzt loh nt sich die Geduld wirklich ... es sind ja nur ein paar Minuten" }} {PARA 0 "" 0 "" {TEXT -1 44 "(Die underflow-Fehlermeldung ist unkritis ch)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "animate(epaks,x=-20. .20,t=-1..1,numpoints=200);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "#csgn(-I-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#epaks;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 476 "Die Anteile mit kleinen Wellenl\344ngen bewegen \+ sich schneller als die mit gro\337en Wellenl\344ngen. Man spricht imme r vom *Zerflie\337en* eines Paketes durch Dispersion, weil man darauf \+ fixiert ist, vorauszuberechnen. Aber wenn Sie die Animation laufen las sen, *sehen* Sie, da\337 sich das Paket auch *bildet*. Im Gegensatz zu m Wellenpaket \"der Wellengleichung\", dessen Bild sich wie ein starre r K\366rper gleichf\366rmig bewegt, entsteht und vergeht hier Informat ion, weil die Koh\344renz fehlt. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "Dreidimensional:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "t:='t':s:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "apaks;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&-%$exp G6#,$*&*$),&%\"tG\"#C%\"xG!\"\"\"\"#\"\"\"F3,&F3F3*&\"$w&F3)F.F2F3F3F1 !\"%F3%#PiGF3F3*$-%%sqrtG6#F4F3F1\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot3d(apaks,x=-40..40,t=-1..1,grid=[50,50],style=hid den,axes=framed);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "(Ve rwenden Sie auch den contour-style im letzten Plot.)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "Die Geschwindigkeit, m it der die Information verlorengeht, h\344ngt davon ab, wie genau man \+ das Paket lokalisieren m\366chte: je geringer die Paketbreite 1/s, des to schneller das Zerflie\337en -- aber auch das Entstehen. Zun\344chst eine Momentaufnahme" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "apak s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**-%$expG6#,$*&*&)%\"sG\"\"# \"\"\"),&$\"#C!\"\"F/%\"xGF4F.F/F/,&F/F/*&$\"#O!\"#F/)F-\"\"%F/F/F4F4F /%#PiGF/-%%csgnG6#*$)-%*conjugateG6#F-F.F/F/F,F/F/*$-%%sqrtG6#F6F/F4F. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "s:='s':t:=0.1:x:='x': # f\374r x=-40..40 'exponent too large'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot3d(evalf(apaks),x=-20..20,s=0.1..2,grid=[50,50],s tyle=hidden,axes=framed);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "#eval(apaks,\{s=0.1,x=40\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "#s:='s':x:=30:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "#plot(evalf(apaks),s=0.1..2,axes=framed);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Und die Anima tion" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "s:='s':t:='t':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "animate3d(apaks,x=-20..20,s=0.1..2, t=-1..1,axes=framed,style=hidden);#grid=[50,50]);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "apaks;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&**-%$expG6#,$*&*&)%\"sG\"\"#\"\"\" ),&%\"tG\"#C%\"xG!\"\"F.F/F/,&F/F/*(\"#OF/)F2F.F/)F-\"\"%F/F/F5F5F/%#P iGF/-%%csgnG6#*$)-%*conjugateG6#F-F.F/F/F,F/F/*$-%%sqrtG6#F6F/F5F." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# auch hier wegen csgn expo nent too large f\374r frames=20 (t=0?)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "F\374r kleine s (etwa 0.4) beh\344lt das Paket seine Form fast bei. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Anm.: Die Gruppengeschwind igkeit h\344ngt nur von k0 ab. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni-tuebingen.de" }}}}{MARK "0 2 0" 19 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }