{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "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 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Helve tica" 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 {EXCHG {PARA 258 "" 0 "" {TEXT 256 25 "Moderne Physik mit Mapl e " }}{PARA 258 "" 0 "" {TEXT 257 9 "PDF-Buch " }{URLLINK 17 "Moderne \+ Physik mit Maple" 4 "http://mikomma.de/fh/modphys.pdf" "" }}{PARA 258 "" 0 "" {TEXT -1 19 "Update auf Maple 10" }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 13 "Kapitel 3.3.1" }}{PARA 258 " " 0 "" {TEXT -1 24 "Worksheet wellen4_10.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "c International Thomson Publishing Bonn 1995 filename: wellen4.ms" }}{PARA 0 "" 0 "" {TEXT -1 102 "Autor: Komma \+ Datum: 5.9.94" }} {PARA 0 "" 0 "" {TEXT -1 52 "Thema: Interferenz, Standardbeispiele in \+ neuem Licht" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 191 "[Interferenz,Spalt,Doppelspalt,Gitter, Vielstrahlinterferenz,N ahzone, Fernzone,`Aufl\366sung`,Polarplot,`Intensit\344tsverteilung`], [limit,polarplot,cylinderplot],[`Superposition und Interferenz`];" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 408 "Wir haben gesehen, da\337 Kugelwellen (Kreiswe llen) die elementaren Bausteine der Wellenphysik sind. Mit ihnen l\344 \337t sich jede Wellenfront aufbauen. In diesem Worksheet werden sie d azu ben\374tzt, die Standardthemen \"Doppelspalt\", \"Vielstrahlinterf erenz\" und \"Spalt\" zu behandeln. Wir werden diese Themen aber nicht nur pflichtgem\344\337 abhandeln, sondern unser CAS so einsetzen, da \337 dabei neue Aspekte sichtbar werden." }}{PARA 0 "" 0 "" {TEXT -1 13 "=============" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Doppelspalt" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "Die Phasen der beiden Wellen lassen sich so schrei ben:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "alpha:=k*r1-omega*t; \+ beta:=k*r2-omega*t;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 320 "Mit r[i] \+ als Abstand des Aufpunktes zum Zentrum 1 bzw. 2. Wenn wir sinusf\366rm ige Wellen ansetzen (die in der Fernzone nicht merklich abklingen), da nn k\366nnen wir die Summe der beiden Wellen auch als Produkt schreibe n, wobei der Faktor \"am\" f\374r die ortsabh\344ngige Amplitude steht und der Faktor \"pha\" f\374r die Phasenfunktion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "am:=2*cos((alpha-beta)/2); pha:=sin((alpha+ beta)/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Zentren bei x1 und x 2 auf der x-Achse:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "r1:=s qrt((x-x1)^2+y^2); r2:=sqrt((x-x2)^2+y^2); " }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 15 "Zahlenbeispiel:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "k:=2*Pi/4: x1:=5: x2:=-5:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Amplitudenquadrat" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot3d(am^2,x=-20..20,y=-20..20,axes=boxed,orientation=[40,7],numpoin ts=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Es entstehen also die bekannten Interferenzhyperbeln k*(r1-r2)= const" }}{PARA 0 "" 0 "" {TEXT -1 152 "Ebenso k\366nnen wir die Kurven gleicher Phase darstellen, also die Wellenfronten k*(r1+r2)=const. Da s ist die zur Hyperbelschar orthogonale Ellipsenschar:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot3d(pha,x=-10..10,y=-10..10,axes=boxed,orientation =[40,7],numpoints=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Mit geeigneter Orientierung \+ des Plots kann man sich viel Rechenzeit sparen und bekommt vor allem b esseren Einblick in die T\344ler." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 32 "Demonstration der Orthogonalit\344t" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "cp1:=contourplot(am^2,x=-20. .20,y=-20..20,axes=boxed,numpoints=5000," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "scaling=constrained,contours=4,color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=0:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "cp2:=contourplot(pha,x=-20..20,y=-20..20,axes= boxed,numpoints=5000," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "scaling=co nstrained,contours=4,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "cp1;cp2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(\{cp1,cp2\});#, orientation=[-90,0]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "Wenn di e Interferenzhyperbeln nicht w\344ren, w\374rde von den beiden Zentren aus eine elliptische Welle mit richtungsunabh\344ngiger Amplitude lau fen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "t:='t': omega:=2: n :=20:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "animate3d(pha,x=-1 0..10,y=-10..10,t=0..2*Pi/omega*(1-1/n),axes=boxed,orientation=[40,7], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "style=hidden,color=blue,frames= n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Aber wir m\374ssen ja das \+ Produkt bilden:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "t:=0:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "plot3d(am*pha,x=-10..10,y=- 10..10,axes=boxed,orientation=[40,7]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "#plot3d(\{am,pha+20\},x=-5..5,y=-5..5,axes=boxed,orie ntation=[40,45],numpoints=1000);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Wo liegen die Minima?" }}{PARA 0 "" 0 "" {TEXT -1 37 "In der Beweg ung sieht man sie besser:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "t:='t': # omega:=2: n:=20: x1:=5: x2:=-5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "animate3d(am*pha,x=-10..10,y=-10..10,t=0..2*Pi/o mega*(1-1/n),axes=boxed,orientation=[40,7]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "style=hidden,color=blue,frames=n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "... wenn man sich etwas M\374he gibt. Man kann sich die Suche erleichtern, wenn man den Amplitudenfaktor einblendet: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "animate3d(\{am*pha,am\} ,x=-10..10,y=-10..10,t=0..2*Pi/omega*(1-1/n),axes=boxed,orientation=[4 0,7]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "style=hidden,color=blue,fr ames=n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Untersuchen Sie: Was \+ \344ndert sich mit x1, x2, k, omega?" }}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Vielstrahlinterferen z" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 199 "Auf der x-Achse seien n Zentren \344quidistant mit der Gitterkonstanten g angeordnet. Im Abstand b zur x-Achse soll die Intensit\344tsverteilun g l\344ngs einer zur x-Achse parallelen Geraden bestimmt werden. " }} {PARA 0 "" 0 "" {TEXT -1 250 "Neben der Erh\366hung der Zahl der Zentr en gibt es noch die zwei Aspekte \"Nahzone\" und \"Fernzone\". Wir bes ch\344ftigen uns zun\344chst mit der Intensit\344tsverteilung in der N ahzone (was nicht hei\337t, da\337 b im Folgenden nicht beliebig gro \337 gew\344hlt werden kann). " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rj:=sqrt((x-xj)^2+b^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xj:=g*j-(n+1)/2*g;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "phj:=k*rj;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#Ac:=(sum(cos (phj),j=1..n))^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=sum (exp(I*phj),j=1..n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "(Anm.1: \+ Die Summe mu\337 nicht mit indizierten Variablen gebildet werden. Anm. 2: Wenn man nur Cosinuswellen aufsummiert, bekommt man nicht das zeitu nabh\344ngige Amplitudenquadrat)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=2*Pi/lambda;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "n-loop:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "n:=4: # muss \+ vor evalc stehen, und evalc wird fuer plot benoetigt" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g:='g': lambda:='lambda': b:='b':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "intens:=evalc(abs(A))^2:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "#reint:=evalc(Re(A)):# verg l. Anm. zu Cosinuswellen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "#A;intens;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Parameter-loop" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b:=5: lambda:=2: g:=5:# b:=0 zur Ko ntrolle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "plot(intens,x=-1 5..15);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Wenn Sie mit den Para metern und der Zahl der Zentren spielen, werden Sie einige \334berrasc hungen erleben ..." }}{PARA 0 "" 0 "" {TEXT -1 127 "Wie anders als mit einem CAS k\366nnte man mit dieser Geschwindigkeit und dieser Pr\344z ision die Antwort auf solche Fragen erhalten?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "#plot(reint,x=0..300); # Cosinuswellen" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "A ber mit der Intensit\344tsverteilung l\344ngs einer Geraden sind die M \366glichkeiten von Maple nat\374rlich nicht ersch\366pft." }}{PARA 0 "" 0 "" {TEXT -1 111 "Richtungsverteilungen sind fast noch informative r. Man beachte die \304nderung der Strahlungscharakteristik mit r!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Wie \344 ndert sich also die Intensit\344t, wenn man auf einem Kreis mit Radius r um die Zentren heruml\344uft?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Abstand zum Zentrum rj" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "rj:=sqrt((r*cos(phi)-xj)^2+(r*sin(phi))^2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Position der Zentren, zugeh\366rige Phase" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "xj:=g*j-(n+1)/2*g;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "phj:=k*rj;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Summe der Amplituden" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "A:=sum(exp(I*phj),j=1..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "k:=2*Pi/lambda;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "n-loop: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n:=4: g:='g':lambda:='l ambda':r:='r':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "intens:=e valc(abs(A))^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "#reint:= evalc(Re(A)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Parameter-loop: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g:=5: lambda:=4:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "r-loop:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "r:=8: # Octopussy?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "polarplot(intens,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#r:= 'r':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "#plot3d([r*cos(phi) ,r*sin(phi),intens],r=0..30,phi=0..2*Pi,axes=boxed,grid=[20,100]);" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Und f\374r den totalen \334berbl ick kann man die Richtungsverteilungen in einem Zylinderplot \374berei nander schichten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r:='r' : #Nautilus!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "cylinderplo t(intens,phi=0..Pi,r=0.1..4,axes=boxed,grid=[100,10],style=hidden);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "(Nach r = 4 geht es noch weiter \+ und weiter und ...)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "\334bergang zur Fernzone. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 308 "Wir betrachten nach wie vor punktf\366rmige Zentren oder \326f fnungen. Ihre Anzahl sei p. In der Fernzone haben wir parallele Strahl en, was die Berechnung der Intensit\344tsverteilung stark vereinfacht. Je zwei benachbarte Strahlen haben die gleiche Phasendifferenz d, die in den Plots als unabh\344ngige Variable dient." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=sum(exp(I*k*d),k=0..p-1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Die Summe kann in diesem Fall (im Gegensatz zur \+ Nahzone) kompakt dargestellt werden und Maple macht das auch (geometri sche Reihe):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A:=simplify (A);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Ein erfreuliches Ergebnis . Wie sieht es aus?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:='p ':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "intensg:=evalc(abs(A) )^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#rea:=evalc(Re(A)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(intensg,d=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Wie \344ndert sich die Intensit\344tsverteilung mit der Anzahl \+ p der Strahlen?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "p:='p':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(\{seq(intensg/p, p=2 ..6)\},d=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 68 "Einzelspalt endlicher Breite (= oo viele \+ Zentren auf endlichem Raum)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 501 "Die Intensit\344tsverteilung am Einzelspalt er h\344lt man nun, wenn man auf der Spaltbreite b, eine wachsende Anzahl p von Strahlenb\374ndeln endlicher Breite beta=b/p unterbringt. Sinnv ollerweise gibt man diesen Strahlenb\374ndeln die Amplitude beta. Mit \+ der Phasendifferenz Delta der Randstrahlen, bzw. delta = Delta/p f \374r benachbarte Strahlenb\374ndel, kann die Intensit\344tsverteilung durch Aufsummieren der Amplituden der parallelen Strahlen bestimmt w erden, wenn man den Grenzwert f\374r p gegen Unendlich bildet." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b:='b':p:='p':Delta:='Delta' :delta:='delta':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "beta:=b /p;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "delta:=Delta/p;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "As:=beta*sum(exp(I*k*delta), k=0..p-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(As); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Grenzwert" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "Asl:=limit(As,p=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Betragsquadrat" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Asq:=evalc(abs(Asl))^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Asq:=simplify(Asq);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 235 "Also Minima f\374r eine Phasendifferenz Delta=k*2*Pi oder eine n Gangunterschied von k*lambda f\374r die Randstrahlen. Eine g\344ngig ere Version ist (Mit Delta = 2*Pi*b/lambda*sin(phi). Es ist aber f\374 r das Folgende bequemer Delta beizubehalten):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Asd:=subs(cos(Delta)-1=-2*sin(Delta/2)^2,Asq);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "b:=10: Delta:='Delta':" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(Asd,Delta=-15..15);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 434 "Schlie\337lich k\366nnen wir die Strahlenb\374ndel aus p Einzelspalten zur Interferenz bringen (p hat jetzt wieder die gleic he Bedeutung wie beim Gitter). Die \326ffnungen seien mit der Gitterko nstanten g angeordnet. Von oben haben wir noch die Intensit\344tsverte ilung \"intensg\" f\374r punktf\366rmige \326ffnungen zur Verf\374gung , die wir nur mit der Intensit\344tsverteilung des Einzelspaltes multi plizieren m\374ssen. Es gibt zwei M\366glichkeiten der Darstellung: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "1.) Un abh\344ngige Variable ist die Phasendifferenz d der Zentralstrahlen zw eier \326ffnungen. Dann gilt f\374r Delta (Strahlensatz):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d:='d': b:='b':g:='g':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Delta:=b*d/g;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "b:=2: g:=4: p:=4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "plot(\{Asd/b,Asd*intensg/b/p^2\},d=-30..30,numpoint s=500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plot(\{Asd/b,As d*intensg/b/p^2,intensg/p^2\},d=-30..30,numpoints=500); # Einblenden d er \"Gitteverteilung\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Was geschieht, wenn man die Bre ite der \326ffnungen \344ndert?" }}{PARA 0 "" 0 "" {TEXT -1 82 "Simula tion der simultanen \304nderung der Breite der p \326ffnungen (default : frames=16)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:='b':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "animate(\{Asd/b,Asd*intensg/ b/p^2\},d=-100..100,b=g/10..g,numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "===================================" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Aufgabe: \+ Polarplot der Intensit\344tsverteilung." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lambda:='lambda':b:='b ': p:='p':g:='g':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "d:=2*P i/lambda*g*sin(phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "#As d;intensg;Delta;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "lambda: =5:g:=11: b:=3:p:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "pol arplot(Asd*intensg/b/p^2,phi=0..2*Pi,numpoints=500,scaling=constrained );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Zur Animation wird mit 1/b^ 2 normiert, um die Amplitude konstant zu halten. (Bitte warten)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:='b':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "animate(\{[Asd*intensg/b^2/p^2,phi,phi=0..2*P i]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "[Asd/b^2,phi,phi=0..2*Pi]\}, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "b=g/10..g,numpoints=200,scaling =constrained,coords=polar);" }}}{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 70 "2.) \+ Zweite M\366glichkeit der Darstellung. Unabh\344ngige Variable ist Del ta" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "b:='b':g:='g':Delta:= 'Delta':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "d:=Delta*g/b;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "b:=2: g:=4: p:=2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot(\{Asd/b,Asd*intensg/b/p ^2\},Delta=-15..15,numpoints=200);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Simulation de r simultanen \304nderung der Abst\344nde der p \326ffnungen" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:='g':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "animate(\{Asd/b,Asd*intensg/b/p^2\},Delta=-15 ..15,g=b..10*b,numpoints=500);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "In Polarkoordinaten:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "lambda:='lambda':b:='b': \+ p:='p':g:='g':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Delta:=2* Pi/lambda*b*sin(phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "la mbda:=5:g:=11: b:=3:p:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "polarplot(Asd*intensg/b/p^2,phi=0..2*Pi,numpoints=500,scaling=cons trained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Animation (Bitte warten)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "g:='g':" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "animate(\{[Asd*intensg/b^2/p^2,phi,phi=0..2*Pi]," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "[Asd/b^2,phi,phi=0..2*Pi]\}," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "g=b..b*10,numpoints=200,scaling=con strained,coords=polar);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Und das Schirmbild auf dem Bild schirm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 66 "##### Densityplot in R5 und Maple 6 nicht mehr zu ge brauchen......" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "g:=15:" }} }{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 122 "#densityplot((Asd*intensg/b /p^2),phi=-Pi/2..Pi/2,y=0..0.25,axes=none,grid=[1000,2],style=patchnog rid,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "plot3d((Asd*intensg/b/p^2),phi=-Pi/2..Pi/2,y=0..1,axes=none,grid= [1000,2],style=patchnogrid,orientation=[90,0],shading=ZGREYSCALE);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 81 "#plot((Asd*intensg/b/p^2),phi=-Pi/2..Pi/2,style=pat chnogrid,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 31 "### Release 5 ver tauscht Achsen" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 111 "#densityplot((Asd *intensg/b/p^2),etwas=0..5,phi=-Pi/2..Pi/2,grid=[30,3],style=patchnogr id,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 118 "#densityplot((Asd*intensg/b/p^2),y=0..1,phi=-Pi/2..Pi/2,axes=none,gri d=[200,2],style=patchnogrid,scaling=constrained);" }}}{EXCHG {PARA 0 " " 0 "" {MPLTEXT 0 21 101 "#densityplot((Asd*intensg/b/p^2),phi=-Pi/2.. Pi/2,y=0..0.25,axes=none,grid=[1000,2],style=patchnogrid," }}{PARA 0 " " 0 "" {MPLTEXT 0 21 16 "colorstyle=HUE);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Aber bitte \+ in Farbe!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot3d((Asd*intensg/b/p^2),phi=-Pi /2..Pi/2,y=0..0.25,axes=none,grid=[1000,2],style=patchnogrid,shading=Z HUE,orientation=[90,0]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "#plot3d(ln(Asd*intensg /b/p^2+1),phi=-Pi/2..Pi/2,y=0..0.25,axes=none,grid=[1000,2],style=patc hnogrid,color=phi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "Ka nn man auch eine realistische Einf\344rbung erreichen?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Im density-Plot nicht. Also R\374ckgriff auf plot3d:" }} {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda:=5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "rot:=plot3d((Asd*intensg/b/p^2),ph i=-Pi/2..Pi/2,y=0..0.25,axes=none,grid=[500,2],style=patchnogrid,color =[Asd*intensg/b/p^2,0,0],orientation=[90,0]): rot;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "lambda:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "blau:=pl ot3d((Asd*intensg/b/p^2),phi=-Pi/2..Pi/2,y=0..0.25,axes=none,grid=[500 ,2],style=patchnogrid," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "color=[0, 0,Asd*intensg/b/p^2],orientation=[90,0]): blau;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "display(\{rot,blau\});" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 " lambda:='lambda':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#Asd;i ntensg;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "licht:=Asd*inten sg/b/p^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "lrot:='lrot': \+ lblau:='lblau':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rotlicht :=subs(lambda=lrot,licht);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "blaulicht:=subs(lambda=lblau,licht);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "lrot:=5: lblau:=3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(rotlicht+blaulicht,phi=-Pi/2..Pi/2,y=0..0.25,a xes=none,grid=[500,2],style=patchnogrid," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "color=[rotlicht,0,blaulicht],orientation=[90,0]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni-tuebingen.de" }}}}{MARK "0 5 0" 17 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }