From 68eb64b51fcbcd64e4d554896f4535187d1792d3 Mon Sep 17 00:00:00 2001 From: Jonas Hahn Date: Thu, 6 Nov 2025 02:45:01 +0100 Subject: [PATCH] auto up 02:45:00 up 0:00, 2 users, load average: 1.03, 0.24, 0.08 --- S3/ExPhyIII/VL/ExIIIVL2.typ | 235 ++++++++++++++++++ .../drawing-2025-11-05-11-18-39.rnote | Bin 0 -> 12978 bytes .../drawing-2025-11-05-11-18-39.rnote.svg | 4 + book.typ | 1 + 4 files changed, 240 insertions(+) create mode 100644 S3/ExPhyIII/VL/ExIIIVL2.typ create mode 100644 S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote create mode 100644 S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote.svg diff --git a/S3/ExPhyIII/VL/ExIIIVL2.typ b/S3/ExPhyIII/VL/ExIIIVL2.typ new file mode 100644 index 0000000..8afdf49 --- /dev/null +++ b/S3/ExPhyIII/VL/ExIIIVL2.typ @@ -0,0 +1,235 @@ +// Main VL template +#import "../preamble.typ": * + +// Fix theorems to be shown the right way in this document +#import "@preview/ctheorems:1.1.3": * +#show: thmrules + +// Main settings call +#show: conf.with( + // May add more flags here in the future + num: 2, + type: 0, // 0 normal, 1 exercise + date: datetime.today().display(), + //date: datetime( + // year: 2025, + // month: 5, + // day: 1, + //).display(), +) + += Uebersicht + +Wenn man N Oszillatoren koppelt, erhlaet man ein System mit wieviel Eigenfrequenzen? + +gekoppelte Oszillatoren und Eigenmoden + +$ + psi_(n) (t) = x_(n) exp(i omega t) \ + ==> m/K (dif psi_(n) ) / (dif t) = psi_(n + 1) - 2 psi_n + psi_(n - 1) \ + ==> - omega^2 m/k x_n = x_(n + 1) - 2 x_n + x_(n - 1). +$ + +Dabei wird unter festen und periodischen Raendern unterschieden. + +- Randbedinungen selektonieren die Loesungen $omega''$ als Eigenwertproblem +- Matrizen symmetrisch $==>$ $N$ Eigenwerte + +$ + partial _(t) ^2 overline(psi) = underline(M) overline(psi) \ + vec(psi_1 (t), psi_2 (t), ..., psi_(n) (t)) = overline(psi) (t) = sum _(j = 1) ^(N) underbrace(A_(j) e^(i omega_(j) t), Q_(j) \ "Normalmoden") arrow(e)_(j) + B_(j) e^(- i omega_(j) t) arrow(e)_(j) \ + arrow(psi) (t) = sum _(i = 1) ^(N) Q_(j) (t) arrow(e)_(j) \ + arrow(e)_(j) = e ^(i k a n) hat(e)_(z) .. k = (2 pi)/ lambda. +$ + +Die Dispersionsrelation ist gegeben durch +$ + omega (k) = sqrt((2 k)/m) (1 - cos (k n))^(1/2). +$ +Und in linearer Form als +$ + omega = c k. +$ +Die erste Brillouin-Zone gibt an, welche Wellen sich ausbreiten koennen. Wie kann man dort sehen, dass dies diskret ist. + +Wir sagen +$ + psi_1 = psi_(n + 1) \ + e ^( i k n) = e ^( i k (n + 1)a) \ + e ^(i k n a) = 1 \ + ==> 2 n a = 2 pi i \ + k = (2 pi)/a i/n +$ + +Gegenueberstellung von Welle und Schwingung + +Notationen +$ + psi (t) = psi_0 sin (omega t + phi) \ + psi (x, t) = psi_0 sin (k x - omega t + phi) \ + psi (r, t) = psi_0 sin (k * r - omega t + phi). +$ +Relationen +$ + v = phi / (2 pi) \ + T = 1/nu \ + lambda = (2 pi)/k \ + I prop abs(psi (x, t))^2. +$ + += Die Wellengleichung + +$ + (diff^2 psi) / (diff t^2 ) = c^2 (diff^2 psi) / (diff x^2 ) \ + dot.double(psi) = c ^2 psi'' +$ + +$ + psi (x, t) = psi_0 sin (k x - omega t) \ + partial _(t) ^2 psi = omega^2 psi_0 sin (k x - omega t) \ + omega/k = c = lambda nu \ + partial _(x) ^2 psi = k^2 psi_0 sin (k x - omega t) +$ + +Als homogene DGL +$ + dot.double(psi) - c^2 psi'' = underbrace(0, "Quelle"). +$ +Die allgemeine Loesung ist +$ + psi (x, t) = f_(1) (x + c t) + f_2 (x - c t) , space f_1, f_2 in C^2 (RR). +$ +In mehereren Dimenisonen +$ + dot.double(psi) - c^2 arrow(nabla) ^2 psi = 0 , space psi = psi (arrow(r), t). +$ + +Kann ein Sperarationsansatz auch Loesungen liefern? + +$ + psi (arrow(r), t) = u (arrow(r)) e ^(i omega t) \ + u (arrow(r)) ( - omega ^2 ) e ^( i omega t) = c ^2 e ^(i omega t) arrow(nabla) ^2 u (arrow(r)) \ + ==> arrow(nabla) ^2 u + k^2 u = 0 , space "stationaere Wellengleichung (Helmholtzgleichung)". +$ + +Die ruecklaufende Welle wird benoetigt um alle Randbedinungen abzudecken. + +Wellen breiten sich mit einer linearen Superposition aus. + +$ + xi (x, t) = sin (x +- v t) = sin (k x - omega t) +$ + += Skalare Wellen und Vektorwellen + +$ + psi (arrow(r), t) = arrow(A) e ^(i (arrow(k) arrow(r) - omega t)) +$ + +Abhaengig von $arrow(A)$ ist die Welle entweder eine Skalarewelle oder eine Vektorwelle. + +$ + arrow(k) * arrow(r) = phi , space "Normalendarstellung einer Ebene" \ + psi (arrow(r), t) = A e ^(i (arrow(k) arrow(r) - omega t)) , space "ebene Welle" \ + psi (arrow(r), t) = A e ^(i (k r - omega t)) . + +$ + +== Kugelwellen + +Wir betrachten Kugelkoordinaten. + +Betrachte eine Loesung +$ + psi (r, theta, phi, t) = psi (r, t) \ + dot.double(psi) = c^2 arrow(nabla) ^2 psi \ + arrow(nabla) ^2 psi , space "fuer kugelsymmetrisen Fall" \ + arrow(nabla) ^2 psi = 1/r diff / (diff r^2 ) (r psi) \ + ==> dot.double(psi)= c^2 1/r diff / (diff r^2 ) (r psi) \ + diff / (diff t^2 ) (r psi) = c^2 diff / (diff r^2 ) (r psi) ==> psi (r, t) = cases( + 1/r f (r - c t) \, space "auslaufen" , 1/r f(r + c t) \, space "" + )\ +$ + +== Zylinderwellen + +$ + psi (rho, t) = A 1/sqrt(rho) e ^(i (k s +- c t)) \ + dot.double(psi ) = e ^2 r 1/rho diff / (diff rho) (rho (diff psi) / (diff rho) ) \, space "loesung durch Besselfunktion" +$ + +Energiedichte +$ + I prop abs(psi)^2 \ + abs(psi)^2 prop 1/r^2 \, space "Energiesatz" \ + I prop E^2 +$ + += Stehende Welle + +Nur bestimmte Frequenzen erlauben fuer eine stehende Welle. + +#figure( + image("typst-assets/drawing-2025-11-05-11-18-39.rnote.svg"), +) + +$ + psi (x, t) = A_0 sin ( - k x - omega t) + A_(r) sin (k x - omega t) +$ +Was ist die Randbedinungen bei $x = 0$? + +$ + psi (x = 0, t) = - A_0 sin (omega t) - A_(r) sin (omega t) = ^(!) 0 \ + ==> A_(0) = - A_(r) = A \, space "Vorzeichenwechsel der Amplitude oder einen Phasensprung" \ + psi (x, t) = A [sin ( - k x - omega t ) + sin ( - k x + omega t)] \ + sin alpha + sin beta = 2 sin ((alpha + beta)/2) cos ((alpha - beta)/2) \ + ==> psi (x, t) = - 2 A sin (k x) cos (omega t) ==> "zeitliche und raeumliche Abhaengigkeit werden separiert". +$ + += Resonator + +Ein Resonator wird erzeugt, wenn zwei Waende gegenuebergestellt werden. Das heisst es gibt Randbedinungen an beiden Enden. + +Eigenwertproblem +$ + L = n lambda/2 \, space n in NN \ + lambda_(n) = (2 L ) / ( n) \, space k_(n) L = n pi \, space c = lambda nu = nu (2 L) / (n) \ + ==> Delta y = (c) / (2 L). +$ +Falls ein Ende offen ist, dann +$ + L = (2 n + 1) lambda/4. +$ + +$ + psi (x, t) = u (x) f (t) \ + u'' + k^2 u = 0. +$ + += Schwingung einer rechteckigen Membran + +Die Kantenlaengen sind $a, b$. Es ergibt sich +$ + arrow(nabla) ^2 u = - k ^2 u \ + u_(n, m) = u_0 sin ((n pi x) / (a)) sin ((m pi y)/b)) \, space n, m in NN \ + ==> k ^2 _(n, m) = pi^2 (n^2 /a^2 + m^2 /b^2 ) \ + omega ^2 _(n, m) = c^2 pi^2 (n^2 /a^2 + m^2 /b^2 ) ==> "Raumfrequenzen" +$ + += Schallwellen + +Starten mit der idealen Gasgleichung. Es schwingt der Druck +$ + p (x, t) = p + tilde(p) (x, t) \ + rho (x, t) = rho_0 + tilde(rho) (x, t) +$ +und die Auslenkung mit der Geschwindigkeit +$ + xi (x, t) \ + v (x, t) = dot(xi). +$ + +$ + A Delta x rho (diff v) / (diff t) = ^("Taylor") - A (diff p) / (diff x) Delta x \ + (diff v) / (diff t) = - 1/rho (diff p) / (diff x) +$ diff --git a/S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote b/S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote new file mode 100644 index 0000000000000000000000000000000000000000..223508ef6489c94ee92da6152585b5fda4462cbc GIT binary patch literal 12978 zcmV;jGEL1NiwFP!00000|LvXavYoeWw$DYsch~R>1nE=hXgrFo#GcBMN0Qs78J+u! z53a4P+EsMl&pUBb`A_87x)w+f{^bYzo3*xfllJiJ&u`xT^diu|`&YmJ`NwD9?CC%I=IxIr@WuOV zZ@nE$?dSdy|1{P9PW+R|!!zqqy_F9S z&#Wu|@olp2o_+KF?JqCSzcWGB-*10>{r!*My?F1x>HpD)KfUx1WFY;;(OhF{M5G^N;`e^85Tz>xzxmw?Dsm_oFM!I{4=ouikklzI*Z0%eOJC zZ~pw}cQ1|m*MAv$n|toMc5Ah-KIg-~%yR3ilu}1+qxIFSai_i2{-!mdmXdA$uQzXg zdiG7bwJu|oF=fua#&R9MfBWX&URq~<`q_H^`n|Q|U;p;(-(S4?#eV1YFRxxbe6%~$ z)Z>5p+p|Bt{pH;ciTnRGwfx|JzI*@TZCH-=%v&<$R)#z*>wviS-oAYI?w7YOU034r zU;gE%N3pNWh~K|SgZ}JRdtj%-fqs5siEqnkmSKCETaVBu(;j?SVuTKdD< zCcJ~_xb3wa9aC{s#nIgaN9V(~#bx8*2yzM0++H7cz4g{`w5p@J2&pzS+t$T__jRr1 zsMd<3!y4Rg((wkbW(^*_P6$u-4-08@5HWqKsS#R;7`wZDE=Dci1**bPS={SvSq<8Z^5H%eGsk z$xf3}c^2QgePs7y&C!zXg#b8G{T&}fvH1l5Kno@U$4BPEDLaOhM zjNKi#$W?5FhWixpMPvCa8fKY;V4KZ^F>iI_Bec4YkP;4N$=$E9(kjjFS;;n=_op*# z3w8<9Uqj6w*@YM)4VGBvR+B(gH^Giz_hOoC-4(koo`hAk7HqGRU~8~Lm1fOJPO`RE z>7-dD*4JdshqEiq4rw&2^ccA>hV5_1-FqL=RoUSkHhVYCjxlL=ZMMz!>8DM@KP*DB z_Ce83vt!XHv)RA1i}`LJc5ikT*)gyQ%?{Z#D`nEG(*0exN|SvJnw8yYRy*I@Re8Bf zCeP})z^)6dTwrY7JqX=C(5o&e~@+6w+X=qkC zp;_ewnLQ2s`ty&kUVYJDFBvCxTK4UCzr6i-Yfkj(#oK2eyoVnHrK)!q4V3cli)}Ec z2lxH@GxLR!QlI~k&kn;=bmHjThaCjF4ojdj(Mt5zZ;^v#;N zG7rnh?a3}iZapj^ccfJnCFBl0EF)Kvott2ft|l=P#?h4!a{JMh5OVvmjlswrhh^lp z!xCtc=t17mc4iS_80{#kF=CIRh~=goJyk~RK`V~dqo_iB5bBvig^}xS3y-v6)}t4Zi;Wq;u>g5Q6wKovcTi{BVzlPBu?GqxR8 zq{3`pX&PN|x#IVI#x~TWWo(428Cx6P=J*oXu6m<4?#6E4rTB>pJw>?7p(Q&z&UXJI z7&|(QngcBlr|6}*v7+Zl46hpIuE>-4798EapNmsGIukUnWW|H;1!k-6V5KdXZNBAp zk4M4fcE5-6-+=dmCU0b$D|!{|I@04Efj(*Uz*AxSjDN@lzQe2x*Ofn)z~xTA3XZ)fjILe- zwV#05JtbJ!DOKVrN}%vnVewe{U>kgQW`84C^ZLVH@Zy1{!A>|zr^NdXX9pZ`TJ2k~ zgDq>r?AYGshR5>T{gcN98VZNldHAse3NMkJ$!iw zD-G|jmXzD!%ZBEkkznyxIA%>FuVnF1)o%A*GuR2cc8%SE3c9f}v{eljkK^$8jg>5X zL+O#;HQ2rfAvgQm;mA#A=LNQ(HWTQ*yUY-s?ruWObsctHV2S?T&59>W{e%5lu1ESg zEj3$0`TLR2|H@|vBG`Rc!P$AxN}w&-&WO!DL4l7i&gx-tRC5D_yB(b!$x?DPKFu4x z=Eu&2$iL5_2Lsf0(%+0*7q zpr^@%FS~e9sIhA|f|X>6tFv^nRy_Q`*>>6tTjV(B`0A-#2==sj;Qsg0b5FitlXP|@ z%a~+8ZDvezoHi3?C8=g*%xb4tC2k5^YHAXfSWQx_5HoO|5rW^BB4gnc6Mj|T3*AtEdkVAav&h(v`gr^%DC6}j?j{}aE@I*@^y1EJ20ID45lvz2 zfM2-#g?A4YPpDzDXQf;?uE^N#i4%t4SEKH*^dr|0;_;ZfX9ArI_s_;{uCcOlAM4K7 zCjvXP?5x*PQ?T>h-J~gYz&>}7Si~-LBXXjxr zw0Q}^_QOtst%oJXNO;sQ3Tbxz!G5*u^ABH%C#^=_3U@vHN(*1^Nc+r>e0CTD%{&rY zahFG(hn={aCgL8PCH!}FXa0)aaJj9X+!utdnjbKoRoolaj@W^FrCPlB_jg*jYpItU zBKSVgS$}2eEFtzt?YN@4W^c9Nn(8~{PNK=9W^d=fH8pw~^TfM5W=>t%1RHrxj3wNa zQqmxW@Ec0Yc_46jmoYg)a1?-}$1;ZCmLbP7M#C**9?R&&k};2E;73zWbCssi2foa$ zL@Vr#9_NeDZsBQOPS6_AF zhj@>m8_9Mw_P`xd9@JTxh{13)MzXmLd+j~?F3^R$(A53?RVZiHj?uMK0O8AujDBJU zLin;g=Y_Ei4p(wNSjw$%28m+3&+Kj$yuT68;_P79dDup<jh+{mT#kpZk^1FTck2gH{5a$yWRTQO2Fc zQI8pNv=FQxAdFp%+?7ox&e4%|&rw`awH~R0pR{T}y(OYb>PVYf9X~GBdE^pDt*S?E zWxPU1F5x(6CFLp+tJGS`RpLn7`jJZ<51SmhxSHEMVh3(k;f->(<1?Y{Be&rundblx z9w6AWL~6lx)kn(3FVo!(mU8F9O|qZGYASA$eI2=sL#iCPf}3Rc(457)JI-oFxRS?N ztl(7h&Ys0;7TlW0(Xwp3yK*MJOx&9FjL;yWBjW{+v#_4WeY4Ou?%Wx1hOkW)npw@C zbot~hn^+?yZwics*Eu_p`n_lI7j3v#X}U2N&uxe|dDSe&vEfNH^GIC$sW9%5YJMax z3r|MwFC#qwCklw*Z5BV88aoJ2*mOsqA;rJ7-v(DqlDX_9btyB+1<14Uy7cFz=H#wvP!r{Td}mi_hehRj99jC2 z3ry;0t)Y1mo$Gr`%q@8&t7`9eVXMNZSvwF84f)zp8mM{BD9Oq#6puMB0U4?3{k`>$twl2pPvhdAKw^efF{SW#Asdddbh7$4A zFgr)y9Zgux+p{bMi∓!{bx(9u*ZkYQD}l6pY?(qvd0DN71eN=(oJ=LSGZK;psI&kt;6iR_ZF2a5j2r3 zI_o{t${__N`?=$;1#NN5w*(aH@8Aa6X3uGR*VKgB$f6%Cc{KI#o0N8WY#gzVYHWAgt(8r) zvdAIww74fgg<*wmbI(dY6ls0aYAX)UPT4dYSvM(m zWe#DI9W*I((JXBk`*?~ucGO0*lO7;DBUf_0(UN)lNH+5LknEtz2skF$IVa67nswi6 zquH)Pv$JoST{O8_wE3BlW+(dz)1KK$Nwz*tp$)VBglJYUd4;p{{w+bpdZy1H8|pzZ zw5O6A>RU5ZR?d2~n3A)qIY7PPa`!l{4Dg{(Q?YIS@t}a&eVjet%|14a-goL9tYG1+ z9zPW$clXm~)MKRcR?fD!H-XtI$1)T~?Y{F^V6*mX2H1 zI1|5Af(>g{{HE2}e%MY}WH|FZ6$@^WZ6>R@*hu*15daNmOQKTnr8y&`@ch29tgq~C z!^P$$ZIt(e*;pYboC4j^8h3~0Twi#T%SyQdf3wH&MEXie>`~$_e0o&?n9cA}Pp5>p zIb*-YQ_z5{CrjMH%EULXT9X}Z!R&~;aDl@qoo$-BY6C{2T&tmVp(feybN^V0*Ig zC*Rp)lK2owIi|50cH}H71e+vo%NYqR_=l)v2}5SqH%}Iws1WFpI&oBwR3*?O6}P4g zIW=?>Xi6o<=6r#ACD0?3Fs5;&5)B?lDq&3hNF|J^AE|^fwIfvt6}KaH5o|qjH^Zh} zw8UJnCFPFFutzTO!R%*&8l7Pi9ZZw++k?LU&l!|WmQ5@)J3kmEBcX|(fFnf2ma&s18&jqWP$g8zsLEp_l?GkB@OOO;8R@T{~&%t>2>@>0>H(@)Sv zYiS8>hr7^S-TcY~zsb6z+t$O*qSTi?$*nBf7SrDIZksXTnU69Cap+p-E#%mQ-1DGA zh;Q>gStbPkz(!j9vTww++vEQ98aKv0^Es9lTvJ=gr!UU#^L`qAli_h2Cm+Xxue|zs zld;`0arU4K_nIHO*hI8QVV@!gdmdxKn?+AIoR)zvaUW9^7ueJIgdLM>v$J#a>~1t$ zE8phY`8L;iq0NhFbLP@4n?`%5QQ;fBSh5$Qy)%=2_r|c9uCz1D)JT-k(>N^bt?XGi z+u0>Jds6TAO<2^)>^g{TQQV?B+nCQ^>S_?@#NCcuX0LqG&kN%WI|&yi;p&Y1Ar()Z z?W{<)c@bOfsJ>gP*%!X6s{XQ~dXZODZna}U%I&ZNFXCv897G)LjKmhzFQH&|lw<{7 zT&qWTB|GspM?873=m?d3<8JJP-NGgB_?#u=){$}*b!1PkVVx+thxCwt-w8U44zE8o z@H=v6(JP(lr?++34YPG+n-}^GboNX@w5)-%jmr!v+(|% zU-{fX1S=O<4=*Iy>N_@;jS}P}y9dp-3wmywXX6~evYQ{HDA2;Pn;#>&Fuvjo3&~Di zRlK6qZ^TJ9a$m5l@AXJ_mkTVt;LLP*u&fN4Z9`}_;%q6_qhOR~2fb+MZFSbJ?1Gv{ zqtpq-`f-bAwQ;KQJgtSCMv>!H7R@eJthse+rCDv8X17NdQ*6s3fh614pjn}Z$Xt$m zlj9nfV!dqTqS-zs&C2RDyDBSp*hit#Y-B7Z*;Y5tD&OW_8qZ!p`-LWVUR9mkga#|Z zvvcsQUSQV+cJuumqIKH0s0WMYS+8nMO}kiV zpj`Dj0~##+La*+u@8vA|`koYn~bK(FfmcduVyE3of4FLBwv5 zb{T@^WHNTgje;$*aGjBA$FUY{LxbIVJCn#RVc<6Rv#6_|eqnYWcA}43x_tmUt-sKBTX6ZZ#Io5!O#-TQ6sU8uRd!yenBWynWPEm7(Ye z4O=1Oi%mR59^DfJC5wJ{y);S<6paQR?iTEcPE=8H2G^ejcbHwVKUwgD316T}_O_!7 zhA~mJ%Kt>)NsWENVT>IMX4f%{0hY$}DkbH#S&lBChD+AErDOU8+$@dv`-5Q&_F$-=97ibwDyil&AUPjl%LaN|pcXjTAr%S6vZ zvd+Z!^6=o~6_Xdf0k2w&#vqr>YwawW!NAxZeg&_n4G!=0q~$4hpidX3KTJto4Rm`m zeVfseMW6OhyK=pw6Gpd(KL-oWqA6MVjbd%D>l|2nFsirvW(yHs($!eCpl(;e*fa>-lExbtaI$T8wSk!6;mf! z$y9J8yF1(Fd2Y2icC$mI=akK{8Lq^zp3A1v>^L7!h80~jJ6osOc)&xk_MxZy>nn1j z*(irivQZG5WHYvlV|j<$_##4(_aF@jtT9KYSI>MAYBWn;QCWK0yA`XCj~f zmCp@CvT|6(+4ycaD+Id_+i-Rqwh{fEr_EsQ-eC3UnP{>MCzr+$b+nImF5_wm1 z!{apxM?J%?=*brZjtBZqTbff3!4|W?N}A+hx-1{fj`L=_cjaO?S82#f;Ky zhC%tOngO%3rD+J7H>ljhFuKQ!!|XnK9s~3``U9^cx1(`E#V>>oi&kVPC&!Yp4Rv=G zFOHBbd`DSM9TZ~=dS~wJ%oV3Jyv;K_2%i3O;cNt`dk!y!S&i>lvsDY%mBuIE6QkmC zSIai9j>}!yWF?v$nMs{RogT&0y3JtGLs772h|o=FJRFR}m=}W=oZX4SD~`>)cEqCT zsY}e-UOQ)HPXs;B`4{mqS@+Q;J_ze1JwcD+!QO{g@tLshJvH%nDiYihRu6#wMsG7_O%l%DKhuI7oT=0urXY-nH-P#^7JL-7@GI6;;o#n6%AKsbPtOZx|OklIP5JTNmkhT{f(={t_ZSMZ@qVvNIG&|@ ziR&n}>;5osc)e_F%pv@VjXFe2%|d0%T*a`BY(;CmNmbKcvA1{|W=@;F@zY#Fsh z($c+ceFo=W^-w=U*Z;8F`#d^ppcOU8ns-PW&-~J>r)FLc?(;zRHdWkF@!(##4wIhW zF}%rWS3M7rKDfWX4;WQDJ{Stm*{V@}X*W27k)w7YVq|XVx%bZ%>~ugJ31RFs)9S9i8Z}2j&^8=;kSOwStA6`oT(gtST6o_i>CoZ9jq5bd(ih zh&)!tSoGdI*%j<0tg33D3q9ODvhIp$G+R+`^AolSFSnP;v%eVu^8CYBaWgv2rzHN3 z`mNNjUwSk8*&q4rFa)YJYL}`sx@mMS8dV-$Ji5ug+FDR)vhiqdH0pab5|u%tQ)pBt zjV`6py;#&;fGkQ6t8bQAbnsGzHl{jhW2(xdi?+kMX*=}1oGe;tJ8X@%!`|q=4t{-^ zq#Z8aBJ!@6LU(MO8MNkEjzs(B)MR(HvN<(*!KlL7Vf8a7M|zc3Zr=F)rE#m6lynflO^{g6>Q!Y(t=ybn?w>NNpshbw$$txLK<-Mq&Z}RyWdi==n_)uh_W6e z>KV~VbZ@f3<|8dM>d&@GG-4bW)Jv9$1t-^Eyf% zQv&51Om2mvubW5Z_t5b8OoN|)_!>M)z2+4{^`}+8_M3j8OX=_Y%4f&n6IL#;i)Yd6 zv`^Rza^*sQQP=G!?1eV#g*Kys(VwuJXTgU!A6c8PrSojPfGW?n!L$8Bo9QW7;p&H< zu+lED7uq~8v{^5-SvSwF3sYh9Q-QatTKJhzD?by^=lY+ryj9hHq0Qq$o5zJV3qKR! zh|Nz}ekN4qX967W`INoTYCm)k2Z zw_jN4`nVwX!gsst1-XltJ1$JYabZ6=I={k>aY62dm3`<1xto_O7j_%cF39ct*GPn4 z>ttS#yDrGRa3q_Rmpj`9xfi}E^TG~k@+LVKFP4{zmO1-`<=05PaE8&?2bPg$F1 zd3Q>?aAs~E-$}E))JBg`^Eh;x9rWtjZZZbV&d#&St9cb(>{`5nuBI{=er9hrFmoroO$#Uw(}^xZ8ny+3}f|z+|A3a7t~y7 zceSw>P~laD=A`|E-IZor**r_{e2pFaRJbDc!o+V|Xcm5L{FG%jgDufPsLJ0ZIm;Xp zJkznWj74g;c`*LHopJ~Iyt>=IwI?fuU_G;pv6En{=T3AMKJyy8l2v$NZ=TFHSoj^5 zvBTfWdE%9cvyy0^cz2_4t+9eWqjYxFWMNOYZ+cVi90co6Fr9_nAs&%;0hMH7kMk3@ z6WyIhRh3~IKSc8kuM({1lyX)!&k|~uv}&?HAhID$f>veEU+K zy?|aY$$mvHZyCMtGXZ`s{*;}YXNB)?>lfNwFSNPvZLW)-z07vukrjUS)(fj7tt#7D z(Zf6NciT@`rA@;F;|I--%Ck+CW~+3XErT#C&wc6MBZghcF2&otV*A_SB3r@6iOkC} z;cbp`NX{0V-91p>9+MvbME$nw|U+#1!ThPI&Fs2 z2pYRfwi!;k)nl6wEWG!XELLCf!tc)JQO*mnSD4*ll8v~qou0BJSXj$mOzEHn7dv{Y zO5iBT>2VUlj*YW=Ml7xa=~4|y#oX!LDQ*j(Y8ON6}r3XKO6U&V_fk)x#>@-PTggQ}7{b{ji~tL_C@GwA9mupvKyIX2d=B(hjC{cuB*`wTX9{D zo(o;uxUR-HtP;ACTb-rWpmfqQ&Y(?*cjP;T`&cb2L#lsj4 zE$041yX&z`HCXr>t9TuCufDnQko;oT8@&hH;i6=IOK7N9$r~{0aY=^4b7-0kdwEnZ zYt~^l3eJh*2Z_i<>AkhzfaihQeBZ&s7Zt^8^LeRJ^NPXs6PIV754J3vZNAQl8yC)c zX(VR{`uxwU<3)MY?y1W`c1Pw;XFFV%b4M<~D(L- z!VlSv-7!WUswX&mYB?*M8r}|nS^X9crE*lW`mGdPZ1qpRSIkDQSDbBGk(nr0*Y3r| z(HUs0^knfu5XFvC$)d%@on5D5{f0|?coGN+cAz_}>bry)EF3=Vk7xb!IlLOygjp|l z=WO42oAtaRqzedJRKcN^~3AxM5 zFAl+PN2^~!Yp`g+`=0NAj2*-n^(g-3EWTX1dm-6I$cQ{#<#a^8ify2AJGdWGpcD!c!5?wUFUp< z1I&%x@xgUJIojQ6e(GFSE9}j1Z$d8TcZrkC>>7&8f>6&KuW0Do9y{P2($${(1c)a89SY8O0hmd zg6G2xR)THJEb{%7J3u#qx~lBvlrZM9)87k+IRzUi`~={Rs>hbFs5arye%KPyOb*6Js~AL^j_s zi{Lq`s=qc$R){f@j23DeH;s6xB>pfAMs;LxR9pB6#o+oaIJ#D?V;Bpsq+)FIt+s69 zNV7amMFh)AAlTy6?gR>*qvGEZazP1{x(g=13yWvmgju&jx7OW|R*ks{H z5W%)!RSCH<`(io}tOi>r!%DE7m_@_i%gz{Pg6+WyV^)V%Vj7Kj^kIjSr5vkvIGKxP z8@vPk$o9swqX^G3^T@}@SgAZZlxG)>!j%P{vf~1)jb^oUp6&eT>g4r&fovDpae#N=%i8Kqmro5 zB;Q=+o4W>$dWZ;v_FJhm**cA~_TXudHjVnpokUe=)Z=J4wDN;f>!8s_8`A}$LK~CU zQD;%*$7a)G(+aJo)@e1hP4|_atc#s-UV>q=sIonk$JJ19blsP4I2VQbWoNK(tyF&^ zI&!+gMRNSPWzRMjIK_gUvQun2ZJ zepmkhOqv~&XZO6f89myJ$CaF6wJN9P69OG55e}($qr!u-N>iGZwn&4V=9eEN>k)nj zC9cKJrkY`+gbKy3$8RRXj`I~_*m<{Rg3X8F47)Qc63NcW%iW}Dv|?Xuq1nC~&C252 zye6sURzEJz46BP~ z*3Gk{@vK}x>xCv?>8|`fs_GZo%zjaV{hUcY|L`@qQQOK5%=E%$zuww@p&#|{{L1GC zBC_J_>3jcnS%>Qt2rC- z2?Ngz>1@Oaj2khytEZNQ(Q+zkGD>b~k4$GzToIhU$k}7claQ-B2Wa9-)|KNkH*OcI zIY3HiSyQYM%aV<;4 zGYS4y-PjRdd#@gXTh)+c`@q{g{Eg!Da*9H**})>SHt@|?`+2>kM42eZDDlZRR!-{cIFrfcED^q7Rj)2Hx;bF?5u-km+<}Fh3s!H znVB|B5=Nz-30QG<9;=$@siXTfUanPwZl-(nmo@nr88F(tgp7an(EWe+8ZOync;{rS z9;>Qw{r8#ys>T-dxYO+>uHAP*9cKH=$4Cpke&}A5F}H1Sy}@>y`Sfbyon1R$UJbak zxW|0vt{6QhdF9*l0II^Jr@PmlvKrw5|WuXq|JXW`PsUbZ&SNfi3knqoz9 zvCAX2#-sZiszYR^k;T3noL%E1**~5#!M0v0u^f zD6jO}Q!czh#zj@2@Ex18>v%cf;q{xU^vu9n-MimZWt)oHJq0hrA? zJ)ZdxH|KP!dOa9I?s+jiCO#G7>uGF99q!I*wi!;l)5p2B@=J7l8b#rFcW)B9TA?e! z6BI?^>U0GQQRI8qbjc3Fu6E1*He&9_W211nGwiu)Q+&BP_bG%$9ulwP{IS8U5Dkuu z1@=uMDyi?2jRH_M}GN87G6ix-{0vvyk8i)NP$ zn#B*0AK9^$X4@j=@qZRdO3pIULTrIlG3Lm3z|9F8tVqMo*uH2W2 z>uPGUaCl6Q0F`75ZqG8ZzhmP2g}zemAhzA{uFYBebhFn%3x0N8E!8|=Qb|tYYLIKK zMAc$vak*pM!+;X2UJA8PAzAJb&XgpKg= z>j^4!i{VV3!RFz*z?UV?=Apj8J6&hfnE=<740a_u88z=<;To$?*-fK;cA6c7@2>E@ zh2uz`u)6pr@4~WPNUYIpt%GJ;5uQb>4L@P~?lcSMb$`N6dMeo2Uikhl;q?p$bU(6! zhvhwCE3Ic^>3L*6@i}O=&qlLwoZS-^u4nm#)eHUIoo(|Zzk?uWN;Uca#)|LNy9uOFYyo=3T_e|h!l z2~b==kMDKg+b=vkd-dY4FQZkzynFff+w=d*f0Ogyu*PAX!v=>5hfNMs4qF_yIjk0v oBkiP;hEhpOsidjYFSV8DPv+Y%uiyJ$slIglKS)SPa$6??0QU)sg#Z8m literal 0 HcmV?d00001 diff --git a/S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote.svg b/S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote.svg new file mode 100644 index 0000000..ea5f520 --- /dev/null +++ b/S3/ExPhyIII/VL/typst-assets/drawing-2025-11-05-11-18-39.rnote.svg @@ -0,0 +1,4 @@ + + + + \ No newline at end of file diff --git a/book.typ b/book.typ index 1663fa7..486a95a 100644 --- a/book.typ +++ b/book.typ @@ -17,6 +17,7 @@ - #chapter("S3/ExPhyIII/index.typ")[ExPhy III] - #chapter("S3/ExPhyIII/VL/ExIIIVL1.typ")[Wiederholung Wellen] + - #chapter("S3/ExPhyIII/VL/ExIIIVL2.typ")[ExIIIVL2] - #chapter("S3/KFT/index.typ")[Kft] - #chapter("S3/KFT/VL/KftVL1.typ")[Wiederholung Grundbegriffe]