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Jonas Hahn
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121 lines
4.9 KiB
Typst
121 lines
4.9 KiB
Typst
// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
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// Main settings call
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#show: conf.with(
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// May add more flags here in the future
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num: 5,
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type: 0, // 0 normal, 1 exercise
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date: datetime.today().display(),
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//date: datetime(
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// year: 2025,
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// month: 5,
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// day: 1,
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//).display(),
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)
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= Uebersicht
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Dispersion
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$
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omega &= c k \
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omega &= c (k ) k.
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$
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Die Geschwindigkeit aus der WEllengleichung wird Phasengeschwindigkeit genannt.
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Femtosekundenlaser durch Inteferenz von monochromatischem Licht. Im Vakuum gibt es keine Dispersion.
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Verallgeminerung: Ausbreitung eines Pules
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$
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psi (x, t) &= integral_(- oo)^(oo) underbrace(A (k) e ^(i (k x - omega (k)t)), "Darstellung im Fourierraum") dif k \
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omega (k) &approx omega (k_0 ) + (k - k_0 ) (diff omega) / (diff k) (k_0 ) \
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psi (x, t) &= integral_(- oo)^(oo) dif k A (k) e ^( i (k_0 x - omega (k_0 )t) - (k - k_0 )(diff omega) / (diff t) t) \
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&= underbrace(e ^( i (k_0 x - omega ( k_0) t )), "monochromatische Welle bewegt sich mit" c_(p) ) integral_(- oo)^(oo) dif k A (k) e ^( i [(k - k_0 )(x - (diff omega) / (diff k) t)]) \
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c &= (diff omega) / (diff k) \
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c_(p) &= omega_0 /k_0.
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$
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Dispersionsrelation mit nicht konstantem $c$
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$
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omega &= c (k) k \
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c^2 &= [underbrace(g/h, "grosse Welle") + underbrace((sigma k) / (rho), "kleine Welle")] tan (k h) \
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c &= sqrt(T/rho),
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$
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wobei $h$ die Wassertiefe und $sigma$ die Oberflaechenspannung ist.
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Wo sind die Wellenzahlen gleich gross
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$
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g/h &= (Delta k_(c)) / (rho) \
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==> lambda_(c) &= 2 pi sqrt(sigma/(rho g)) \, space O ("cm").
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$
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In der zum Beispiel Nordsee
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$
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"fuer" lambda >> lambda_(c) \, space k << k_(c) \
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"fuer" h k << 1 \
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==> c^2 approx g h \, space k := "const".
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$
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In der Tiefsee
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$
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h k >> 1 \
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c = sqrt(g/k) \, space omega = c k = sqrt(g k) \
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c_(g) = (diff omega) / (diff k) = 1/2 sqrt(g/k)= 1/2 c_(p).
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$
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#theorem[
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Navier-Stokes Gleichung
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$
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rho (partial _(t) arrow(v) + arrow(v) * arrow(nabla) arrow(v)) = rho arrow(g) - arrow(nabla) p.
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$
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]
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= Von der Maxwellgleichung zur Wellengleichung
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Faradaysche und Amperesche Gesetz
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$
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arrow(nabla) times arrow(E) = - (diff arrow(B)) / (diff t) \, space arrow(nabla) times arrow(B) = epsilon_0 mu_0 (diff arrow(E)) / (diff t) \
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underbrace(arrow(nabla) times arrow(nabla) times arrow(E), = arrow(nabla) (underbrace(arrow(nabla) * arrow(E), = 0\, "da" rho = 0))- arrow(nabla) ^2 arrow(E)) = arrow(nabla) times (- (diff arrow(B)) / (diff t) ) = - diff / (diff t) ( arrow(nabla) times arrow(B))= - epsilon_0 mu_0 (diff ^2 arrow(E)) / (diff t^2 ) \
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==> arrow(nabla) ^2 arrow(E) - epsilon_0 mu_0 (diff ^2 arrow(E)) / (diff t^2 ) = 0 \
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arrow(B) (arrow(r), t) "analog" arrow(nabla) ^2 arrow(B) - epsilon_0 mu_0 (diff ^2 arrow(B)) / (diff t^2 ) = 0 \
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==> c_0 = 1/sqrt(epsilon_0 mu_0 ) approx 2.9 * 10 ^(8) "m"/"s" \
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arrow(nabla) ^2 phi - 1/c^2 (diff ^2 phi) / (diff t^2 ) = 0 .. forall phi in {E_(x) , E_(y) , E_(z) , B_(x) , B_(y) , B_(z) }.
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$
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Spezialfall ebene Welle zum Beispiel Wellenvektor $k$ ist $k hat(z)$. Welle ist konstant in $x$ und $y$ $forall z, t$ $==>$ Ebene senkrecht auf $z$ haben konstante Phase und Amplitude
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$
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arrow(E) = arrow(E) (z, t) .. "aus" arrow(nabla) * arrow(E) = 0 "folgt" partial _(z) E_(z) = 0 \
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==> E_(z) "const". \
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arrow(E)_(0) = vec(E_(x), E_(y) , 0) \, space arrow(E) (z, t) = arrow(E)_(0) e ^(i (k z - omega t)) \
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"mit" partial _(x) ^2 arrow(E) = partial _(y)^2 arrow(E)= 0 .. "wird die WG" \
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partial _(z) ^2 arrow(E) - 1/c^2 partial _(t) ^2 E= 0.
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$
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Allgemeiner
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$
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arrow(E) = arrow(E)_(0) e ^(i (arrow(k)* arrow(r) - omega t)) \, space arrow(k) = vec(k_(x) , k_(y) , k_(z) ) = (2 pi)/lambda hat(k) \
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arrow(nabla) ^2 arrow(E) - 1/c^2 partial _(t) ^2 arrow(E) = 0 \, space "3 WG" \
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"fuer jede Komponente" psi = psi_0 e ^(i (arrow(k)* arrow(r) - omega t)) \
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==> partial _(t) ^2 psi = - omega ^2 psi_0 e ^(i (arrow(k)* arrow(r) - omega)) \
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arrow(nabla) ^2 psi = psi_0 (i arrow(k))^2 e^(i (arrow(k) arrow(r) - omega t)) \
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==> - k^2 + omega^2 /c^2 = 0 ==> c = omega/k.
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$
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Das Magnetfeld der ebenen Welle. Sei
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$
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arrow(E) = underbrace(arrow(E)_(0) hat(x), "Polarisation") e ^(i (k z - omega t)) \
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==> arrow(nabla) times arrow(E) = (diff arrow(B)) / (diff t) \, space arrow(B) = arrow(B) (arrow(r), t) \
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- abs(mat( hat(x), hat(y), hat(z); partial _(x) , partial _(y) , partial _(z) ; E_(x) , 0, 0; )) = - (partial _(z) E_(x) ) hat(y) = (diff B_(y) ) / (diff t) \
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(diff B_(y) ) / (diff t) = - (diff E_(x) ) / (diff z) = - i k E_0 e ^(i (k z - omega t)) \
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==> B_(y) = k/omega E_0 e ^(i (k z - omega t)) = 1/c E_0 e ^(i (k z - omega t))
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$
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#figure(
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image("typst-assets/drawing-2025-11-14-09-19-41.rnote.svg"),
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)
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Es folgt
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$
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arrow(B) = 1/c^2 (arrow(k) times arrow(E)).
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$
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Warum dreht sich die Reflektion um in einem Koaxialkabel?
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