// 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: 5,
	type: 0, // 0 normal, 1 exercise
	date: datetime.today().display(),
	//date: datetime(
	//	year: 2025,
	//	month: 5,
	//	day: 1,
	//).display(),
)

= Uebersicht

Dispersion 
$
  omega &=  c k \ 
   omega &=  c (k ) k.
$
Die Geschwindigkeit aus der WEllengleichung wird Phasengeschwindigkeit genannt.

Femtosekundenlaser durch Inteferenz von monochromatischem Licht. Im Vakuum gibt es keine Dispersion.

Verallgeminerung: Ausbreitung eines Pules
$
  psi (x, t) &=  integral_(- oo)^(oo)  underbrace(A (k) e ^(i (k x -  omega (k)t)), "Darstellung im Fourierraum") dif k \ 
  omega (k) &approx omega (k_0 ) + (k -  k_0 ) (diff omega) / (diff k) (k_0 ) \ 
  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)  \ 
  &=  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)])  \ 
  c &=  (diff omega) / (diff k) \ 
  c_(p) &= omega_0 /k_0.
$

Dispersionsrelation mit nicht konstantem $c$
$
  omega &=  c (k) k \ 
  c^2 &= [underbrace(g/h, "grosse Welle") +  underbrace((sigma k) / (rho), "kleine Welle")]  tan (k h) \ 
  c &=  sqrt(T/rho),
$
wobei $h$ die Wassertiefe und $sigma$ die Oberflaechenspannung ist.
Wo sind die Wellenzahlen gleich gross
$
  g/h &=  (Delta k_(c)) / (rho)  \ 
  ==> lambda_(c) &=  2 pi sqrt(sigma/(rho g)) \, space O ("cm").
$
In der zum Beispiel Nordsee
$
  "fuer" lambda >> lambda_(c)  \, space  k << k_(c) \
  "fuer" h k << 1 \
  ==> c^2  approx g h \, space k := "const".
$
In der Tiefsee
$
  h k >> 1 \ 
  c =  sqrt(g/k) \, space omega =  c k = sqrt(g k) \
  c_(g) = (diff omega) / (diff k) = 1/2 sqrt(g/k)= 1/2 c_(p).
$

#theorem[
  Navier-Stokes Gleichung 
  $
    rho (partial _(t) arrow(v) +  arrow(v) * arrow(nabla) arrow(v)) = rho arrow(g) -  arrow(nabla) p.
  $
]

= Von der Maxwellgleichung zur Wellengleichung 

Faradaysche und Amperesche Gesetz 
$
  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)  \ 
  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 ) \ 
  ==> arrow(nabla) ^2  arrow(E) -  epsilon_0 mu_0 (diff ^2 arrow(E)) / (diff t^2 )  =  0 \ 
  arrow(B) (arrow(r), t) "analog" arrow(nabla) ^2 arrow(B) - epsilon_0 mu_0 (diff ^2 arrow(B)) / (diff t^2 ) = 0 \ 
  ==> c_0 = 1/sqrt(epsilon_0 mu_0 ) approx 2.9 * 10 ^(8) "m"/"s" \ 
  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) }.
$
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 
$
  arrow(E) =  arrow(E) (z, t) .. "aus" arrow(nabla) * arrow(E) = 0 "folgt" partial _(z) E_(z) = 0 \ 
  ==> E_(z) "const". \ 
  arrow(E)_(0)  =  vec(E_(x), E_(y) , 0) \, space arrow(E) (z, t) =  arrow(E)_(0) e ^(i (k z -  omega t)) \ 
  "mit" partial _(x) ^2 arrow(E) =  partial _(y)^2 arrow(E)= 0 .. "wird die WG" \ 
  partial _(z) ^2 arrow(E) -  1/c^2 partial _(t) ^2  E= 0.
$
Allgemeiner 
$
  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) \ 
  arrow(nabla) ^2 arrow(E) -  1/c^2 partial _(t) ^2 arrow(E) = 0 \, space  "3 WG" \ 
  "fuer jede Komponente" psi =  psi_0 e ^(i (arrow(k)* arrow(r) -  omega t))  \ 
  ==> partial _(t) ^2 psi = - omega ^2 psi_0 e ^(i (arrow(k)* arrow(r) - omega))  \ 
  arrow(nabla) ^2 psi = psi_0 (i arrow(k))^2 e^(i (arrow(k) arrow(r) - omega t))  \ 
  ==> - k^2 +  omega^2 /c^2 = 0 ==> c = omega/k.
$
Das Magnetfeld der ebenen Welle. Sei 
$
  arrow(E) =  underbrace(arrow(E)_(0)  hat(x), "Polarisation") e ^(i (k z -  omega t))  \ 
  ==> arrow(nabla) times arrow(E) =  (diff arrow(B)) / (diff t)  \, space arrow(B) =  arrow(B) (arrow(r), t) \ 
  - 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)  \ 
  (diff B_(y) ) / (diff t) = - (diff E_(x) ) / (diff z) =  -  i k E_0 e ^(i (k z -  omega t))  \ 
  ==> B_(y) =  k/omega E_0 e ^(i (k z -  omega t))  = 1/c E_0 e ^(i (k z -  omega t)) 
$

#figure(
  image("typst-assets/drawing-2025-11-14-09-19-41.rnote.svg"),
)

Es folgt 
$
  arrow(B) =  1/c^2  (arrow(k) times arrow(E)).
$

Warum dreht sich die Reflektion um in einem Koaxialkabel?