Ausbreitung einer skalaren Welle im Medium,
Phasen- und Gruppengeschwindigkeit

Brechungsindex

\(c = 1; η = 0.1; k\_0 = 5; 


    n[k_] = 1.2\  + \ 
        η\ \((k - k\_0)\)\ 
          E\^\(-\(\((k - k\_0)\)\^2\/4\)\); 


    Plot[n[k], {k, 0, 10}, PlotRange -> {0.5, 1.5}, 
\ \ \ 
      AxesLabel -> {"k", " "}]; \)

Dispersionsrelation

\(ω[k_] = \(c\ k\)\/n[k]; 


    Plot[ω[k], {k, 0, 10}, PlotRange -> {\(-2\), 10}, 
\ \ \ 
      AxesLabel -> {"k", " "}]; \)

Phasen- und Gruppengeschwindigkeit

\(v\_p[k_] = ω[k]\/k; 

v\_g[k_] = ∂\_k\ ω[k]; 
    

Plot[{v\_p[k], v\_g[k]}, {k, 0, 10}, PlotRange -> {\(-0.5\), 2}, 

      \ \ \ AxesLabel -> {"k", " "}]; \)

spektrale Verteilung der Welle

\(g[k_] = E\^\(-\(\((k - k\_0)\)\^2\/0.25\)\); 


    Plot[{g[k], v\_g[k]}, {k, 0, 10}, PlotRange -> {\(-0.5\), 1.5}, 
\ \ \ 
      AxesLabel -> {"k", " "}]; \)

skalare Welle

\(φ[x_, t_] = 
      NIntegrate[
        g[k]\ E\^\(I\ 
              \((k\ x\  - \ ω[k]\ t)\)\), {k, 0, 2, 100}]; 


    Plot[Re[φ[x, 0]], {x, \(-10\), 10}, 
	
      PlotRange -> {\(-2.5\), 2.5}, 
      AxesLabel -> {"x", "Re[φ]"}]; \)

Zeitentwicklung

\(\(For[i = 1, i < 101, \(i++\), 
	\ t = 0.1\ i; 
\ \ \ \ 
      Plot[Re[φ[x, t]], {x, \(-10\), 10}, 
	\ \ \ \ \ \ \ 
        PlotRange -> {\(-2.5\), 2.5}, 
        AxesLabel -> {"x", "Re[φ]"}]]; \)\)