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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 27 "phaseVgroup.mws 2002/02/07\n" }
{TEXT 256 44 "Phase velocity and group velocity contrasted" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 709 "wave:=(k,x,t)->cos(k*x-omega(k)*t)
;\nomega:=k->sqrt(k);              # DISPERSION RELATION\nmy_k:=2.0;\n
kratio:=1.05;\nwave1:=(x,t)->wave(my_k,x,t);\nwave2:=(x,t)->wave(my_k*
kratio,x,t);\nwaves:=wave1+wave2;\ntmax:=50; nt:=40: dt:=tmax/nt;\nplo
ts[display](\n  seq( plots[display](\n\n          plot([[omega(my_k)*t
/my_k,1],\n                [omega(my_k)*t/my_k,2.5]],color=red,thickne
ss=2),\n          plot([[omega(my_k*kratio)*t/(my_k*kratio),1],\n     \+
           [omega(my_k*kratio)*t/(my_k*kratio),2.5]],color=blue,thickn
ess=2),\n          plot([wave1(x,t),wave2(x,t),waves(x,t)],\n         \+
      x=-100/my_k..100/my_k,color=[red,blue,black],numpoints=100)\n   \+
    )\n     , t=seq(i*dt,i=0..nt)\n     )\n  , insequence=true );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%waveGf*6%%\"kG%\"xG%\"tG6\"6$%)operatorG%&arrowGF*-%$cosG6#,&*&9$\"
\"\"9%F4F4*&-%&omegaG6#F3F49&F4!\"\"F*F*F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&omegaGf*6#%\"kG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#9
$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%my_kG$\"#?!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'kratioG$\"$0\"!\"#" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&wave1Gf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-
%%waveG6%%%my_kG9$9%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&wave2
Gf*6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF)-%%waveG6%*&%%my_kG\"\"\"%'kr
atioGF29$9%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&wavesG,&%&wave
1G\"\"\"%&wave2GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG\"#]" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG#\"\"&\"\"%" }}{PARA 7 "" 1 "" 
{TEXT -1 33 "Warning, computation interrupted\n" }}}{PARA 0 "" 0 "" 
{TEXT -1 178 "The black curve is the sum of the red and blue curves.\n
The red and blue curves are each travelling sinusoidal waves. They hav
e slightly different wave-numbers (i.e. wavelengths)," }}{PARA 0 "" 0 
"" {TEXT -1 123 "and the dispersion relation determines how this affec
ts their speeds. Compare the speed of the red and blue component waves
" }}{PARA 0 "" 0 "" {TEXT -1 54 "with that of the pulse formed by thei
r superposition.\n" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 0" 54 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }