deletions | additions       diff --git a/Appendix.tex b/Appendix.tex  index 325a130..fee9cfa 100644  --- a/Appendix.tex  +++ b/Appendix.tex 
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The phase velocity is \[v_{Phase}=\frac{\omega}{k}\] where $\omega$ is the wave frequency and $k$ is the wave number.  It follows that the group velocity is \[v_{group}=\frac{d\omega}{dt}\]  The code that was used to make the wave simulations is as follows  $  MAKEWAVE.PRO function makewave, k, theta, B0, freq 
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graph1=plot( z, k[*,0], colorline=1, name='40 G')  graph2=plot( z,k[*,1], colorline=100, /OVERPLOT, name='60 G')  graph3=plot( z,k[*,2], colorline=150, /OVERPLOT,name='80 G')  l = legend(TARGET=[GRAPH1,GRAPH2, GRAPH3]) GRAPH3])$