# what is group and phase velocity?

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March 30, 2009, 11:03
what is group and phase velocity?
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Edison
Join Date: Mar 2009
Posts: 40
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what is group and phase velocity in the one dimensional wave equation?for example,
du/dt+a(du/dx)=0
with periodic boundary and initial condition u0(x,0)=sin(kx)exp[-16(x-0.5)^2]

Then, its exact solution is u(x,t)=sin[k(x-at)]exp{-16[(x-at)-0.5]^2}

Does u(x,t) represent a single wave or a wave packet with all the wave numbers equal to k? Is the group and phase velocity equal to a?

thank you.
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 wave equation.gif (1.1 KB, 35 views)

 March 31, 2009, 02:07 #2 Member   Join Date: Mar 2009 Posts: 35 Rep Power: 10 Here a is the phase velocity for a single wave, group velocity is for a group of waves. Only in shallow water the group velocity is equal to the phase velocity.

 March 31, 2009, 05:24 #3 Member   Edison Join Date: Mar 2009 Posts: 40 Rep Power: 10 thank you. But if u(x,t) is the single wave.why does the numerical solution, i.e. the leap-frog scheme, of wave equation have a group velocity?

 April 1, 2009, 02:27 #4 Member   Join Date: Mar 2009 Posts: 35 Rep Power: 10 According to your 1D wave equation, u(x,t) is a wave packet and both the group velocity and the phase velocity are included. Last edited by panda; April 1, 2009 at 19:37.

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