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what is group and phase velocity?

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Old   March 30, 2009, 11:03
Default what is group and phase velocity?
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Edison
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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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Old   March 31, 2009, 02:07
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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.
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Old   March 31, 2009, 05:24
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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?
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Old   April 1, 2009, 02:27
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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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