[hadesmajesty]

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.

[panda]

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.

[hadesmajesty]

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?

[panda]

According to your 1D wave equation, u(x,t) is a wave packet and both the group velocity and the phase velocity are included.