Find the phase and group velocity of three plane waves

[frankR]

Y_i(x,t) = A*Sin(k_ix - {&omega}_it), where i = 1, 2, 3



Phase velocity for two wave can be given by v = (&omega + [&omega]')/(k + k') and group velocity u = (&Omega - &omega')/(k - k') but what about three waves?

I'm not sure what to do for three waves. I've looked all over my text and all over the web and can't find anything, that's why I'm here, really stuck.

Edit: Forget what the code for greek characters is and can't find it anywhere on this site, sorry.

 

[Aaron Barnard]

The phase velocity for three waves is the same as for two waves, v = (&omega + [&omega]')/(k + k'). The group velocity u for three waves is a bit more complicated. It can be expressed as u = (1/3)[(&Omega - &omega) / (k-k)+ (&Omega - [&omega]')/(k-[&k]')+ (&Omega - &omega")/(k-k")].

 

[M98Ranger]

The phase velocity for a single plane wave is given by v = &omega/k, where &omega is the angular frequency and k is the wave vector. In this case, we have three plane waves with different values of i, which correspond to different values of &omega and k. Therefore, we can calculate the phase velocity for each wave separately using the above formula.

For example, for the first wave (i=1), the phase velocity would be v1 = &omega1/k1. Similarly, for the second and third waves, the phase velocities would be v2 = &omega2/k2 and v3 = &omega3/k3, respectively.

As for the group velocity, it represents the velocity at which the envelope of the three waves propagates. It can be calculated using the formula u = (&Omega - &omega')/(k - k'), where &Omega is the average angular frequency and k is the average wave vector of the three waves.

To find the average values, we can use the fact that for three plane waves with different values of i, the average angular frequency is given by &Omega = (&omega1 + &omega2 + &omega3)/3 and the average wave vector is given by k = (k1 + k2 + k3)/3.

Substituting these values into the formula for group velocity, we can calculate the group velocity for the three waves.

In summary, the phase velocity for each wave can be calculated using the formula v = &omega/k and the group velocity for the three waves can be calculated using the formula u = (&Omega - &omega')/(k - k'). I hope this helps clarify the process for finding the phase and group velocities for three plane waves.