Group/Phase velocity - amplitude, frequency, velocity

[tquiva]

Homework Statement

I've been working on a python simulation program that is based on phase velocity and group velocity. It looks like the following screenshot, where three parameters (E02, f2, and v2) can be adjusted. These three parameters are adjusted in terms of E01, f1, and v1.

I adjusted the parameters for E_02, f2, and v2 to get the following cases:

As a result of obtaining these arbitrary parameteres, I need to develop a theory that tells how to adjust these parameters E02, f2, v2 to get the group velocity designated in the cases above.

Homework Equations

I know that:
Vp = ω/β = (2πf)/β
VG = dω/dβ

For the python simulation:
Group velocity is the velocity of the envelopes, phase velocity is the velocity of the resultant wave (white curve)
Positive velocity: Wave is moving from left to right
Negative velocity: Wave is moving from right to left

The Attempt at a Solution

Observing the cases in the above table:
For VG = 0, it appears that the value for E02 and f2 does not matter as long as v2 = 0*v1 = 0.

For VG = VP, it is also knows the E02 = E01, f2=f1, and v2 = v1. So the parameters of one wave are equivalent to the parameters of the other wave.

At this point, I'm a little lost. My TA says that there is an equation to be derived so that I can obtain the above cases by inputting certain values to it. I don't know where to go from here to obtain that equation. Will someone please help me? I'd highly appreciate it.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[Nate810]

From the given parameters, it appears that the group velocity depends on the frequency and phase velocity of the two waves. Thus, the equation for the group velocity can be derived as follows: VG = dω/dβ where ω and β are the frequencies and phase velocities respectively of the two waves. The equation for the group velocity can then be written as: VG = (f2 - f1)/(v2 - v1) This equation can be used to adjust the parameters E02, f2, v2 to get the desired group velocity, as in the cases shown above. For example, to get VG = 0, set v2 = 0*v1 = 0 and the values for E02 and f2 will not matter. To get VG = VP, set E02 = E01, f2=f1, and v2 = v1. Thus, this equation can be used to adjust the parameters E02, f2, v2 to get the desired group velocity.