My work on wave packets in quantum mechanics has led me to something called the stationary phase approximation. The idea is that if you are adding together a lot of sinusoidal waves to make a wave packet but you are doing this across a continuous range of frequencies instead of the discrete frequencies I tried the other day, then you get points where the frequencies change so quickly that they come in and out of phase adding up incoherently. To get a decent superposition, you want the response to be roughly the same over a range of frequencies instead of rapidly varying. I have been working from
https://en.wikipedia.org/wiki/Stationary_phase_approximation I have gone for the section called "An example". So in the following picture, the integral means you are adding up waves across a continuous range of frequencies, w. (Because w=2pi.f) and expi(kx-wt) is a way fo writing sinusoidal waves using imaginary numbers.
kx-wt is the phase and k is actually a function of w, for examples perhaps k=2w.
If you differentiate the phase with respect to frequency, you find how fast it is varying. The stationary point is when the differential = 0.
Rearranging that leads to a relationship. Now x/t has units metres per second, so dw/dk is a velocity. It is called the group velocity of the wave packet.
