How's the superposed waves are moving?

KFC

Hi there,
Suppose there are two plane waves with different wavevector and frequency, the superposition of these waves give
[tex]\phi(x, t) = 2\cos(k'x + w't)\cos(k''x + w''t)[/tex]
with
[tex]k'=(k_1+k_2)/2, w'=(w_1+w_2)/2[/tex]

[tex]k''=(k_1-k_2)/2, w''=(w_1-w_2)/2[/tex]

here [tex]\cos(k''x + w''t)[/tex] gives the oscillation and [tex]\cos(k'x + w't)[/tex] is the envelope. My question is if I look at the very first crest when x=0, t=0 and I attach a "point" to that. How can I trace the movement of that point? Can we just look at [tex]\cos(k''x + w''t)[/tex] and say that the every point is moving at the phase velocity w''/k''? I am not sure the physics behind that but seems it is not correct. But with several trials, I find that it seems the point (crest of the second cosine for example) is moving at the speed [tex](w'+w'')/(k'+k'')[/tex], is that correct? why?

I want to trace the crest point and I know that the magnitude of the crest is changing on the envelope and it is correct at different time and space. So I initially fix the initial position xi = 0; yi = 2; for the crest. Then by enumerating the time, we can find the new x and new y for the crest as

x = (w'+w'')/(k'+k'')*t;
y = 2*cos(k'x + w't);

but when I plot this point at different t, it doesn't really move as I expect. Any idea how to trace a point? I want to plot something like the red point shown here https://en.wikipedia.org/wiki/File:Wave_group.gif

Thanks