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
$$\phi(x, t) = 2\cos(k'x + w't)\cos(k''x + w''t)$$
with
$$k'=(k_1+k_2)/2, w'=(w_1+w_2)/2$$

$$k''=(k_1-k_2)/2, w''=(w_1-w_2)/2$$

here $$\cos(k''x + w''t)$$ gives the oscillation and $$\cos(k'x + w't)$$ 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 $$\cos(k''x + w''t)$$ 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 $$(w'+w'')/(k'+k'')$$, 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

 

[Valerie Park]

in advance!The equation you provided, \phi(x, t) = 2\cos(k'x + w't)\cos(k''x + w''t),describes the superposition of two waves which represent the envelope and oscillation. The crest point (or any other point) can be traced by looking at the oscillation component, \cos(k''x + w''t). This is because it is the oscillation that determines the size, shape, and position of the wave. To trace the movement of the point, we can use the angular velocity equation, w''/k''. This equation tells us that the point is going to move with an angular velocity w''/k''. The speed of the point is then given by the magnitude of the velocity, |v| = |w''/k''|. So to trace the point, you need to solve the equation x = (w'+w'')/(k'+k'')*t;y = 2*cos(k'x + w't);for different values of t. Then you will have the x and y coordinates of the point at different times. With these coordinates, you can then plot the trajectory of the point.