# Group velocity of two cosines

I just encountered the following problem, which is really strange to me. I simply want to calculate the group velocity of the following wave

$$F(x,t) = \cos(k_1x) + \cos(k_2x-w_2t).$$

Now I thought, that's super easy, we have two different $$k$$, which makes the dispersion relation linear and it is

$$w(k) = \frac{w_2}{k_2-k_1}k$$

and so

$$v_G = \frac{\partial w}{\partial k} = \frac{w_2}{k_2-k_1}$$

Now when I plot this, the above formula seems to work, as long as k1 and k2 are relatively close together, but as they get more different, several envelops can be seen and as far as I can tell, it doesn't even look like the absolut value is correct anymore. So I suppose, that either I just judge the plots incorrectly or what I did is actually a limit for small $$k_2-k_1$$.

Below you can see an example for $$k_1 = 10$$, $$k_2 = 18$$ and $$w_2 = 1$$. The left red point has $$v_G$$ from the above formula, but there are two envelops, moving to the right.

I hope you can help me to understand this, or to find the right formula!

• $F(x)=\cos (k_1\,x)$ is not a travelling wave. – Farcher Oct 18 '19 at 7:19
• It is a travelling wave with $w_1 = 0$. You could also replace it by $F(x,t) = cos(k_1 x - \epsilon t)$ for a small epsilon, which shows, that this is not the reason for the problem I have unfortunately. – yanscha Oct 19 '19 at 14:24