There is something that confuses me a bit about how group velocity is sometimes introduced and explained. For example, Walter Lewin (great professor) in his lectures uploaded on youtube, explains the concept in this way. He begins by adding two waves with different wavenumber and angular frequency, and notes that the sum of the two waves $Asin(k_1x-w_1t)$ and $Asin(k_2x - w_2t)$, is equal to: $$2Asin(\frac{k_2+k_1}{2}x-\frac{w_2+w_1}{2}t)cos(\frac{k_2-k_1}{2}x-\frac{w_2-w_1}{2}t)$$ It is then said that the velocity of the sine part of this wave, namely $$(1)\frac{w_2+w_1}{k_2+k_1}$$is the phase velocity.

While the velocity of the cosine part of this wave, namely: $$(2) \frac{w_2-w_1}{k_2-k_1}$$ is the group velocity.

So far so good. Finally, it is said that if $k_1$ is approximately equal to $k_2$, and $w_2$ is approximately $w_1$, then the group velocity is: $$(3)\frac{\partial w}{\partial k}$$ Indeed the correct way to define group velocity is not, after all (2), but (3).

A similar explanation is present in many online introductions to the topic (ex. http://spiff.rit.edu/classes/phys283/lectures/velocities/velocities.html). Now, it is (sloppily) true to say that (2) is almost the same as (3) for values of $k$ and $w$ that are approximately the same. But still, shouldn't be possible to define group velocity even when adding two waves with wavenumbers and frequencies that are not approximately the same?

To explain my confusion with an example. Suppose that $w=k^3$. And suppose that I add two waves $sin(k_1x-w_1t)$ and $sin(k_2x - w_2t)$ such that $$k_1=1, w_1=1, k_2 = 3, w_2 = 27$$ Then the expression (2), namely $\frac{w_2-w_1}{k_2-k_1}$, gives a number, namely 13, while the expression (3), namely $\frac{\partial w}{\partial k}$ gives the function $3k^2$. Now, of course the latter can be made equal to 13 for an appropriate choice of $k$ (if k= 2.081...), but this choice of $k$ seems completely arbitrary.

So, to conclude, the group velocity of the result of adding two waves $\psi_1$ and $\psi_2$ cannot be both the ratio of the difference between the wavenumbers and frequencies of the two waves (2), and the derivative of the dispersion relation (3). But then why introduce the correct definition (3), starting by (2)? Why indeed, if group velocity can be defined also when the two waves added together have wavenumbers and angular frequencies that are NOT approximately the same?

I hope I made my doubt clear, and I hope someone might help.


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