Discussion
Consider two waves \(y_{1} = a\cos(\omega t - kz)\) and \(y_{2} = a\cos[(\omega + \Delta\omega)t - (k + \Delta k)z]\). The group velocity of the superposed wave will be (\(\Delta\omega \ll \omega\) and \(\Delta k \ll k\))
Answer: C
The group velocity is \(\frac{\Delta\omega}{\Delta k}\).
Adding the two waves gives a carrier at the mean frequency, modulated by an envelope:
\[ \begin{aligned} y &= 2a\cos\left(\frac{\Delta\omega\, t - \Delta k\, z}{2}\right) \\ &\quad \times \cos(\bar{\omega} t - \bar{k} z) \end{aligned} \]
The envelope, which carries the energy, moves with speed \(\frac{\Delta\omega/2}{\Delta k/2} = \frac{\Delta\omega}{\Delta k}\), which is the group velocity \(d\omega/dk\) in the limit of small differences.
The options built from \(\omega \pm \Delta\omega\) over \(k \pm \Delta k\) are phase velocities, of one wave or of the carrier, not the speed of the envelope.
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