# Wave packet in dispersive medium, how will the group velocity be affected? [closed]

A wave packet with center frequency ω is propagating in dispersive medium with phase velocity of 1.5 x 10^3 m/s. When the frequency ω is increased by 2%, the phase velocity is found to decrease by 3%. What is the group velocity of the wave packet?

My answer would be that the group velocity should be half the phase velocity but that does not seem to be the correct answer , where am i going wrong ? could you guide me to more sources of similar problems (other than wikipedia, MIT ocw etc. ) ?

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## 1 Answer

I think you need to consider the basic expressions for group velocity and phase velocity.

Phase velocity: $$v_p = \frac{\omega}{k}$$, where $$v_p$$ is the phase velocity, $$k$$ is the wavenumber, $$\omega$$ is the angular frequency.

Group velocity: $$v_g = \frac{\partial\omega}{\partial k}$$, where $$v_g$$ is the group velocity.

Now you can say $$k = \frac{\omega}{v_p}$$ so the new $$k$$ at the shifted frequency becomes $$k' = \frac{1.02 \omega}{0.97 v_p}=\frac{1.02}{0.97}k$$

For the group velocity:

$$\Delta\omega = 0.02\omega$$ and $$\Delta k = (\frac{1.02}{0.97}-1)k= \frac{0.05}{0.97}k$$

$$v_g = \frac{\partial\omega}{\partial k} \approx \frac{\Delta\omega}{\Delta k} = 0.388v_p$$

• @Ijs thanks for clearing that , do you know of the exact reason why my reasoning is flawed , what am i missing , am i over generalizing , if yes then where ? ; a request if you could guide me to relevant sources of this topic at a high school level , i would appreciate that a lot – Anamika Ghosh May 7 at 14:44
• @AnamikaGhosh If you fill in how you reasoned that it should be half the phase velocity I would be happy to look at it. Currently I cannot say where you went wrong though. – ljs May 7 at 14:48
• @Ijs , I am not very good at this particular topic so i used wikipedia which says that in dispersive medium group velocity equals half the phase velocity as in deep water gravity waves . en.wikipedia.org/wiki/Group_velocity . could you please direct me to appropriate sources for further reading preferably an elementary physics textbook with this topic ? – Anamika Ghosh May 7 at 14:52
• @AnamikaGhosh That was for the particular relationship between the frequency and the wavenumber for deep water gravity waves. So it was just an over generalization. For an online resource this could work. I unfortunately don't know of any elementary physics textbooks off-hand. – ljs May 7 at 15:02