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Consider the following equations:

i.) $\frac{\partial \phi}{\partial t} + c \frac{\partial \phi}{\partial x} $

ii.) $\frac{\partial^2 \phi}{\partial t^2} - c \frac{\partial^2 \phi}{\partial x^2} $

a.) Calculate the dispersion relation and group velocity for each equation above and determine whether the waves are dispersive or non-dispersive.

b.) what is the difference between the waves in (i) and (ii)

I know that wave number $k = \nabla \theta =\frac{\partial \theta}{\partial x} $ and that the frequency $\omega = -\frac{\partial \theta}{\partial t}$ and that $c=\frac{\omega}{k}$...I am stuck on what to do next

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  • $\begingroup$ Look at plane wave solutions $\theta = \cos(\omega t - k x)$. Plug in and get the dispersion relation $\omega$ a function of $k$. The group velocity $\frac{d\omega}{dk}$ and the phase velocity $\frac{\omega}{k}$ is easy to compute from this. $\endgroup$ – Winther Jan 16 '17 at 2:27
  • $\begingroup$ I was told that the phase of a plane wave $\theta = (kx-\omega t)$ is this wrong? If it is what you're saying is $\frac{\partial}{\partial t} (kx-\omega t)=-\omega $ and $\frac{\partial}{\partial x} (kx-\omega t)=k $ therefore, i.) $-\omega + c k = 0$, making $\omega = ck $ $\endgroup$ – Abigail Jan 16 '17 at 2:49
  • $\begingroup$ I'm sorry you're right, it isn't the phase I wrote the symbol wrong $\endgroup$ – Abigail Jan 16 '17 at 2:57
  • $\begingroup$ Your result for i) looks correct. $\endgroup$ – Winther Jan 16 '17 at 2:58
  • $\begingroup$ then the group velocity $c_g = \frac{\partial \omega}{\partial k} = c$ therefore it is a shallow water wave and is non-dispersive? $\endgroup$ – Abigail Jan 16 '17 at 3:25

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