# dispersive vs. non-dispersive waves

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

• 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. – Winther Jan 16 '17 at 2:27
• 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 $\omega = ck$ – Abigail Jan 16 '17 at 2:49
• I'm sorry you're right, it isn't the phase I wrote the symbol wrong – Abigail Jan 16 '17 at 2:57
• Your result for i) looks correct. – Winther Jan 16 '17 at 2:58
• then the group velocity $c_g = \frac{\partial \omega}{\partial k} = c$ therefore it is a shallow water wave and is non-dispersive? – Abigail Jan 16 '17 at 3:25