A particular wave has the form
$\phi =Ae^{i\theta (x,t)}$
$\theta = -\frac{gt^2}{4x}$
What is the x-wavenumber?
If the wave number $\vec{K} = \nabla{\theta}$ then $K =\nabla({-\frac{gt^2}{4x}}) $ and if we are just looking at the x-wavenumber then the gradient will only be concerned with the x-direction.
$K =\nabla({-\frac{gt^2}{4x}}) = \frac{\partial}{\partial x}({-\frac{gt^2}{4x}}) = \frac{gt^2}{4x^2}$
Does this make sense at all how I did this, genuine question?
What is the frequency?
If the frequency $\omega = -\frac{\partial \theta}{\partial t } $ then $ \omega = \frac{\partial}{\partial t} ({-\frac{gt^2}{4x}}) = \frac{gt}{2x}$
Does this make sense at all how I did this?
Under what conditions is it sensible to talk about a slowly varying frequency?
At what speed need you move to see a constant frequency and wave number?
(possible answer) If the medium is independent of time and space then both the frequency and wave number will propagate with the group speed. However I am unsure of how to calculate this.
Moving at that speed, what is the relation between frequency and wave number?
At what speed do you have to move at to see a constant phase $\theta$? Is that speed constant with time?