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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?

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    You have a bunch of things missing - namely $k$ is usually the wave number and we have $\omega$ for frequency. But notation aside, what about the governing wave like equation? since a dispersion relation is what relates the parameters that you are trying to determine. – 2017-01-13
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    Is this fluids or MHD your are doing these calculations within? just out of interest! but, you have not given the governing wave equation as it is this thay determines the relationship between frequency and wave number. – 2017-01-13
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    I do not know what MHD is but this is for a geophysical waves class, the question comes out of the book by Joseph Pedlosky "Waves in the Ocean and Atmosphere" – 2017-01-13

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