Frequency wavenumber Usually, when you have to go from frequency to wavenumber you have to adopt the following equation:
$$k = \frac{2\pi f}{U}$$
where k is the wavenumber, f is the frequency and U is the velocity. 
Where does this equation come from? Is this a consequence of the Taylor Hypothesis? Is this formula always reliable?
(I am studying turbulence)
Thanks in advance
 A: Waves are periodic solutions with a phase given by $\phi(x,t) = kx - \omega t$, where $\omega = 2\pi f$. The point of a constant phase is thus moving with a speed that can be determined from this relation: $$d\phi(x,t) = kdx -\omega dt = 0,$$
that is the phase speed is $$v_{ph} = \frac{dx}{dt} = \frac{\omega}{k}=\frac{2\pi f}{k} = \frac{2\pi f}{2\pi/\lambda} =\lambda f,$$
where $\lambda$ is the wave length. This is also where the relation in question comes from.
In addition to the phase speed one often defines defines a group speed, which describes propagation of wave packets, i.e. groups of waves with a distribution of frequencies and wave vectors. I just state the result without the lengthy math:
$$v_g = \frac{d\omega(k)}{dk}.$$
In vacuum and many media the frequency linearly depends on the wavenumber, $\omega = v_{ph}k$, so that the group and the phase speeds are the same. When this is not the case we speak of dispersion and dispersive media. 
Remarks
The group speed is less than the phase speed. It is usually said that information propagates with the group speed, since a plane wave by itself cannot carry any information: it is predictable in all the space. This has to do with some paradoxes in relativity.
