Understanding Dispersion Relations in Fluid Dynamics

[nickthequick]

I am having trouble understanding a basic problem in fluids that came up during an exam I took last quarter. Namely, we are given a dispersion relation and asked to quantify how a one dimensional surface disturbance propagates in space. (The disturbance is initially an approximate delta function at the origin).

The problem was the following.
Given
(1) $$\sigma^2 = c^2k^2$$
(2) $$\sigma^2=c^2(k^2+\epsilon k^4)$$
(3) $$\sigma^2=c^2(k^2-\epsilon k^4)$$
(4) $$\sigma^2=c^2k^2 +f^2$$
where k is the wavenumber and c is the phase speed, f and $$\epsilon$$ are constants and $$\sigma$$ is the angular frequency

We are then asked to identity, with proper justification, which of the following plots go with which dispersion relation. (file is attached)




I was able to properly match up pairs by considering relationships between phase and group velocities. Analytically I know that you can attack this problem by analyzing the stationary phase of the inverse Fourier transform of the initial PDE; however, I am looking for a more physical approach to the problem.


Basically my question is this: what information can we gather about the propagation of the disturbance through analysis of the dispersion relationship without actually fully solving the PDE?

(I think I am confused about the mapping (through the FT) between k-space and x-space)

 

[0xDEADBEEF]

Look at the plots. Where do we have no dispersion, and thus no change in the shape of the wave packet. A peak contains all frequencies, which plot looks as if strongly oscillating waves travel faster, which one looks as is they travel slower, and which one looks as if very slowly oscillating waves don't travel at all?