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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) [tex]\sigma^2 = c^2k^2 [/tex]

(2) [tex]\sigma^2=c^2(k^2+\epsilon k^4) [/tex]

(3) [tex]\sigma^2=c^2(k^2-\epsilon k^4)[/tex]

(4) [tex]\sigma^2=c^2k^2 +f^2 [/tex]

where k is the wavenumber and c is the phase speed, f and [tex]\epsilon[/tex] are constants and [tex]\sigma [/tex] 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)

The problem was the following.

Given

(1) [tex]\sigma^2 = c^2k^2 [/tex]

(2) [tex]\sigma^2=c^2(k^2+\epsilon k^4) [/tex]

(3) [tex]\sigma^2=c^2(k^2-\epsilon k^4)[/tex]

(4) [tex]\sigma^2=c^2k^2 +f^2 [/tex]

where k is the wavenumber and c is the phase speed, f and [tex]\epsilon[/tex] are constants and [tex]\sigma [/tex] 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)