Dispersion of waves - wavephysics

[Nikitin]

Hey! Two questions:

1) How exactly does the proof go for the formula for the group-velocity:
$$v_g = \frac{d \omega}{dk}$$
Is it something along the line of using the superposition principle to add all the (arbitrarily numerous) wave-functions making up the wave-packet together, then using trigonometric identities to discover that the wave-packet is like a "beat-phenomenon" moving with a velocity equal to the group velocity through the medium? Ie, it's just function of the type
$$A \sin(\omega_0 t -k_0 x) \cos(\omega_{beat} t -k_{beat} x)$$
? And then finding out that $$v_g = \frac{\omega_{beat}}{k_{beat}} = \frac{d \omega}{dk}$$?
2) How do you actually calculate the group velocity? What wave-lengths and frequencies do you insert into the formula? I mean, there can be hundreds of different waves making up the wave-packet, all with different frequencies and wavelengths. Do you just use the averages of their wavenumbers, or something?

3) What exactly does the formula for the phase-velocity in a wave-packet mean?
$$v_p = \frac{\omega}{k}$$
Is this the general formula for the velocity of each wave of the packet? Or is it the velocity of the sine-term in the equation above, assuming the waves making up the wave-packet have nearly identical frequencies and wavelengths? So:
$$v_p = \frac{\omega_0}{k_0}$$

?

 

[arildno]

1. Well, I've seen it done in potential theory (i.e, inviscid, irrotational fluid), in which a necessary condition for energy conservation in the fluid is, indeed, that the energy flux goes by the speed dw/dk.

2. " How do you actually calculate the group velocity? What wave-lengths and frequencies do you insert into the formula? I mean, there can be hundreds of different waves making up the wave-packet, all with different frequencies and wavelengths. Do you just use the averages of their wavenumbers, or something?"

You might possesses the function w(k). Then you differentiate it.

If we look at linearized, potential theory, for the case of a flat bottom (at depth z=-h from still water level z=0), we have, when neglecting effects of surface tension:
$$\frac{\omega}{k}=\sqrt{gh}\sqrt{\frac{Tanh(\kappa)}{\kappa}}, \kappa\equiv{kh}$$
This is called the linear dispersion relation for waves.
We see that in the shallow water limit, when the nondimensionalized wavenumber $\kappa\to{0}$, the wave speed is INDEPENDENT of the wave number, and equals $\sqrt{gh}$
On deep water (kappa to infinity), Tanh(kappa) goes to 1, and the deependency on "h" disappears, yielding
$$\frac{\omega}{k}=\sqrt{\frac{g}{k}}$$

 

[Nikitin]

thx for the reply :)

2) I know you differentiate it, but after that you need to insert a wave-number into the formula (for the group velocity, v_g = dw/dk) to get the group-velocity. I thought the group velocity was the velocity of the highest amplitude of the wave-packet.. So how do you find this amplitude?

EDIT: are wave-packets ALWAYS made up of waves which have circa the same wavelength and frequency? In that case things will get allot clearer to me.

 

[Nikitin]

bump? Anyone willing to help me out?

 

[arildno]

Well, it is in general NOT true that the wave packet contains waves of only a SINGLE frequency, but if we look at a particular frequency w_0 for which the frequencies can be regarded as locally SLOWLY VARYING (i.e, in terms of change in wavenumber) about that fundamental frequency, THEN we get the nice properties of the wave packet that we want.
However, for nasty dispersion relations, where for certain frequencies it cannot be regarded as slowly varying with respect to the wave number, the simplest derivations and interpretations of the group velocity must be nuanced a bit.
I made a quick glance at the Wikipedia on "group velocity", and it is an accessible source that seems fairly well balanced and solid. (It's been a while since I was into this, so I cannot be regarded as giving "expert testimony" here!)

 

[jtbell]

I thought the group velocity was the velocity of the highest amplitude of the wave-packet.. So how do you find this amplitude?

Generically, how do you find the maximum of any function A(k)?

 

[Nikitin]

Well, it is in general NOT true that the wave packet contains waves of only a SINGLE frequency, but if we look at a particular frequency w_0 for which the frequencies can be regarded as locally SLOWLY VARYING (i.e, in terms of change in wavenumber) about that fundamental frequency, THEN we get the nice properties of the wave packet that we want.

Yeah that's what I hoped: that the frequency/wavelengths are circa/almost the same for all the waves in the packet. If that is the case, and if the wave-packet is symmetrical, you can calculate the group-speed for the wave-packet by inserting a suitable k-value? What's this k-value, though?

However, for nasty dispersion relations, where for certain frequencies it cannot be regarded as slowly varying with respect to the wave number, the simplest derivations and interpretations of the group velocity must be nuanced a bit.
I made a quick glance at the Wikipedia on "group velocity", and it is an accessible source that seems fairly well balanced and solid. (It's been a while since I was into this, so I cannot be regarded as giving "expert testimony" here!)

I tried looking at wikipedia, but I didn't understand much.

Generically, how do you find the maximum of any function A(k)?

Ahh, so you are saying one should take a fourier-analysis of the wave-packet, and then find for which k the amplitude is greatest? Then insert the k value into the formula for v_g? I assume a prequisite for this is that the wave-packet is built up by waves which only have wavelengths around a central one?

 

[arildno]

"If that is the case, and if the wave-packet is symmetrical, you can calculate the group-speed for the wave-packet by inserting a suitable k-value? What's this k-value, though? "

That depends on the dispersion relation, and are implicitly defined through that to be those intervals of k over which w doesn't change much. (It might be more than one k in which this holds)
In such regions, we have a very good approximation of the wave frequencies with the linear terms in the Taylor series,
$$\omega=\omega_{0}+\frac{d\omega}{dk}|_{k=k_{0}}(k-k_{0})$$

(Clearly, slowly varying can be taken to mean that the interval of k's here about k_0 is not ridiculously small, and essentially depends upon the smallness of magnitude of the higher-order derivatives of w with respect to k)

Thus, wave trains predominated by such wave numbers will exhibit nice group velocity properties, wave trains dominated by a range of nasty k's won't have such nice properties.

 

[Nikitin]

ok, I understand it now. thanks :)

but one last thing, as a bonus: what about those wave-packets where each wave have widely different k-values? Like water-waves, maybe? What do you do with those?

 

[arildno]

Well, if you go now back to the Wikipedia article, it writes that you can experience, that the fastest component waves will move ahead of the shortest wave components within identifiable wave packets, resulting in a general distortion phenomenon of the wave packet.

Quote:
"If the wavepacket has a relatively large frequency spread, or if the dispersion \omega(k) has sharp variations (such as due to a resonance), or if the packet travels over very long distances, this assumption is not valid. As a result, the envelope of the wave packet not only moves, but also distorts. Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back. Eventually, the wave packet gets stretched out."