Dispersion of waves - wavephysics

In summary: Taylor series expansion of w(k), which are neglected in the right hand side of the above)In summary, the group velocity is given by the formula v_g = \frac{d \omega}{dk}, where \omega is the wave frequency and k is the wavenumber. This formula can be derived by using the superposition principle to add all the wave-functions making up the wave-packet and then using trigonometric identities to find that the wave-packet is like a "beat-phenomenon" moving with a velocity equal to the group velocity through the medium. To calculate the group velocity, one needs to differentiate the function w(k) and then insert a suitable k-value, which is defined by the dispersion relation
  • #1
Nikitin
735
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Hey! Two questions:

1) How exactly does the proof go for the formula for the group-velocity:
[tex] v_g = \frac{d \omega}{dk}[/tex]
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
[tex] A \sin(\omega_0 t -k_0 x) \cos(\omega_{beat} t -k_{beat} x)[/tex]
? And then finding out that [tex] v_g = \frac{\omega_{beat}}{k_{beat}} = \frac{d \omega}{dk} [/tex]?
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?
[tex] v_p = \frac{\omega}{k}[/tex]
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:
[tex] v_p = \frac{\omega_0}{k_0}[/tex]

?
 
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  • #2
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. :smile:

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:
[tex]\frac{\omega}{k}=\sqrt{gh}\sqrt{\frac{Tanh(\kappa)}{\kappa}}, \kappa\equiv{kh}[/tex]
This is called the linear dispersion relation for waves.
We see that in the shallow water limit, when the nondimensionalized wavenumber [itex]\kappa\to{0}[/itex], the wave speed is INDEPENDENT of the wave number, and equals [itex]\sqrt{gh}[/itex]
On deep water (kappa to infinity), Tanh(kappa) goes to 1, and the deependency on "h" disappears, yielding
[tex]\frac{\omega}{k}=\sqrt{\frac{g}{k}}[/tex]
 
  • #3
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.
 
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  • #4
bump? Anyone willing to help me out?
 
  • #5
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!)
 
  • #6
Nikitin said:
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)?
 
  • #7
arildno said:
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.

jtbell said:
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 vg? I assume a prequisite for this is that the wave-packet is built up by waves which only have wavelengths around a central one?
 
  • #8
"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,
[tex]\omega=\omega_{0}+\frac{d\omega}{dk}|_{k=k_{0}}(k-k_{0})[/tex]

(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.
 
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  • #9
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?
 
  • #10
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."
 

1. What is dispersion of waves?

Dispersion of waves refers to the phenomenon where waves of different frequencies or wavelengths travel at different speeds in a medium. This leads to the separation of a wave into its individual frequency components, resulting in a change in the shape of the wave.

2. How is dispersion of waves different from refraction?

Dispersion and refraction are related but different concepts. Refraction refers to the change in direction of a wave as it passes through a boundary between two different media, while dispersion specifically refers to the separation of waves based on their frequencies in a medium.

3. What causes dispersion of waves?

Dispersion of waves can be caused by a variety of factors, including the properties of the medium through which the wave is traveling, such as density and elasticity. It can also be caused by interactions between the wave and particles in the medium, such as scattering or absorption.

4. How does dispersion affect the speed and wavelength of a wave?

Dispersion can cause different wavelengths of a wave to travel at different speeds, resulting in a change in the overall speed of the wave. This can also lead to a change in the wavelength of the wave, as different frequencies are affected differently by the dispersion.

5. What are some real-life examples of dispersion of waves?

Dispersion of waves can be observed in various natural phenomena, such as the separation of colors in a rainbow, the different speeds at which different frequency radio waves travel in the ionosphere, and the distortion of sound waves in a concert hall due to different frequencies traveling at different speeds.

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