Understanding the dispersion of waves

  • #1
I am trying to learn about the dispersion of waves and used one of Walter Lewin's lectures (see below) as a source. I understand phase and group velocity and dispersion relations, but I don't understand when/what kinds of waves are prone to dispersion.
For example, a simple wave in the form $$y(x,t)=A_{0}sin(kx-wt)$$ will never disperse no matter what medium it's in because there are no "groups" to have a group velocity, right?

As I understand it, to have any dispersion you need a wave in the form $$y(x,t)=A_{0}sin(k_{1}x-w_{1}t)+A_{0}sin(k_{2}x-w_{2}t)=2A_{0}sin(k_{3}x-w_{3}t)cos(k_{4}x-w_{4}t)$$
But this is just the interference of two waves, so can you only have dispersion when you have more than one wave (of different frequency) interfering? So do pulses disperse because, looking at it from a Fourier analysis perspective, they are built from a bunch of waves of different frequencies?

thanks



 

Answers and Replies

  • #2
hutchphd
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Only a pure frequency harmonic wave of infinite extent is a "simple" wave. For finite extent, there are more wavelengths involved (indeed they conspire to suppress the wave envelope at the ends) We always deal with finite extent, particularly in communications where it is vitally important.
The dispersion is as you describe it and Prof. Lewin can tell you the rest far better than I.
 
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  • #3
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My understanding is that waves disperse because of dissipative processes like viscosity that are present in all real fluids.
 
  • #4
hutchphd
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My understanding is that waves disperse because of dissipative processes like viscosity that are present in all real fluids.
I don't think that is strictly true.
The proximate theoretical cause is that ω=ω(k) or equivalently that different wavelenths move at different speeds. The most common example is deep water waves where v=√(λg /2π). Also the dispersion in optical glass (which causes chromatic aberration) is present without concomitant dissipation.
Dissipative process can cause dispersion, but are not the most typical cause.
 
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  • #5
256bits
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concomitant
I had to look that word up.
As an adjective " natural, or associated".
I will have to use it 5 times, as they say, to burn it into my memory.
 
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  • #6
sophiecentaur
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As I understand it, to have any dispersion you need a wave in the form
y(x,t)=A0sin(k1x−w1t)+A0sin(k2x−w2t)=2A0sin(k3x−w3t)cos(k4x−w4t)​
y(x,t)=A0sin(k1x−w1t)+A0sin(k2x−w2t)=2A0sin(k3x−w3t)cos(k4x−w4t)y(x,t)=A_{0}sin(k_{1}x-w_{1}t)+A_{0}sin(k_{2}x-w_{2}t)=2A_{0}sin(k_{3}x-w_{3}t)cos(k_{4}x-w_{4}t)
But this is just the interference of two waves, so can you only have dispersion when you have more than one wave (of different frequency) interfering?
If you take a (temporal) trace of that wave at some point, it will have a certain shape. Move along the path a bit and the (temporal) shape will not change if the two waves have the same speed (c). When c varies, the shape of the wave will change and that's dispersion. I think the devil is in the detail of what must be happening to the relative phases of the two waves as they progress in a non-dispersive medium ( the k's and the ω's). That's a sort of reality check with the evidence that pulse shapes don't change with distance without a dispersive medium.
 
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