# Understanding the dispersion of waves

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

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hutchphd
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.

• stephen8686
Chestermiller
Mentor
My understanding is that waves disperse because of dissipative processes like viscosity that are present in all real fluids.

hutchphd
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.

• Merlin3189
256bits
Gold Member
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.

• hutchphd
sophiecentaur
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