Is dispersion a property of medium or wave I got a problem in learning Lewin's 'Vibration and Waves'. He first solve the eigenvalue of 5 beats connecting together and with fixed boundary condition. Then the velocity for different modes are calculated, and found that velocity depends on the wave length.
   My question is, for me, it looks like 'dispersive' is a property of the medium. But here, in calculating the dispersion relation of the 5-beats, We didn't give any constraint on what kind of beats are they. It seems to me that 'dispersive' is a property of the wave itself in this case. So can anyone give me some help on it. Thanks!
In my own opinion, I think in this case, the dispersion relation comes from the boundary condition we use (fixed boundary condition), that is the only place that can be correlated with the property of the medium. 
 A: Dispersion in waves arises from both material property variation with frequency and from the geometry of the fields in question.
That wave dispersion will arise from material property variation is obvious.
But wave geometry and boundary conditions also matter. Simple example: a conductive waveguide with rectangular cross-section with sidelengths $a$ and $b$ and vacuum within. With perfectly conducting walls, the electromagnetic boundary conditions enforce the dispersion relationship:
$$k =\sqrt{\omega^2\,\mu\,\epsilon - \left(\frac{\pi\,n}{a}\right)^2- \left(\frac{\pi\,n}{b}\right)^2}$$
where $m,\,n$ are integers greater 0 representing the number of half cycles of the transverse wave. Work out the group speed $\mathrm{d}\omega/\mathrm{d} k$ and you'll very quickly see it depends on the waveguide's dimensions (and thus geometry).
This is well illustrated further by an optical waveguide; specifically a single mode waveguide. Even with dispersionless materials, there is a complicated eigenvalue equation (see Snyder and Love, "Optical Waveguide Theory", Chapter 12) that means the dependence of the wavenumber on frequency is geometry dependent. Indeed, special refractive index profiles can be engineered to shift the zero dispersion point exactly into the lowest loss, 1550nm wavelength band, even though the natural zero dispersion point for a silica step index profile waveguide is roughly 1300nm wavelength.
