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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.

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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.

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  • $\begingroup$ Thanks a lot for your help! Your instruction really helps. You gave two examples, both of them show that the geometry of medium can cause a dispersion. On thing I'm not sure about is the meaning of 'wave geometry' in the 3rd paragraph. Does that mean the nature of the wave, like what kind of wave, sound wave or lights? Or it means the geometry of the medium as you talked about later after that words. $\endgroup$
    – Wu Jeremy
    Jun 29, 2016 at 16:54
  • $\begingroup$ @WuJeremy It means the shape of the total field distribution: the detailed vector fields as a function of position. For example, in a TEM mode in a two-conductor transmission line, both $\vec{E}$ and $\vec{H}$ fields are completely transverse, there is no component in the propagation direction. The modes of hollow waveguides and optical fibers always have a component of at least one of the field vectors in the propagation direction complicated relative orientation. $\endgroup$ Jun 29, 2016 at 23:43
  • $\begingroup$ @WetSavannaAnimalakaRodVance - You should also mention that waves can modify/affect a medium (e.g., EM waves can induce fields in a plasma that react with the original incident wave, thus affecting its propagation). That "communication" with the medium effectively "slows down" the wave and can alter its propagation direction. $\endgroup$ Jun 30, 2016 at 0:55

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