Another mechanical analogy of an optical property. Should this analogy
also be dismissed as usless in teaching elementary physics?
Consider a long flat elastic bar. Hit it in the direction of the easiest
bending to generate a mechanical wave. Hit it the direction of the hardest
bending to generate another wave. For obvious reason the phase velocities
of mutually perpendicular waves are different (unless the cross section of
the bar is a square).
Hit this bar diagonally and two waves are generated; they travel in the
same direction with two different velocities. It is a mechaical analog of
a birefrigent crystal. Particles along the blade oscillate according to
the principle of superposition. Their trajectories are ellipses. At some
distances from the origin the ellipses become circles, at others they
degenerate into straight lines, depending on phase differences. These are
well known Lissajous patterns resulting from adding simple harmonic
oscillations at the right angle, as described, for example, in Jenkins
and White.
The analogy does not address the issue of the origin of differences
between v1 and v2 for the electromagnetic waves. But it does show what
should be expected when differences between v1 and v2 are in existence.
I used this analogy to explain quater-wave plates in teaching optics.
The input light is polarized linearly, the output light can be polarized
in many ways, depending on the plate thickness. And vice versa.
I know that differnces between v1 and v2 (for light) are dues to
peculiarities of arrangments of atoms in some crystals. But I do not
know how to explain this clearly. Any help will be appreciated.