At 9:16 PM on Sat 11/26/2005 Brian Whatcott wrote:
"If a transverse excitation can lead to longitudinal vibrations,
why couldn't a longitudinal stimulus give rise to transverse vibratio=
ns?"
I think this is precisely what happens in the discussed experiment. =
Rubbing a rod along its length produces longitudinal waves propagatin=
g along the symmetry axis, but it also gives rise to transverse wave =
propagating along the radial directions, that is perpendicular to the=
symmetry axis. As a result we actually have a system of longitudinal=
and transverse standing waves. Of course the fundamental frequency o=
f the transverse wave would be much higher (even though it propagates=
slower than the longitudinal wave) due to the geometry of the rod, a=
nd its coupling with surrounding air is generaly weaker, so the domin=
ating contribution to the audible sound would come from the longitudi=
nal wave. But this situation can be reversed, for instance, by changi=
ng geometry, - say by increasing the thickness of the rod and dicreas=
ing its length so it becomes a disk. If its center is fixed and we ru=
b as before (which now is rubbing against the edges along the symmetr=
y axis), then, I think the transverse wave will be overwhelmingly pre=
dominant. Thus, either one of these two opposite situations can be ob=
tained as continuous extention from the other. So in principle we sho=
uld be aware of the possible transverse contributuion as well. Partic=
ularly, I wonder - would it be experimentaly feasible to "amplify" th=
e transverse contribution in the rod by attaching rigidly a thin plat=
e to one of its edges, and then rubbing the rod as before, but close =
to this edge, without touching, however, the plate? Then we could use=
for detection a high frequency tuning fork? This (or similar) kind o=
f experiment could demonstrate that the actual picture is, as John De=
nker correctly said, rather tricky, and generally we have a superposi=
tion of the two types of waves.=20
Another aspect of this trickiness involves allowed propagation modes=
. Even if we separate (or consider only) one of the states, say, pure=
longitudinal vibrations in the rod, they will not be longitudinal as=
one might think due to loose use of the language. That is, they are =
longitudinal with respect to local direction of propagation of the c=
orresponding momentum eigenstate in Cartesian coordinates, but the mo=
mentum itself, due to diffraction, will not be along the rod, and so =
the vibrations will not generally be along the rod, either. So when I=
said in the beginning "longitudinal waves propagating along the symm=
etry axis", I was not exactly accurate. It is only the net momentum r=
esulting from the superposition of all allowed plane waves, that will=
be parallel to the rod, but even so the local "vibrations" may gener=
ally be elliptical. =20