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Re: Beats



Perhaps the longitudinal and transverse vibrational modes are very close
in frequency???

I think that's unlikely. The *two* lowest transverse modes are probably
very close in frequency. I am guessing that the rod was struck from the
side and not on its end. Such a system could be made by having a rod that
deviates slightly from having a circular cross-section. I would also
suggest that perhaps the rod was hollow rather than solid. Though I have
not yet done the calculation, I will be able to test that by doing so
given the 900 Hz frequency and the approximate dimensions of the rod (or
someone else could do it first). The ends were perhaps then closed and
the whole polished to look solid.

I was once asked by the Vancouver Opera Association if I could fabricate
a single chime tuned to (I think I remember) a low F. It is used in the
Opera "I Puritani" just once! There is already one such chime in North
America and it travels from threater to theater where the opera is
produced. They got a commitment to fly the chime to Vancouver in time
for the performance, and I didn't have to come through, though it would
have been easy to do so**. That chime would have been a hollow tube open
at both ends and suspended vertically on two taut strings through small
holes in the tube*. Beats can be a problem with such chimes unless they
are carefully made. I suspect they can be produced at will using a
beatless chime and squashing it slightly. The mass is the same for
transverse oscillations, but the stiffness differs slightly in the two
axes of the elliptical cross-section, leading to two normal frequencies.
The bar is suspended in such a manner that when it is struck from the
side both modes are excited with roughly equal amplitudes. If the struck
point is moved around by 45 degrees the beat should be less noticeable,
I guess.

Leigh

*See Harry F. Olson "Music, Physics and Engineering". The strings are
located at 0.2242 of the bar's length from each end for a thin bar,
well within the tolerance of "about the 5 cm and 15 cm points from one
end" of a 20 cm bar.

**Evelyn and I did receive comps and had a nice backstage tour anyway.
There are perqs associated with being an off-diagram physicist.