Re: Beats

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Nick G. writes:
> ...
>     At a retreat over the weekend, I came across a device (which may have
> once served as the cabin's dinner bell) that really has me puzzled.  It
> consists of what looks to be a solid aluminum rod, about 20 cm in length
> and 1 cm diameter, suspended by a pair of fine strings (looped around the
> rod) at about the 5 cm and 15 cm points from one end.  When struck with the
> supplied rubber-headed mallet, the rod emitted a nice high-pitched tone -
> and beats!  The beat frequency was in the 1-2 Hz range and was both clear
> and reproduceable.  In fact, the beats were considerably more noticeable
> than the ones I *try* to produce in class with a pair of few-Hz-apart
> tuning forks.
>
>     I can't for the life of me figure out why a single rod would produce
> two strong tones a Hertz or two apart in frequency when the dominant tone
> was in the 800-900 Hz range.  Do the strings introduce some weird boundary
> conditions?  Is there a plausible explanation if the rod were, in fact,
> non-uniform?  Could temperature differentials across the rod (in a
> non-uniformly-heated cabin - but it's a short rod compared to the length
> scale of the temperature differences) have anything to do with it?  I
> didn't have the opportunity to do extensive tests on the device, but this
> really has me bothered.
>
>     Any insights?  Or am I missing something completely obvious?

If the rod had perfect axial symmetry its bending modes perpendicular to its
long axis would be degenerate when bending in either the x-z plane or in the
y-z plane (with the z-axis along the rod's symmetry axis).  If the rod was not
exactly axially symmetric, or if the strings (which break axial symmetry) 
detuned one mode more than the other, these two modes would be become weakly
split causing the beats if the mallet strike excited both of them at once
(which would be pretty hard to avoid).

David Bowman
dbowman@gtc.georgetown.ky.us