Re: Resonance frequencies

Resonance is a fascinating topic -- examples are found everywhere
in nature.  The topic is not without some complexity, however.
For example, there is a considerable difference between the resonance
of something like a violin string, say, and of an atom or molecule.
Violin strings, tuning forks, bells, pan lids, etc have acoustic
resonances.  One way to view this kind of resonance is in terms of
waves ("sound") traveling through the object.  The wavelength is
just (speed of sound in material)/(frequency of vibration).  For
a simple object, like a taut string, the condition for resonance
is that an exact number of half-wavelengths fit into the length of the
string.  The "fundamental" resonace is when just one half-wave fits:
the ends of the strong don't move, but the middle has maximum
motion.  The next resonace up has two half-waves (that is one wave).
Again the ends don't move, but now the middle doesn't move either!
Instead, there are two places where the motions are largest, (1/4) of
the way in from each end. A skilled violinist can use the bow to
excite these higher resonances.  Now for two- or three- dimensional
objects (pan lids, bells, etc), the situation becomes rather more
complex.  The motions of the object have an intricate two- or three-
dimensional structure: pretty to look at, but kind of hard to describe
in text.  Still, the basic idea is "fitting" the wavelength into the
available space.  And, again, there will be a "fundamental" resonace
and many others at higher frequencies.  To add to the fun, resonances
of this kind can occur not only with sound waves, but pretty much
any wave.  An example is the microwave oven.  The device that produces the 
microwaves (which are just high frequency radio waves, i.e electro-
magnetic waves) is called a magnetron.  The frequency is set, basically,
by the size of a cavity inside this gadget -- analagous to the
length of a string in the case I first discussed.

For atoms and molecules, a different kind of resonance is involved.
This resonance is more like that of a pendulum than a violin string.
The resonant frequency is set by physical parameters of the system.
For a pendulum the only parameters that matter turn out to be the
length and the strength of the local gravity.  (That this is so is
_far_ from obvious!) Since the strength
of local gravity is fairly constant everwhere on the surface of the
earth, the length pretty much determines the frequency.  This handy
fact is why clocks used to have pedulums to set the rate at which
they "ticked".  Nowadays, we have much more accurate clocks that use
atomic vibrations.  Using these, we can turn the pedulum "clock"
around and use it to look for small variations in the strength of
gravity (useful for geologists).  For atoms and molecules, the
various resonant frequencies are (I simplify here a bit) set by
fundamental physical constants -- they can't be "tuned" like a
violin string.  The 2.3 GHz resonance of water is due to flexing
motions in the water molecules (at least I think so...I don't have the
appropriate reference handy as I write).  Because the water molecules
are packed together rather tightly in a liquid, the resonace is
fairly broad (as a guess, something like 10%).  I can't think of
a simple way to explain this -- it's a consequence, basically, of the
uncertainty principle: (fuzziness in time)*(fuzziness in frequency)
is always more than one.  The molecules get banged together at short
times, and the frequency will be broadened to something of order
1/(time between collisions).

Hope this helps.