Conventional belief is that an acoustic resonator will amplify sound
at its resonance frequency.  Lots of applications tend to lead your
thoughts that way, but in general this is a _misconception_ :

A free resonator in a sound field _eats_ energy!

The classical example is the Helmholtz resonator, or 'tone bottle',
a cavity with a hole or a throat.  The resonance frequency of that
device is F=(0.5*c/pi)*sqrt(S/LV).  pi=3.14...  You can use any units
that are coherent, same length unit in all four parameters, two
alternatives given here:

speed of sound  c    343   meters/sec     1125  feet/sec,
throat length   L          meters               feet,
hole area       S   square meters        square feet,
cavity volume   V    cubic meters         cubic feet.

L includes end corrections.  These are difficult to compute, but are
totally something like the hole diameter.  Basically the enclosed
volume constitutes a compliance (spring) element and the air in the
throat a mass element.  Caveat: for the formula to be valid all
dimensions must be appreciably smaller than the actual wavelength.

When you put this device in a sound field at the resonance frequency
it will expose a low acoustic impedance at the hole and the sound level
will drop in the vicinity, sound pressure is 'shorted out'.  But inside
the bottle, where you don't hear it, sound level goes up and the energy
is dissipated into heat.  Helmholtz resonators are a standard means for
selective sound absorption in architectural acoustics.

Paradox example: In several idiophones like xylophone, Vibraphone,
celesta, the vibrating bar is basically a dipole, two close sound
sources (upper and lower sides) in opposite phase such that they almost
cancel (forget about the weak sources at the ends of the bars).  That
you hear anything at all is because these sources are generally at
somewhat different distances from the listening point -- maximally in
the up/down direction, then the distance difference is about the width
of the bar.  When you reduce _one_ of those opposing sources with a
resonator (or just by boxing it in, as with a conventional closed-box
loudspeaker) then there is less cancelling and the total radiated sound
increases.  Just because you have taken some sound away, not added any!

Helmholtz also used another trick.  He drilled a little extra hole into
the cavity and put a short tube in it connecting to his ear.  Thus he
could listen to that amplified internal sound and use the device as a
frequency analyzer.

This is a very important variation, the use of the resonator as a
two-port, with entirely different impedance characteristics at the two
ports.  (Acoustic impedance = pressure divided by flow).  In principle
a low impedance 'open end' and a high impedance 'closed end'.  Then its
action should really not be described as that of a resonator, but
rather a filter or a transformer.  You input something at one of the
ports and this will come out at the other port, modified some way.

Example: Bass reflex loudspeaker.  At resonance the back side of the
speaker cone meets a high impedance reducing the cone excursion.  The
high pressure inside the box is transformed into high velocity in the
port which then delivers a greater part of the total flow from the
complete speaker device.

In the organ pipe field we mostly use tube (line) resonators rather
than Helmholtz cavities.  A tube closed at one end behaves very much
like a Helmholtz resonator at its fundamental quarter wave resonance.
But then you additionally have the characteristic phenomenon that it
has periodic higher resonances -- it is no more small compared to
wavelength.

In a flue pipe a primary function of the resonator is to provide a
reactive impedance, pressure out of phase with the flow, and arranged
such that its (mouth) flow controls the direction of the driving flue
jet.  This establishes the oscillation where the jet supplies a
generator signal similar to a square wave.  This is then further
modified by the filter action of the resonator tube.

Johan Liljencrants
Stockholm, Sweden