What properties define a good resonator?

In my question "Resonance in a 1 ft granite box", someone answered that my granite box (a one foot cube) would make a very poor resonator, which makes me ask what characteristics make a good resonator? Is there a difference between a cube or other rectangular box and a ball (like a Helmholtz resonator)? Also, what defines the pitch? Size?

-
The figure of merit here is $Q$ in the mode or modes you are interested in, and more or less represents the inverse of the fractional energy loss per cycle. – dmckee Dec 28 '12 at 0:31
Perhaps the question of pitch should be deleted from this post, since it is your other question: physics.stackexchange.com/questions/47750/… – Chris White Dec 28 '12 at 1:30
Then I'd be happy to move the pitch part of my answer over to that other question – Chris White Dec 28 '12 at 1:30

"Good" can mean a couple different things. The first thing that comes to my mind is not leaking energy. If you can hear sounds from inside your granite box, then clearly energy is escaping. There is no such thing as perfectly reflecting surfaces for sound (or light or anything else), so there will always be some losses - it is just a question of degree.

Related to this, acoustic energy can be transformed into heat, which would also make for a bad resonator. If you have ever been in a room with exposed insulation in the walls (or lots of draperies as you find in theaters), then you may notice how "dead" sounds are - there are very few echoes, since the sound energy is absorbed by the walls.

As for shape, this comes down to solving a partial differential equation - the wave equation, which may be written as $$\frac{\partial^2u}{\partial t^2} = c_\mathrm{s}^2 \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) u$$ for some scalar quantity $u$, which in the case of sound could be the local pressure. Here $c_\mathrm{s}$ is the speed of sound, which you can often approximate as constant across the small pressure and density variations found. Solutions to this equation satisfying the boundary conditions of your particular geometry give the resonant modes and their frequencies.

With rectangular prisms, solving is easy, since the equation "separates." The solutions can all be written in the form $$u(x,y,z,t) = \sin(k_xx) \sin(k_yy) \sin(k_zz) \sin(\omega t),$$ where the spatial wavenumbers are given by $k_x = \pi n_x/L_x$ for any choice of integer $n_x$ and length in the $x$-direction $L_x$, and similarly for $y$ and $z$. The frequency of the mode (for $L_x$, $L_y$, and $L_z$ fixed by your setup and whichever choices of $n_x$, $n_y$, and $n_z$ you like) is $$\omega = c_\mathrm{s} \sqrt{k_x^2 + k_y^2 + k_z^2}.$$ This is in radians per unit time; divide by $2\pi$ to get cycles per unit time: $$f = \frac{c_\mathrm{s}}{2} \sqrt{\left(\frac{n_x}{L_x}\right)^2 + \left(\frac{n_y}{L_y}\right)^2 + \left(\frac{n_z}{L_z}\right)^2}.$$ If the longest dimension is $L_x = 12'' = 0.305~\mathrm{m}$, then the lowest resonant frequency comes from setting $n_x = 1$, $n_y = n_z = 0$, which gives (assuming $c_\mathrm{s} = 340~\mathrm{m}/\mathrm{s}$) $f = 557~\mathrm{Hz}$.

If you decompose the sound with which you are driving the resonator into its Fourier components, anything with a frequency not matching an $f$ you can get by choosing $n_x$, etc. will not drive a resonance. In particular, anything below $557~\mathrm{Hz}$ will not drive the system. (I'm being somewhat loose here - resonances are naturally "broadened," so you can be close and still have some effect. This is the other meaning of "good" - having narrow response functions.) Intuitively, frequencies that are off-resonance do not have wave patterns that echo constructively, and so there is no build-up of energy in a single mode.

Other geometries can resonate, but they are more complicated. Spheres and cylinders can be done with Bessel functions in place of sines, for instance. However, you will always find that the lowest resonant frequency is something like $c_\mathrm{s}/2L$, where $L$ is the longest dimension of the resonator.

In summary: If your box is a poor resonator it is probably because it is too "leaky." Even if made of a better material (a very stiff plastic may work, or the right kind of glass), you need to drive it at an appropriate frequency to have an effect. Also, it is very hard to demolish a structure with sound. If you want anything spectacular to happen, I suggest looking into treating a crystalline solid as the resonator itself (rather than having an air cavity be the resonator).

-

Wind instruments are all long tubes.

The tube shape is one that gives the instrument builder very predictable control over the pitch characteristics. Longer tube, lower sound. It doesn't have to be a tube. Tube shape is just more predictable.

The opening(s) to the outside must be relatively small in proportion to the volume of vibrating air.

Too small an opening and the sound simply cannot get out. To propagate away from the instrument you need good transfer to the outside.

With a very large opening there is very little internal reflection, hence very little opportunity for resonance.

Generally:
Any form of wind instrument needs a continuous source. With a wind instrument the vibrating mass weighs very little, so any vibration dies away very fast.

At the other extreme there is a church bell. Because of its huge mass the vibrations of a church bell take a long time to decay. Striking a church bell once per second is quite enough, whereas in a wind instrument - even a very big one - the sound will decay in just tenths of a second.

-

I believe resonance occurs when a wave reflects back and forth in a medium whose length and rate of propagation match the frequency of the wave source. In this way the amplitude of the wave and it's reflection are in phase and additively increase.

For best effect, I believe the medium in which the wave propagates has to have some elastic properties and needs to be composed of reasonably uniform material.

This is true (at least approximately) of piano-strings, organ-pipes, tuning forks, bells and glockenspiel keys.

a pile of one-inch thick slabs of granite has a large number of (potentially) internally reflecting surfaces rather than a single reflecting surface. It is unlikely that sound waves propagate from slab to slab in the way they would through a solid cube.

Granite consists of a variety of materials, some crystalline, in a granular mixture. This probably isn't optimal for resonance as I suspect it will cause scattering and irregular attenuation of the wave-front.

To tune your resonant cube I guess you'd have to work out the speed of sound in granite and adjust the resonator's dimensions and/or the frequency to arrange that the waves and their reflections are in phase at the reflecting surfaces.

-