A while since this question has been asked, but none of the above responses really captures the physics, in my opinion.

@Evert is most definitely correct in the 'reversal' argument, i.e. excite the eigenmodes and then use this to determine the resonant mode.  However this doesn't enable you to predict what will happen.

The key to answering this problem is to realise that the speed of sound for the glass you have described is not the excitation that is causing the glass to break.  You have quoted the bulk speed of sound in glass, which is the correct speed for determining propagating of sound waves, for example through a large block of glass.

Instead, the wine glass is being destroyed by resonance in the flexural modes, and so the correct treatment is due to cantilever equations.  By looking at the equations and treating the wine glass as a cantilever, one can see that the resonance frequency should be proportional to the thickness divided by length to the power of (3/2). (Here I'm ignoring the change in geometry from linear to circular).

So using the standard reference, Wikipedia (https://en.wikipedia.org/wiki/Cantilever), we can write down the resonant frequency of a cantilever as

\omega = \sqrt(E w t^3/(4 L^3 m))

where \omega is the resonant frequency, E  as the Young's modulus of glass, w the effective width of the cantilever, t as the thickness, L the length and m the mass.

Now of course, I don't know most of these numbers, but I can do an order of magnitude calculation, so:

E = 50 \times 10^9 Pa (http://www.engineeringtoolbox.com/young-modulus-d_417.html)

Only part of the glass is oscillating, so let's choose an effective width as 1 cm, so 

w = 1 \times 10^-2

thickness will be closer to 1 mm than anything else, length of order 10cm

mass is also unknown, but let's assume 10 g (remember this is the effective mass of the resonant part).

Caveats here: of course any real calculation must take into account boundary conditions, as discussed by Kai Fauth

Anyway, plugging these numbers into the formula gives \omega = 100 Hz.  Now this is close enough for an order of magnitude calculation to explain the discrepancy between the values that you've quoted, and also helps to explain why the mode shown in the Youtube clip appears to be the fundamental mode of the clip.

All the best

Andy

I'd guess that the resonce frequencies are determined rather by boundary conditions (material shape) and elastic properties than by intrinsic materials constants (speed of sound (acoustic phonon dispersion relation)). Now of course elastic constants are related to the phonon properties and probably links can be stablished.

I think it's much the same as with string instruments: why can you tune them with tension on the strings. You certainly do not alter the phonon dispersion relation by pulling on the strings (harmonic/elastic regime). Yet, you change the restoring force for a lateral displacement of the string. And as we know, the mass of the string does play a role, not just intrinsic material properties.

Now how to properly set up that interrelation is an interesting question!!

I guess it has to do with the fact that we are dealing with transverse modes and finite material thickness. As opposed to how we set up our equations for homogeneous materials, at finite thickness the restoring force is very small for an in phase movement of all constituents (volume elements or atoms if you like). So there probably are low frequency modes for finite volumina which don't correspond with what we consider speed of sound.

You could also link this to the modes of the Tacoma bridge - take typical frequencies there and compute the wavelength based on the speed of sound of concrete. In that case, torsional modes were active, too.

This is an intriguing question. I went to Wikipedia and found this:

http://en.wikipedia.org/wiki/Acoustic_resonance

I think a wine glass as shown in the video can be approximated as a sphere with an opening, as discussed in the Wikipedia article. The equation they show is

D = 17.87 * cubed root (d / freq^2)

where: (in meters)

D = diameter of sphere

d = diameter of sound hole

Taking their word for it, and assuming a diameter of the open part of the glass to be 8 cm, the speed of sound though air 343 m/sec at 20 deg C, and 300 Hz to be the resonant frequency, the diameter of the wine glass would have to be 17 cm, for it to resonate at 300 Hz. That's a little larger than most wine glasses, but hardly a meter wide or more.

I think the glass material has nothing to do with the problem, except that it will break more easily than many other materials.

So, it's not so far fetched to think that perhaps the first or second harmonic is causing the break.

Take a glass, crystal preferably, and touch it with a small metal stick. You will hear a tone: the eigenfrequency of the glass.

If you produce the same frequency singing (or with a tone generator), the glass will resonate, may even break.

The glass harmonica (a musical instrument) shows that the eigenfrequencies depend on the size and properties of the glass; not on the volume of air enclosed in the glass.

I think the sound speed involved should be that of the bending wave in the thin-walled glass. It can be much smaller than that in the air.

I found this other source, which may explain how the 300 Hz sound impacting on the glass can cause it to shatter.

Going back to my previous point that the sound is causing the glass to resonate like a sphere, or now let's say a bell instead of sphere, the next question is what are the harmonics created by such devices? Well, it seems there are a bunch of harmonics:

http://computermusicresource.com/Simple.bell.tutorial.html

The example they use is a bell whose fundamental is 200 Hz. Not terribly different from our glass. It turns out, bell sounds are very difficult to model with an analytical math model, however to emulate the sound of the bell, the harmonics you need vary from 0.56 * the fundamental frequency, to 4.07 * the fundamental frequency, with many others between these two.

PLENTY of opportunity to create one or more standing waves around the circumference of the glass, at various locations, which could break the glass if one or more of these are intense enough.

Elastic waves (longitudinal, extensional, torsional, flexural and several superficial waves) move forward and backward in a solid, even if the bulk of the material is in a state of rest. In this case (if the shape of your object is small) you cannot hear anything. On the contrary if the object is big enough, you can hear the sound due to the flexural waves.

If you have a force enough to "displace" the bulk of the material, it moves following its natural frequency modes, among them the resonance frequency (and the harmonics) of the whole solid (the volume sound source). Rayleigh integral allows to define the amount of sound pressure level due to this vibration.

Colliding forces such as the sound waves onto the glass are additive, therefore greater than the glass alone.

A while since this question has been asked, but none of the above responses really captures the physics, in my opinion.

@Evert is most definitely correct in the 'reversal' argument, i.e. excite the eigenmodes and then use this to determine the resonant mode.  However this doesn't enable you to predict what will happen.

The key to answering this problem is to realise that the speed of sound for the glass you have described is not the excitation that is causing the glass to break.  You have quoted the bulk speed of sound in glass, which is the correct speed for determining propagating of sound waves, for example through a large block of glass.

Instead, the wine glass is being destroyed by resonance in the flexural modes, and so the correct treatment is due to cantilever equations.  By looking at the equations and treating the wine glass as a cantilever, one can see that the resonance frequency should be proportional to the thickness divided by length to the power of (3/2). (Here I'm ignoring the change in geometry from linear to circular).

So using the standard reference, Wikipedia (https://en.wikipedia.org/wiki/Cantilever), we can write down the resonant frequency of a cantilever as

\omega = \sqrt(E w t^3/(4 L^3 m))

where \omega is the resonant frequency, E  as the Young's modulus of glass, w the effective width of the cantilever, t as the thickness, L the length and m the mass.

Now of course, I don't know most of these numbers, but I can do an order of magnitude calculation, so:

E = 50 \times 10^9 Pa (http://www.engineeringtoolbox.com/young-modulus-d_417.html)

Only part of the glass is oscillating, so let's choose an effective width as 1 cm, so 

w = 1 \times 10^-2

thickness will be closer to 1 mm than anything else, length of order 10cm

mass is also unknown, but let's assume 10 g (remember this is the effective mass of the resonant part).

Caveats here: of course any real calculation must take into account boundary conditions, as discussed by Kai Fauth

Anyway, plugging these numbers into the formula gives \omega = 100 Hz.  Now this is close enough for an order of magnitude calculation to explain the discrepancy between the values that you've quoted, and also helps to explain why the mode shown in the Youtube clip appears to be the fundamental mode of the clip.

All the best

Andy