The breaking of a glass or beaker does not occur at its "Helmholtz
resonance frequency". The Helmholtz resonator is (ideally) a vibrational
mode of the
air in a rigid-walled vessel like a narrow-neck bottle, not the oscillatory
motion of the cylindrical wall itself. In the Helmholtz oscillation the air
in the neck can be treated approximately as a rigid plug having mass and
appreciable velocity, while the rest of the air in the bottle acts as the
restoring spring against which the neck-plug moves, but which has negligible
velocity itself. When you do the calculation remember that it is the
adiabatic gas law that is appropriate here. (This idealization is
inappropriate to calculating Helmholtz oscillators' overtones, of course.)
I used to do a demonstration for my physics of music students where I blew
the fundamental and three Helmholtz overtones on a "stubby" beer bottle. I
didn't always point out to the students that I had modified the embouchure
by grinding a sharp wedge into one side of the lip of the bottle. This makes
the feat of blowing the overtones easier for an untalented klutz like me.
If one wishes to refer eponymously to the resonances of a cylindrical wall
loaded by surrounding air, then I suspect Lord Rayleigh worked that out.