Re: why can't you break a square sheet of glass with resonance

Dear Ralph,
A fine question, indeed.  At first it was not obvious that you absolutely 
cannot shatter a square piece of glass using resonance.  But I will 
attempt to shed some light on the subject.

The resonant frequency of glass depends on a number of factors, including 
shape, size, and chemical composition, but also on the physical stresses 
that are present in the glass crystal itself.  All these will contribute 
to defining the natural frequency of a particular glass sample.  The shape 
of the glass really relates to the boundary conditions of the glass.  When 
a glass piece is vibrating at it's natural frequency, it means that there 
exist a number of preferred complex vibrational modes that are active on 
the surface.  In order to use the resonant frequency like an antenna to 
transfer acoustic energy into the crystal structure of the glass, a large 
subset of those resonant modes needs to be free to vibrate.

A slight diversion: consider a radio antenna.  An optimized antenna is a 
whole fraction of a wavelength, preferably 1, 1/2, or 1/4.  The smaller 
the antenna gets, the fewer vibrational modes are availble for the 
incoming radio wave to excite in the antenna, so the weaker the signal.  
Also, the harmonic waves (less than 1 wavelength) amplitudes decay rapidly 
with decreasing wavelength, so not only are there fewer modes, but the 
permitted modes are the weaker ones.

In solving the 3D wave equation, the solution for a circular boundary 
(such as a round drum) are Bessel functions, while the solution for a 
square boundary is limited to sines and cosines.  This limitation of the 
square drum head to vibrating in sines and cosines may be why breaking a 
square piece of glass is more difficult (but probably not impossible given 
enough input acoustic energy).  Read on:

Consider the case where a wine glass is gently struck and allowed to 
ring.  Then, if you place your finger on the glass somewhere and strike it 
again, the resonant pitch is the same, but the amplitude is reduced.  This 
is because you are forcing a node (point at which no surface motion is 
allowed) by placing your finger on the glass.  This will prohibit certain 
vibration modes that require an antinide at the point where your finger is 
on the glass, so you are restricting the available vibrational modes of 
the surface, and reducing the overall energy in the surface vibrations 
making the amplitude of the sound less.  This is much like the square 
boundary condition limiting the vibrational modes to sines and cosines- 
there are a large number of other modes that are not possible given the 
boundary condition.

Back to coupling acoustic energy into the glass.  If there are fewer 
permitted vibrational modes, then the ammount of availble acoustic energy 
that will couple into the glass will also be reduced.  In order to couple 
enough energy into the crystal structure to break it, you would need to 
increase the input energy accordingly.  There comes a point, however, 
where the input energy is so great that it may no longer be resonance that 
is causing the shattering, but just the compression of the airwaves 
against the surface (i.e., the sledge-hammer approach).

So it _may_ be possible to break square glass given the right conditions, 
but these will change from sample to sample, and is much more difficult 
than for other glass shapes such as a round piece or a bell-shape (wine 
glass).  This sounds plausible to me, but I don't have any direct 
experience with coupling acoustic energy into glass of various forms.

Good luck,
Drew Procyk