before I get into this, I will state some basic assumptions of the conditions involved.
These assumptions are to give quantities to the unknown factors necessary for such discussion to take place.
We will first assume that our wine glass is void of all defects, bubbles, cracks, stress-lines, and other such imperfections in the crystals
and since either two of the common methods of calculation are dependent upon temperature and other such factors,
(Young's Modulus or Shear Modulus)
We will further assume that these calculations take place at Sea Level, at SI "standard" temperature, pressure, humidity, etc.
We will also assume there is "standard" ratio of the components of the "air", such that the mathematical constants applied thereto,
do not change throughout the mathematical analysis of the glass' resonance.
I will use Young's Modulus, as it pertains, with respect to glass-ceramics, directly to the velocity and propagation of the wave function.
This is derived using the Bulk Modulus K, and by a proportional constant of nature, both Shear Modulus and Poison's Ratio are defined.
Young's Modulus (defined by a capital "E", not to be confused with Energy) is therefore modified as:
E = pv^2
p is the density of the glass, and v is the longitudinal velocity of the pulse, or wavefront.
as you can see here, we must hold temperature constant, or we change the value of the modulus.
like sticking a tuning fork in the oven
To further simplify the discussion, it will be assumed that the particular wine glass is of known composition.
let us assume this particular wine glass to be composed of a silicate
with a Young's modulus of 75
We will assume the speed of sound in air to be 343 Meters per Second
We will also assume that the wine glass is (mostly) cylindrical, and without odd curvatures.
[Note: there may be other obvious assumptions I may have forgotten,]
[ so the above list may be added to over the next short time]
[ while I attempt to fulfill what has been requested of me.]
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Now, what represents the resonant standing wave inside an (empty) glass, or glass filled with only air:
is considered in two parts.
There is the resonance of the cavity
and the shift in frequency caused by the elasticity of the glass material.
I will deal with the easy part first, the other half would probably take me most of a day to put into words.
The above Young's Modulus equation is already crunching half a page of math into a single line.
Do it again for the resonant cavity, and forgive me for needing a little time to plug both of them together.
it has been the better part of 15yrs since I've dealt with this particular issue.
Such is generally only required in theory, not in practice....
better yet, I have already done way more than necessary I think.
So let me just set this mess on my desk for now, I think you can see where it is leading.
I am not prepared at this moment to fully describe the wave interactions between the soundwave
and the vibrational wave through the glass material
in a coherent mathematical equation that represents both features.
this would be a lot easier if it were a flat, thin rectangle of glass.
Not a round goblet shape, its a mathematical blunder taking the tensor forces over a curvature.
(I think you have gotten the best of my patience on this one!! grr)
But what I will do instead is give you a footing by which you can change your own diaper.
(sorry, my babies are grown, I don't do that anymore)
So,. let us begin:
the frequency of the resonant cavity, formed by the dimensions of the wine glass is defined as:
f = v / 4(h + 0.4d)
where h is the height of the glass from the inside bottom to the rim
d is the diameter
v is the speed of sound
coincidentally:
v = f * wavelength
wavelength = 4(h + 0.4d)
v= f(4(h +0.4d))
and so forth and so on...
If the f represented by the resonant cavity were labeled as F1
the actual frequency resonating from the glass labeled as F2
then,
F1 - F2 = the shift in frequency caused by the glass material, as defined by Young's Modulus.
What you will find with your microphone or other testing apparatus, is both F2 resonating from the glass material
as well as F1 resonating inside the glass (with traces of F2 observable at certain locations therein)
as well as other (lower intensity) odd number harmonics.
these are defined as:
f = N(v) / 4(h + 0.4d)
where N is an odd number.
Why an odd number?
Because, much like our Joule Thief, the standing wave in the wineglass is a Half-Wave.
therefore, only the odd harmonics are not destroyed.
go ahead and test for these harmonic frequencies with your wine glass microphones.
If you find yourself unable to grasp these simple concepts,
you cannot understand what resonance "is".
everything that resonates, does so according to simple natural laws.
distances, velocities, intervals of time, rates of propagation
the things that define frequency
electricity and magnetism do not get a free pass on this one.
it is all the same
