Raise a Glass

Last night, we drank champagne to celebrate the end of my treatment.  The flutes we used got me thinking about the old trick of running a wet finger around the rim of a wine glass to make a tone.  I wondered whether it would be possible to predict the frequency of the tone based on the shape of the glass.  I tried it out on three different glasses:

According to Wikipedia, the resonant frequency of a cone (which we'll approximate the leftmost two glasses as) is the same as that of an open cylinder, which is given by

where n is any positive integer, v is the speed of sound, L is the length of the tube (in this case the height of the inside of the glass), and d is the diameter of the tube (in this case the diameter halfway up the glass).  The rightmost glass we'll approximate as a closed cylinder, whose resonant frequency is

I tabulated the results of these equations, and the frequencies I actually measured below:

Glass	Small Cone	Large Cone	Flute
n	1	1	3
L (cm)	4.62	7.61	11.3
d (cm)	7.23	7.5	5.49
Theoretical Freq. (Hz)	1914.49	1416.18	1906.12
Measured Freq. (Hz)	1850.3	1548.9	1704.7
Percent Error	3.47	8.57	11.82

I'm impressed by how well the results fit the theory, given all the approximations involved.  Even a perfect cone or cylinder would not be exactly described by the equations above, but it's nice to see that they work well on even an imperfect example.