Author: Bassie
Date: 2008-07-08 08:40
It took me ages to get my head around /why/ a conical closed tube overblows an octave:
In a /straight closed pipe/ the pressure distribution looks like a sine wave. You need a maximum at the closed end and a minimum at the open end, so you can fit in a quarter wavelength, or 3/4 of a wavelength, or 5/4 of a wavelength... leading to the odd harmonic series of the clarinet (and its curiously low fundamental notes)
In a /straight open pipe/ you need a minimum at both ends, so you can fit in half a wavelength, or a whole wavelength etc.
In a /cone/ the pressure distribution looks like a modified sine wave known as a 'sinc' function. Sinc is defined as sin(x)/x and looks very much like a sine wave except that it doesn't go to zero at zero, but instead has a maximum. So half a wavelength of sinc has a maximum at one end and a minimum at the other. So does a whole wavelength, and so on and so forth. The frequencies at which a cone will resonate turn out to be the same as those of a straight open pipe, though the cone has a pressure maximum at the closed end and a pressure minimum at the open end just like a clarinet.
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The Clarinet BBoard
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