Pipes and Harmonics

Why do closed conical bores have the same set of resonances as open cylindrical bores of the same length, whereas closed cylindrical bores of the same length have only odd harmonics starting one octave lower?

The bores of three woodwind instruments are sketched below. The diameters are exaggerated. The flute (top) and clarinet (middle) are nearly cylinders. The oboe (right) is nearly conical (as are the saxophone and bassoon). The clarinet is about the same length as the flute, but plays nearly an octave lower. The oboe is closed like the clarinet, but its range is close to that of the flute.

     photo of flute, clarinet and oboe

For a background to this discussion, it is worth looking at the difference between closed and open pipes, which is explained in Open vs closed pipes (Flutes vs clarinets), which compares them using wave diagrams, air motion animations and frequency analysis.

To compare cylindrical, conical, closed and open pipes, let's look first at diagrams of the standing waves in the tube.
 

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