Reeds must vibrate faster than the air column to reinforce resonance

Evidence

When two oscillators have unequal frequencies before they are coupled, as we learned in connection with figure 10.9, one of the two new modes has one of the oscillators mostly excited, and the second mode has the other mostly excited. In the higher frequency of the two modes, both oscillators move with that same frequency, and likewise in the lower frequency mode. There is a mostly air column mode and a mostly reed mode. … We shall therefore seek a solution in cases A or a—that is, supposing the free vibration of the reed to be higher frequency than the air column. … This leaves only one last possibility, case a. This fortunately works well: the mode is mostly air column vibration and the reed opens when the pressure is high in the mouthpiece (resonance enhancement, adding amplitude to amplitude of the same sign). The flow (velocity) does maximum work against the high pressure in the mouthpiece and reinforces the pressure there by constructive interference. In practice, the blown closed reed frequency is set quite high compared to the playing frequencies. The natural frequency of the reed can be heard in the following way: The mouthpiece is removed, the reed is attached, and the open end is placed within sealed lips. Inhaling strongly generates a tone around 2200 Hz for a clarinet reed—very high indeed, above any playing frequencies, and in accord with the conclusions just presented. — Eric J. Heller, Why You Hear What You Hear- An…, 1613

Categorization

• QUESTION: Why exactly 2200 Hz for a clarinet reed — what determines this natural frequency across different reed instruments and materials?
• XREF: connects to: Positive feedback drives the reed to slam shut — David Howard, Jamie Angus, coustics and Psychoacoustics, 1227