Narrower pipes raise acoustic impedance through in-phase wall reflections

Evidence

The physical reason for the increase of specific impedance as the pipe diameter decreases is understandable from the cellular picture. The higher specific impedance of a small pipe implies that if a small cell of air is pushed, a neighboring cell will push back harder than it would in a larger pipe. Why should this be? All the pushing and pushing back is of course communicated by the air in the pipe from cell to cell at the speed of sound. Suppose a given cell is being pushed to the right for a time τ; in free space, that push would be communicated in all directions a distance x = cτ in the time τ, where c is the velocity of sound. In the pipe, most of those directions lead to the walls of the pipe, where the pressure pulse created by the push is reflected. Some of the reflected pressure returns fast enough to the cell that was originally disturbed that it leads to an increased pushback, while the original push is still happening and therefore in phase with the pushing, thus increasing the impedance. “Fast enough” is in relation to the frequency of pushing. This suggests the wall needs to be within an eighth of a wavelength or so, to return in phase. Most musical instruments are operating at frequencies such that the wall is always this close or closer. In fact, the pressure pulse doesn’t reflect just once, but many times, depending on the diameter of the pipe. Thus the narrower the pipe, the higher the impedance. — Eric J. Heller, Why You Hear What You Hear- An…, 121

Categorization

• DEFINE: Explains specific impedance via a cellular, reflected-pressure-pulse mechanism rather than as a given.
• XREF: Connects to why brass/wind instruments’ bore shapes affect tone — the physical basis of instrument design.