A mouthpiece’s phase shift compresses tube resonances into a playable harmonic series

Evidence

The third effect of the mouthpiece is to shift the tube resonance frequencies so as to pack them closer together. Explanation: Across a resonance, the phase of the drive relative to the velocity shifts 180 degrees. The consequence for the mouthpiece Helmholtz resonance is that the air just downstream of the mouthpiece changes from an antinode to a node and back to an antinode of the opposite sign as frequency goes across the resonance from below to above. This is equivalent to a shift of half a wavelength in effective length of the trumpet! (Any phase shift near the end of the tube is equivalent to a change in the length of the tube, since the same phase shift could also be obtained by lengthening the tube; the air next to the mouthpiece would be none the wiser.) Changing an antinode to a node at the constriction of the mouthpiece is the same as adding 1/4 wavelength to the tube length (see figure 16.14). The growth of the effective length means that the next resonance shows up early in frequency, causing the resonances to pack closer together. The lowest peak is unaffected by the Helmholtz resonance (it is too far below it) and will end up being discarded as a playing frequency. Benefit: This is a significant step toward fixing the half-open tube spectrum of the trumpet missing half its overtones. The idea is to compress and shift the existing partials so that they form a harmonic series based on every partial of a fundamental f0, except the first (see figure 16.11). — Eric J. Heller, Why You Hear What You Hear- An…, 1494

Categorization

• DEFINE: Explains the mechanism by which effective tube length changes across a resonance — an antinode-to-node shift equals a quarter-wavelength addition.
• XREF: The ‘sculpting principle’ and effective-length tricks echo earlier instrument acoustics chapters; connects to why trumpets have mouthpieces at all.