§5 · The traveling-wave envelope

The traveling-wave envelope

The dispersive traveling-wave envelope — the seed’s deepest [O] — is characterised without tuning: it is a one-parameter family in the sharpness Q. Every LOCATION is forced and Q-invariant; every MAGNITUDE scales with Q and is fixed by no inherited constant, so a number for Q would be tuning.

Modelling the partition as a driven damped resonator on the frozen Greenwood CF(x), the velocity peak sits at ω0=CF for every Q (= the §1 place), the reactance flips sign at CF forcing an apical cutoff, and the −3 dB bandwidth is exactly ω0/Q — 0.1000 vs the law 0.1000 at Q=10. The group delay peaks at CF with magnitude 2Q/ω0.

The peak sits at CF for every Q — the keystone

The envelope is a one-parameter family in Q. The single-pole velocity peak is at ω=ω0 independently of damping, and ω0(x)=CF(x) is the inherited place — so the peak IS the §1 forced place to |Δ|=0 [F][V].

The near-peak FORM is the universal resonance |V(ω)|² ∝ ω²/[(ω0²−ω²)²+(ω0ω/Q)²]; its location is forced and carries no free choice but Q.

The asymmetry and the bandwidth are forced

The asymmetry is forced by the reactance sign. χ = 1−(f/CF)² is positive basal (propagating) and negative apical (evanescent), so a tone is cut off apical of its place — the SIGN robust to the slope scale [F].

The −3 dB bandwidth is exactly ω0/Q: numeric 0.3333 / 0.1000 / 0.0333 / 0.0100 versus the law 0.3333 / 0.1000 / 0.0333 / 0.0100, to machine precision [V]. Only the width carries Q; the place is Q-invariant.

The active amplifier is negative damping

The active process enters as negative damping, Q_eff = Q0/(1−G), and at the critical point it imposes the §2 cube root [F]. The direction (active → sharper, taller, → cube-root compression) is forced; the gain magnitude G is the §2 [O].

The group delay peaks at CF with magnitude 2Q/ω0: 10.0 / 40.0 / 160.0 for Q=5/20/80 [V]. The location of the delay peak is forced; its absolute value in ms is [O].

One knob, and fixing it is tuning

Varying Q ten-fold leaves the peak place invariant while the bandwidth scales exactly ∝1/Q. No inherited constant sets Q — the √-law fixes only the place, and γ is structure only, not a damping [F].

A closed numeric envelope = a choice of Q = tuning (forbidden). §5 does not close the [O]; it characterises it: FORM forced, exactly one scalar Q open, with a proof that writing it down would be tuning.

Honest negatives — what is not claimed

  1. N1. The absolute sharpness is [O] — Q, the −3 dB bandwidth in Hz, the peak gain in dB. Only the FORM, the exact bandwidth law (=ω0/Q), and the peak place are forced.
  2. N2. The apical cutoff SLOPE (dB/oct) is [O] — it needs Q plus the fluid mass-loading prefactor; only the cutoff’s existence and side (apical) are forced.
  3. N3. The active gain MAGNITUDE (the fraction G, the proximity to the bifurcation) is the inherited §2 [O]; §5 forces only the direction.
  4. N4. The absolute group delay (ms), the phase in cycles, and the traveling-wave speed are [O] (they need the fluid hydrodynamics); only that the delay peaks at CF is forced.
  5. N5. The model is the long-wave (1-D, WKB) approximation; the full 2-D/3-D fluid problem, the short-wave region at the peak, and the “second filter” are the deeper [O].
  6. N6. Two-tone suppression, distortion products, and combination tones are downstream of the cubic but are not derived here — [O].
  7. N7. The felt percept of pitch and timbre is the mind volume’s (firewall); §5 moves only the physical envelope.