A card from a free library — ask anything, no account, works offline. Every card carries its source.
Resonators — Helmholtz, the cavity, Fabry-Perot: one shape, a Q that fades
instruments
Three instruments, one shape. The HELMHOLTZ RESONATOR (1863) is a volume of air on a neck - the air in the neck is the mass, the air in the body the spring - f = (c/2 pi) sqrt(A/(V L')), the lumped limit where the wavelength is far longer than the body: a litre bottle with a 2 cm neck sings at 118 Hz (sealed on the resonators stick; run tick.py resonators). The CAVITY RESONATOR (Hansen 1938, the rhumbatron) is a hollow conductor whose modes carry three integers, f = (c/2) sqrt((m/a)^2 + (n/b)^2 + (p/d)^2): a ten-centimetre copper cube rings at 2.12 GHz (sealed) - the shape of every klystron, magnetron and microwave filter. The FABRY-PEROT CAVITY (1897-1899) is two mirrors: a mode fits when L = n lambda/2, so the modes form a comb spaced c/2L - 1.499 GHz for 10 cm (sealed); the mirrors' reflectivity sets the finesse, pi sqrt(R)/(1-R) = 312.6 at R = 0.99 (sealed), the linewidth 4.80 MHz (sealed), Q = 4.0e7 at 1550 nm (sealed). All three are standing waves on a BOUNDED DOMAIN counted by integers; the Earth-ionosphere cavity at 7.83 Hz is the planet-sized one. And every one of them FORGETS: energy decays as e^(-t/tau), tau = 1/(2 pi linewidth) = 33 ns for that cavity (sealed), about a hundred round trips (sealed) - the ring-down is the echo state property, which is why a delay loop or a laser with a feedback cavity is a reservoir computer (Appeltant 2011; Brunner 2013): the cavity is the fixed substrate, the finesse its memory depth, only the readout is trained. Our place: the engine is a receiver tuned to a band - a question excites the modes that fit and nothing else, and a high-Q instrument refuses off-band, which is the zero-false-positive discipline. Rests on the instruments, the logarithm (the logarithmic decrement delta = pi/Q), the Schumann resonance and the reservoir.
- part of → The instruments — every verifier and validator, and where they join — three resonators, one shape: standing waves on a bounded domain, counted by integers
- connects at → The Schumann resonance — the planet's cavity tone, sealed — the Earth-ionosphere cavity is the planet-sized cavity resonator; same count of modes, sam
- connects at → Reservoir computing — a fixed substrate, a trained readout; our use of it — the ring-down is fading memory: a cavity with a delay loop or a feedback laser is a reserv
- connects at → The logarithm - the instrument the joints share — decay per cycle is the logarithmic decrement delta = pi/Q; a ring-down is read on a log sc
- connects at → Our place — a deterministic instrument you read, not a program you run — our place: the engine is a receiver tuned to a band - high Q answers on resonance and refu
Is this card incomplete? Tell the library — it will call out for more ↗