{"id": "card_tool_resonators", "kind": "reference", "title": "Resonators — Helmholtz, the cavity, Fabry-Perot: one shape, a Q that fades", "body": "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.", "source": {"label": "Narrow Highway — the instruments", "url": "", "authority_tier": "engine_derived", "ref": "card_floor_the_instruments"}, "shelf": "codex", "box": "instruments", "bands": ["resonator", "resonators", "helmholtz resonator", "helmholtz", "cavity resonator", "cavity", "rhumbatron", "fabry-perot", "fabry perot", "interferometer", "free spectral range", "finesse", "linewidth", "quality factor", "q factor", "ring-down", "standing wave", "mode", "resonance", "echo state", "reservoir", "receiver", "band", "instrument", "no-code", "one map"], "subject": "Resonators — Helmholtz, the cavity, Fabry-Perot: one shape, a Q that fades", "connections": [{"to_card_id": "card_floor_the_instruments", "relationship": "part_of", "evidence": "three resonators, one shape: standing waves on a bounded domain, counted by integers"}, {"to_card_id": "card_tool_schumann_resonance", "relationship": "connects_at", "evidence": "the Earth-ionosphere cavity is the planet-sized cavity resonator; same count of modes, same fit"}, {"to_card_id": "card_tool_reservoir_computing", "relationship": "connects_at", "evidence": "the ring-down is fading memory: a cavity with a delay loop or a feedback laser is a reservoir computer"}, {"to_card_id": "card_floor_logarithm", "relationship": "connects_at", "evidence": "decay per cycle is the logarithmic decrement delta = pi/Q; a ring-down is read on a log scale"}, {"to_card_id": "card_tool_the_engine_among_the_instruments", "relationship": "connects_at", "evidence": "our place: the engine is a receiver tuned to a band - high Q answers on resonance and refuses off-band"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.instruments", "facets": {"subject": ["band", "cavity", "cavity resonator", "echo state", "fabry perot", "fabry-perot", "finesse", "free spectral range", "helmholtz", "helmholtz resonator", "instrument", "interferometer", "linewidth", "mode", "no-code", "one map", "q factor", "quality factor", "receiver", "reservoir", "resonance", "resonator", "resonators", "rhumbatron", "ring-down", "standing wave"]}, "presentation": {"glyph": "•", "kind_label": "instruments", "by": "Narrow Highway — the instruments", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_floor_the_instruments", "title": "The instruments — every verifier and validator, and where they join", "relationship": "part of", "why": "three resonators, one shape: standing waves on a bounded domain, counted by integers", "href": "/card/card_floor_the_instruments", "resolved": true}, {"id": "card_tool_schumann_resonance", "title": "The Schumann resonance — the planet's cavity tone, sealed", "relationship": "connects at", "why": "the Earth-ionosphere cavity is the planet-sized cavity resonator; same count of modes, same fit", "href": "/card/card_tool_schumann_resonance", "resolved": true}, {"id": "card_tool_reservoir_computing", "title": "Reservoir computing — a fixed substrate, a trained readout; our use of it", "relationship": "connects at", "why": "the ring-down is fading memory: a cavity with a delay loop or a feedback laser is a reservoir computer", "href": "/card/card_tool_reservoir_computing", "resolved": true}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "relationship": "connects at", "why": "decay per cycle is the logarithmic decrement delta = pi/Q; a ring-down is read on a log scale", "href": "/card/card_floor_logarithm", "resolved": true}, {"id": "card_tool_the_engine_among_the_instruments", "title": "Our place — a deterministic instrument you read, not a program you run", "relationship": "connects at", "why": "our place: the engine is a receiver tuned to a band - high Q answers on resonance and refuses off-band", "href": "/card/card_tool_the_engine_among_the_instruments", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_tool_resonators", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_tool_resonators"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}