Acoustics · the geometry of a room

Heard All the Way Round

How does a whispering gallery work? Two answers were given for St Paul's Cathedral, and they are different physics. The Astronomer Royal said the dome gathers the whisper to the opposite point, as a mirror gathers light; Rayleigh went up, found that any two positions would do, and said the curved wall holds the sound against itself so it cannot spread and fade. Try each geometry, then run the wave: with a hard wall, a quarter of the way round, the ear at the wall hears about 17 dB more than an ear 2.6 wavelengths further in, and a narrow lath against the wall cuts it by about 19 dB.

Climb 257 steps inside St Paul's Cathedral in London and you come out on a narrow round balcony at the foot of the dome, the Whispering Gallery. Whisper against its wall and someone far round the circle can hear you. The cathedral's own website puts it as a whisper "audible on the opposite side".

Two explanations have been given for this, by an astronomer and a physicist, and they are different physics. One says the curved surface gathers the sound to a point, the way a concave mirror gathers light. The other says the curved wall holds the sound against itself, so that it cannot spread out and fade. Each is exactly right for some shape of room. Below you can try both shapes, then watch a wave do what the rays only suggest.

The first answer: sound gathered to a point

In 1868 George Airy, the Astronomer Royal, gave the first account. In his book On Sound (the same words stand in its 1871 second edition) he imagines a fan of rays from the speaker's mouth running round a hemispherical dome in "successive equal chords", and he notes that "the reflexions tend towards the point of the hemisphere opposite to the speaker; to which point there is consequently a general convergence of reflexion-paths." And because every path is nearly the same length, "the different waves from different parts meet at that opposite point in nearly the same phase". He called it, fairly, "only a popular and imperfect account".

The cleanest room for that idea is not a dome but an ellipse, because an ellipse has a property no other curve has: a ray leaving one focus reflects off the wall, from any point on it, straight through the other focus, and every such path has the same length. Put the whisperer at one focus and every part of the whisper arrives at the other focus at the same moment. Slide the whisperer and watch what happens to that promise.

An oval room, seen from above

Whisperer's position along the long axis (the listener mirrors it)

Off the focus the promise breaks at once: the rays scatter, and their paths differ in length, so what arrives is smeared. Focusing is a point effect. It works for one pair of positions and it depends on the exact shape. The Architect of the Capitol says that in the half-domed National Statuary Hall in Washington, "in some spots, a speaker many yards away may be heard more clearly than one closer at hand", and then adds: "The modern-day echoes occur in different locations from those in the 19th century, when the floor and ceiling of the hall were different."

The second answer: sound held against the wall

John William Strutt, Lord Rayleigh, went up to the gallery himself. In The Theory of Sound (1878) he summarised the Astronomer Royal's view and then disagreed with it:

“Judging from some observations that I have made in St Paul's whispering gallery, I am disposed to think that the principal phenomenon is to be explained somewhat differently. The abnormal loudness with which a whisper is heard is not confined to the position diametrically opposite to that occupied by the whisperer, and therefore, it would appear, does not depend materially upon the symmetry of the dome. The whisper seems to creep round the gallery horizontally, not necessarily along the shorter arc, but rather along that arc towards which the whisperer faces.”Rayleigh, The Theory of Sound, vol. 2, §287 (1878; the same words in the 1896 second edition, pp. 126 to 128)

His reason is a piece of geometry you can check with a ruler. In a round room, a ray that leaves the wall at some angle to it meets the wall again at exactly the same angle, bounce after bounce, forever. So it can never come nearer the centre than the radius times the cosine of that angle. A ray that sets off nearly along the wall is trapped in a thin ring against it. Nothing is gathered to any point; the whisper just cannot get away from the wall.

A round room, seen from above

Steepest launch angle, measured from the wall (degrees)

What that trapping buys, Rayleigh said, is a slower fading. Sound in the open spreads over the surface of a growing sphere, so its intensity falls as the square of the distance. Sound held against a wall can only spread one way, up and down it, so it falls as the distance itself: “The less rapid enfeeblement of sound by distance than that usually experienced is the leading feature in the phenomena of whispering galleries.” He did not throw Airy out altogether. For a nearly spherical dome like St Paul's, he wrote, “a part of the observed effect depends upon the symmetry, though perhaps the greater part is referable simply to the general concavity of the walls.”

Twenty-six years later, lecturing at the Royal Institution, he said it outright. “I think that Airy's explanation is not the true one; for it is not necessary, in order to observe the effect, that the whisperer and the listener should occupy particular positions in the gallery. Any positions will do equally well.” And he showed it with a model: a strip of zinc “about 2 feet wide and 12 feet long” bent into a semicircle, a bird-call making sound of wavelength 2 cm aimed along it, and a flame that flares when sound reaches it. An obstacle put in the straight line between them made no difference. But “if a lath of wood W, which need not be more than 2 inches wide, is placed against the inner surface of the zinc, the flame recovers, showing that the sound has been intercepted.”

Rays are not sound: the wave

Rays are a picture. Sound is a wave, and a whisper's wavelength (about 34 cm at 1,000 Hz) is not tiny beside a gallery a few tens of metres across. So here is the wave itself: the pressure of a short whisper, computed on a grid, spreading through a round room seen from above. The whisperer stands just off the wall at the bottom, facing along it. Gold dots are ears against the wall; blue dots are ears 26 grid cells (2.6 wavelengths) further in. Change the wall, the way the whisperer faces, or put Rayleigh's lath against the wall an eighth of the way round.

A round gallery, 15 wavelengths in radius

wall

faces

obstacle

Loudest moment each ear hears, round the room from the whisperer (0°) in the direction it faces, relative to the loudest ear:
at the wall 2.6 wavelengths further in

The grid in your browser is coarse: ten cells to a wavelength, so that it can run here, with a round wall built as a staircase of square cells. That is coarse enough to change the answer. Run the same room on grids two, three and four times finer and the figures move by several decibels, closing in, though not smoothly, on a second calculation that has no grid at all: the round room solved exactly, as a sum of its own Bessel-function modes. Those are the figures given here; the check below reruns the exact solution every time, and the finer grids when asked. A quarter of the way round, with a hard wall, the ear against the wall hears the whisper 17.0 dB louder than the ear 2.6 wavelengths into the room (18.2 dB on the finest grid; the coarse grid in your browser says 11.1); half way round, 12.4 dB louder (12.5 on the finest grid; the coarse grid says 5.5). These two are delicate. The ear 2.6 wavelengths in stands near a quiet place in the pattern, and moving the wall out or in by an eightieth of a wavelength puts the exact figure between 15.6 and 18.6 dB a quarter round, and between 11.7 and 13.1 dB half way round: every one of them far above what the coarse grid says. Make the wall soft, so that whatever reaches it is swallowed, and the same two ears differ by only 0.4 dB, on the coarse grid and the finest alike (the ear in the room, if anything, is the louder): the advantage of standing at the wall has gone, because nothing is holding the sound there. The hard wall is also 14.4 dB louder at the wall than the soft one, a quarter round, on the finest grid (8.1 on the coarse one).

Rayleigh's lath, a rigid strip reaching 1.2 wavelengths in from the wall an eighth of the way round, cuts the whisper a quarter round by 18.5 dB on the finest grid (12.9 on the coarse one). A screen four wavelengths wide standing across the straight line between whisperer and listener changes it by under 0.1 dB (0.2 on the coarse grid). That is his flame experiment of 1904, done on a grid: what reaches the listener comes along the wall, not across the room.

One thing did not come out the way the story tells it. Turning the whisperer to face into the room instead of along the wall changes what the ear at the wall hears a quarter of the way round by only 0.3 dB in the exact solution (0.5 on the finest grid, 0.9 on the coarse one), because a mouth this close to the wall spills sound along it whichever way it points. Which way round the whisper goes is another matter: the ear a quarter round in front of the whisperer hears 13.2 dB more than the ear a quarter round behind (14.1 on the finest grid). But the soft-walled room shows a similar gap, 11.0 dB, so most of that is the whisper's own lopsidedness, not the wall's doing. That is Rayleigh's own explanation for the direction: it is “a consequence of the very unequal audibility of a whisper in front of and behind the speaker”.

Ten Cells to a Wavelength. Companion film, 3:16, sound on, headphones if you can. Every sound in it is the pressure the wave above computes at two ears, played at the room's own speed: the room 30.0 m across, the whisper a six-cycle burst at 343 Hz, a hum far below a real whisper. The left ear is against the wall, the right ear 2.6 wavelengths further in. On the coarse grid the left ear is 11.1 dB louder a quarter round; with no grid, 17.0.

What the film adds. Heard by turns at 90 degrees, the coarse grid and the exact solution agree on the ear in the room to within 0.1 dB, and the ear at the wall hears 6.2 dB more with no grid. The coarse grid does not put sound into the room; it fails to hold the whisper against the wall. (The grid four times finer lands within 0.1 dB of the exact solution there, which is what makes the comparison fair.)

You can check what the ears hear. listen.mjs (with spine.mjs beside it, and ffmpeg) reads the film's soundtrack and nothing else, and hears the figures above from the delivered file, each to within 0.3 dB. On a control soundtrack, the hard wall made soft and the exact solution swapped for the coarse grid, it hears the difference.

How thick is the band?

In 1910 Rayleigh returned to the question “on distinctively wave principles”. He took the wave that runs round a circular wall, whose strength at distance r from the centre is the Bessel function J_n(kr), with n whole waves fitting round the circumference, and a hard wall where the function stops changing (J_n′(kR) = 0). For large n this wave is almost nothing in the middle of the room and lives in a thin ring against the wall. His example: with n = 1,000 the vibrations “are practically limited to an annulus of width 20, or one fiftieth part only of the radius.”

The same calculation, done here for a gallery 100 feet across, gives the width of the band that holds 90% of the whisper's energy. The band narrows as the pitch rises, in proportion to the wavelength to the power two-thirds.

The whispering band against a wall 100 feet across

Pitch of the whisper (Hz)

At 1,000 Hz the band is about 40 cm deep; at 4,000 Hz, about 16 cm. So at St Paul's the whispering wave lives in a layer a few tens of centimetres deep against the wall, and an ear a metre out from it is outside the band at every pitch above about 250 Hz. In Rayleigh's zinc model, 12 feet of strip bent into a semicircle with a 2 cm wavelength, the band comes out at 2.6 cm: his 2-inch lath was about twice as wide as the whole band, which is why so narrow a strip was enough. And the wave on the grid above agrees with the Bessel functions independently: along the quarter-round line its whisper falls to half its wall loudness 8 cells in, and the 90% band for a room of that size is 8.0 cells deep. The exact solution falls to half at the same place.

The wave model shows one more thing Rayleigh's single ring does not. Its loudness does not fall smoothly as you move in from the wall; past the first band it rises again. The exact solution shows it sharply: an ear moving in from the wall a quarter of the way round hears the whisper fall to 18.9 dB below what the ear 4 cells from the wall hears, 1.4 wavelengths in, then rise again to 12.5 dB below, 2 wavelengths in. In 1921 C. V. Raman and G. A. Sutherland took a steady source of sound, blown from a bellows, up to the gallery and found this in the real room (they wrote it up the next year): “Exceedingly well-marked fluctuations of intensity were perceptible as the observer's ear was moved radially away from the wall, sound alternating with comparative silence at distances of the same order as the wave-length.” Rayleigh's theory, they concluded, “is undoubtedly on the right lines”, but “it does not offer a complete explanation”: a real whisper does not run perfectly along the wall, and the waves that run a little inward make further belts.

The same thing with light

Rayleigh noted that the same mathematics describes a vibrating drumskin, and in a note added in 1911, electrical waves. In 1961 C. G. B. Garrett, W. Kaiser and W. L. Bond saw stimulated emission of light from a small sphere of samarium-doped calcium fluoride and explained it “in terms of the electromagnetic analog of the Rayleigh theory of the whispering gallery”. Light going round inside the rim of a glass bead or a microscopic ring on a chip, trapped against its curved edge, is now routinely called a whispering-gallery mode. The word comes from a cathedral in London; the physics is the one the ruler shows in the round room above.

The check

The geometry: 3,600 rays from one focus of an ellipse all pass through the other (worst miss about 10⁻¹⁵), every path the same length; the mirror law re-derived at 1,000 wall points without the engine; in a circle, rays launched at 5°, 10° and 30° keep their angle for 200 bounces and never come closer to the centre than R cos α.
The Bessel functions: the engine's J_n (Bessel's integral) agrees with an independent Miller backward recurrence to 10⁻¹⁰ for orders up to 1,000; the first zeros of J_n′ match the printed table in Abramowitz and Stegun (Table 9.5, orders 1, 2 and 5) to five decimals and approach n + 0.8086 n^1/3; the band at 500 Hz is about 4 times the band at 4,000 Hz, as the two-thirds power says.
The wave: a leapfrog finite-difference solution of the two-dimensional wave equation, 340 by 340 cells, a round room of radius 150 cells, a whisper of wavelength 10 cells (a Gaussian-enveloped tone burst from a pair of opposed point sources, which points it), 1,300 time steps, the loudest moment at each ear. The same runs on grids two, three and four times finer (680, 1,020 and 1,360 cells across, with every length and the number of steps scaled), and the exact solution with no grid: the room's own modes J_n(j′_nmr/R) cos nφ, for every j′_nm up to 165, 3,466 of them, 6,880 modes counting their sine partners, as Weyl's law predicts (their first zeros match the printed table), each driven by the same whisper and summed; the finest grid comes closer to it than the coarse one on every figure both compute, and the exact figures are rerun with the wall an eightieth of a wavelength out and in. The check prints every figure in the paragraphs above and looks for each in the sentence that states it on this page, and the check breaks the engine on purpose (a hard wall made soft; a wrong law of reflection) to prove it can fail.
The sources, all read in the original: Airy, On Sound and Atmospheric Vibrations (1868), §67, pp. 137 to 138, quoted from the 2nd ed. (1871), pp. 145 to 146, where the words are the same; Rayleigh, The Theory of Sound, vol. 2, §287 (1878; 2nd ed. 1896, pp. 126 to 128); Rayleigh, “On Shadows”, Proc. Roy. Inst., 15 January 1904 (Scientific Papers vol. 5, pp. 171 to 172); Rayleigh, “The Problem of the Whispering Gallery”, Phil. Mag. 20, 1001 to 1004 (1910; Scientific Papers vol. 5, pp. 617 to 620); Raman and Sutherland, “On the Whispering-Gallery Phenomenon”, Proc. R. Soc. A 100, 424 to 428 (1922), doi:10.1098/rspa.1922.0007, and their letter in Nature 108, 42 (1921); Garrett, Kaiser and Bond, Phys. Rev. 124, 1807 (1961), doi:10.1103/PhysRev.124.1807, quoted from its abstract. The 257 steps and “audible on the opposite side” are from the cathedral's visitor pages; the Statuary Hall sentence is from the Architect of the Capitol's page on the hall.
Run it yourself: node verify-how-does-a-whispering-gallery-work.mjs in an empty folder downloads this page's engine and text from the site and redoes all of the above; add --mutate for the planted faults (the script). It takes about a minute, the exact solution included; add --grids to rerun the three finer grids as well (the finest took eleven minutes on the machine that wrote this).

What this does not show. The wave model is flat: a slice through the gallery at ear height, so it cannot show Rayleigh's central point, that the sound spreads only up and down the wall rather than in every direction; it shows the other half, that it does not spread inward. It is scaled down, 15 wavelengths in radius where St Paul's at 1,000 Hz is about 44, and its wall is a staircase of square cells; on the grid in your browser that coarseness, of the wall and of the wave alike, moves the figures by several decibels, which is why the ones quoted come from finer grids and from the exact solution. Its walls lose nothing, and its soft wall is a numerical sponge, not a curtain. The size of the gallery is not settled either: a 1900 guide gives “Height of Whispering Gallery about 100 feet, and same diameter”, and a guide of the same period gives 112 feet for the drum beneath the dome; the band depends on the cube root of the radius, so the larger figure would widen it by less than 4%. Nothing here measures St Paul's; it measures the geometry that the explanations rely on.

Corrections

Corrected 2026-10-07: The page gave the wave's figures from the coarse grid that runs in your browser, ten cells to a wavelength: 11.1 dB a quarter round and 5.5 dB half round between the ear at the wall and the ear in the room, 12.9 dB for the lath, 8.1 dB between the hard and soft walls, 9.4 dB between ahead and behind, 0.9 dB for facing into the room and 0.2 dB for the screen. Those figures are not converged. On grids two, three and four times finer, and in an exact solution with no grid, the hard wall holds more: about 17 and 12 dB in the exact solution (18.2 and 12.5 on the finest grid), and on the finest grid 18.5, 14.4, 14.1, 0.5 and under 0.1 dB. The soft wall's 0.4 dB stands; found while building this layer's companion film, 2026-10-07.

The claims record 56 claims re-read against their sources, 7 October 2026

Written 2026-09-27. Claims re-read against their sources on 2026-10-07: 56 checked, 53 confirmed, 3 wrong, 0 unverifiable. By claude-charming-edison-go7smd, one agent on oversight/claims-pass.md before publish (repo files, layer not yet deployed); the 1868 finding re-read by the instance on the page image, the rest against the saved texts; the wave's figures re-read 2026-10-07 by claude-determined-keller-fu2yq4 (grid refinement and an exact mode sum), with one fix-checker agent on the correction.

Reader check from an empty directory (verifier plus the page's engine.mjs and index.html): `--mutate` 43/0 before the fixes; 45/0 after. The agent also broke the prose on purpose (three pairs of figures swapped, the dek's lath figure changed) and the verifier still passed 41/0: its prose checks looked for each number anywhere on the page. Fixed below; the same swap now fails 3 checks.

Claims

CONFIRMED Airy: rays run round in "successive equal chords"
https://archive.org/details/onsoundatmospher00airyuoft : 2nd ed. §67 p. 145
CONFIRMED Airy: "the reflexions tend towards the point of the hemisphere opposite to the speaker; to which point there is consequently a general convergence of reflexion-paths."
https://archive.org/details/onsoundatmospher00airyuoft : verbatim, p. 145
CONFIRMED Airy: "the different waves from different parts meet at that opposite point in nearly the same phase"
https://archive.org/details/onsoundatmospher00airyuoft : pp. 145-146
CONFIRMED Airy: "only a popular and imperfect account"
https://archive.org/details/onsoundatmospher00airyuoft : "we can therefore offer only a popular and imperfect account of it"
CONFIRMED Airy, On Sound and Atmospheric Vibrations, 2nd ed. 1871, §67, pp. 145-146
https://archive.org/details/onsoundatmospher00airyuoft : title page "SECOND EDITION ... 1871"
WRONG "In 1871 George Airy ... gave the first account" (body and plain)
https://echo-old.mpiwg-berlin.mpg.de/ECHOdocuView?url=/permanent/library/V6S8K1ZQ/pageimg : the same §67 words stand in the 1st ed., Macmillan 1868, pp. 137-138 (page image pn=158 read by the instance)
CONFIRMED Airy was Astronomer Royal
https://archive.org/details/onsoundatmospher00airyuoft : title page
CONFIRMED Rayleigh blockquote "Judging from some observations ... towards which the whisperer faces."
https://archive.org/details/theorysound08raylgoog : verbatim, 1878 vol. 2 p. 115; 1896 p. 127
CONFIRMED Theory of Sound vol. 2 §287 (1878); 2nd ed. 1896 pp. 126-128
https://archive.org/details/theoryofsound02raylrich : §287 runs pp. 126-128
CONFIRMED the same words in the 1896 second edition
https://archive.org/details/theoryofsound02raylrich : editions compared
CONFIRMED Rayleigh summarised the Astronomer Royal's view and disagreed
https://archive.org/details/theoryofsound02raylrich : page said "quoted"; see findings
CONFIRMED "The less rapid enfeeblement of sound by distance ... whispering galleries."
https://archive.org/details/theoryofsound02raylrich : verbatim p. 128
CONFIRMED Rayleigh: intensity r^-2 in the open, r^-1 against the wall
https://archive.org/details/theoryofsound02raylrich : "varying as r^-2 is replaced by ... varying as r^-1"
CONFIRMED "a part of the observed effect depends upon the symmetry, though perhaps the greater part is referable simply to the general concavity of the walls."
https://archive.org/details/theoryofsound02raylrich : verbatim
CONFIRMED "a consequence of the very unequal audibility of a whisper in front of and behind the speaker"
https://archive.org/details/theoryofsound02raylrich : verbatim
CONFIRMED circle: angle kept at every bounce; never nearer the centre than R cos alpha
https://archive.org/details/theoryofsound02raylrich : "continues to touch a concentric sphere of radius OP cos θ"; verifier drift 1e-12 degrees
CONFIRMED ellipse: every ray from one focus reflects through the other, all paths 2a
public/checks/verify-how-does-a-whispering-gallery-work.mjs : standard geometry; worst miss 1.2e-15, re-derived without the engine
CONFIRMED "Twenty-six years later, lecturing at the Royal Institution"
https://archive.org/details/scientificpapers05rayliala : 1878 to 15 Jan 1904
CONFIRMED 1904: "Airy's explanation is not the true one; ... Any positions will do equally well."
https://archive.org/details/scientificpapers05rayliala : Sci. Papers v5 pp. 171-172; "I think that" had been dropped, see findings
CONFIRMED zinc strip "about 2 feet wide and 12 feet long" bent into a semicircle
https://archive.org/details/scientificpapers05rayliala : verbatim p. 172
CONFIRMED bird-call wavelength 2 cm; sensitive flame; obstacle on the straight line makes no difference
https://archive.org/details/scientificpapers05rayliala : "[λ = 2 cm.]"; "the flame flares as before"
CONFIRMED lath quote "if a lath of wood W ... the sound has been intercepted."
https://archive.org/details/scientificpapers05rayliala : verbatim
CONFIRMED "On Shadows", Proc. Roy. Inst., 15 Jan 1904; Sci. Papers v5 pp. 171-172
https://archive.org/details/scientificpapers05rayliala : running head "ON SHADOWS"
CONFIRMED 1910: "on distinctively wave principles"
https://archive.org/details/scientificpapers05rayliala : verbatim p. 617
CONFIRMED rigid wall J_n'(kR) = 0; waves J_n(kr)
https://archive.org/details/scientificpapers05rayliala : verbatim condition
CONFIRMED n = 1000: "are practically limited to an annulus of width 20, or one fiftieth part only of the radius."
https://archive.org/details/scientificpapers05rayliala : verbatim p. 619
CONFIRMED Phil. Mag. 20, 1001-1004 (1910); Sci. Papers v5 pp. 617-620
https://archive.org/details/scientificpapers05rayliala : header
CONFIRMED drumskin; note added 1911 on electrical waves
https://archive.org/details/scientificpapers05rayliala : "stretched membrane"; "[1911. Electrical vibrations ...]"
CONFIRMED Raman and Sutherland in 1921 took a steady source blown from a bellows to the gallery
https://doi.org/10.1098/rspa.1922.0007 : p. 426
CONFIRMED "Exceedingly well-marked fluctuations ... of the same order as the wave-length."
https://doi.org/10.1098/rspa.1922.0007 : verbatim p. 426 (the 1922 paper; now said)
CONFIRMED "is undoubtedly on the right lines" / "it does not offer a complete explanation"
https://doi.org/10.1098/rspa.1922.0007 : p. 428
CONFIRMED inward-running waves make further belts
https://doi.org/10.1098/rspa.1922.0007 : "slight inevitable deviation from ... strictly circumferential wave-propagation"
CONFIRMED Proc. R. Soc. A 100, 424-428 (1922)
https://api.crossref.org/works/10.1098/rspa.1922.0007 : Crossref
CONFIRMED Nature 108, 42 (1921)
https://api.crossref.org/works/10.1038/108042a0 : Crossref, 8 Sep 1921
CONFIRMED Garrett, Kaiser, Bond, Phys. Rev. 124, 1807 (1961)
https://api.crossref.org/works/10.1103/PhysRev.124.1807 : Crossref
CONFIRMED "in terms of the electromagnetic analog of the Rayleigh theory of the whispering gallery"
https://api.openalex.org/works/doi:10.1103/PhysRev.124.1807 : abstract, verbatim
CONFIRMED samarium-doped calcium fluoride sphere
https://api.openalex.org/works/doi:10.1103/PhysRev.124.1807 : "CaF2:Sm++"
CONFIRMED 257 steps
https://www.stpauls.co.uk/explore-our-map : "Climb up 257 steps"
CONFIRMED "audible on the opposite side"
https://www.stpauls.co.uk/explore-our-map : verbatim
CONFIRMED AOC: "in some spots, a speaker many yards away may be heard more clearly than one closer at hand"
https://www.aoc.gov/explore-capitol-campus/buildings-grounds/capitol-building/house-wing/statuary-hall : verbatim
CONFIRMED AOC: "The modern-day echoes occur in different locations ..."
https://www.aoc.gov/explore-capitol-campus/buildings-grounds/capitol-building/house-wing/statuary-hall : verbatim
CONFIRMED half-domed Statuary Hall
https://www.aoc.gov/explore-capitol-campus/buildings-grounds/capitol-building/house-wing/statuary-hall : "The half-dome shape"
CONFIRMED a 1900 guide gives the gallery about 100 ft across
https://www.gutenberg.org/ebooks/25266 : "Height of Whispering Gallery about 100 feet, and same diameter"; wording now quoted whole
CONFIRMED other figures run to 112 ft
https://www.gutenberg.org/ebooks/48491 : Gilbertson: "diameter of the drum beneath the Dome, 112 feet"; now said as the drum
CONFIRMED A&S Table 9.5: 1.84118, 3.05424, 6.41562
https://archive.org/details/HandbookOfMathematicalFunctions : printed values
WRONG j'_{10,1} = 11.77088 "from Abramowitz and Stegun, Table 9.5" (verifier)
https://archive.org/details/HandbookOfMathematicalFunctions : the table covers n = 0 to 8 only
CONFIRMED asymptote n + 0.8086 n^(1/3)
https://archive.org/details/HandbookOfMathematicalFunctions : A&S 9.5.16
CONFIRMED dek: the dome gathers the whisper "as a mirror gathers light"
https://archive.org/details/scientificpapers05rayliala : Airy's convergence; the mirror image is Rayleigh's 1904 gloss of Airy
CONFIRMED Rayleigh went up and found any two positions would do
https://archive.org/details/scientificpapers05rayliala : "Any positions will do equally well"
CONFIRMED band scales as wavelength^(2/3)
public/checks/verify-how-does-a-whispering-gallery-work.mjs : ratio 3.93 against 4
CONFIRMED wavelength about 34 cm at 1,000 Hz
public/checks/verify-how-does-a-whispering-gallery-work.mjs : c = 343 m/s
WRONG the wave's figures as the page gave them: 11.1 dB a quarter round, 5.5 half round, 12.9 the lath, 8.1 hard against soft, 9.4 ahead against behind, 0.9 facing, 0.2 the screen
research/how-does-a-whispering-gallery-work/convergence/grid-k4.json : the coarse grid's (10 cells a wavelength), not converged: exact mode sum 17.0 and 12.4; finest grid (k = 4) 18.2, 12.5, 18.5, 14.4, 14.1, 0.5, under 0.1 (research/how-does-a-whispering-gallery-work/convergence/)
CONFIRMED the soft wall: the two ears differ by 0.4 dB, the room ear the louder
research/how-does-a-whispering-gallery-work/convergence/grid-k4.json : −0.4 on every grid once the sponge is the same in time; the fix-checker's exact open-space soft room gives −0.41
CONFIRMED past the first band the loudness rises again
research/how-does-a-whispering-gallery-work/convergence/modes.json : exact, one ear at 90°: 18.9 dB below at 1.4 wavelengths, 12.5 below at 2
CONFIRMED half its wall loudness 8 cells in
research/how-does-a-whispering-gallery-work/convergence/modes.json : the exact solution's mean over 80 to 100° falls below half at 8 cells too
CONFIRMED the soft room's ahead-behind gap (11.0 dB) is similar, so most of it is the whisper's lopsidedness
research/how-does-a-whispering-gallery-work/convergence/grid-k4.json : exact 13.2, finest grid 14.1, within the layer's 1,300-step window (with 1,400 steps the exact figure is 10.9)

What was done

[fixed] Airy dated 1871; the same §67 is in the 1868 first edition (pp. 137-138, page image read). Body and plain now say 1868, the sources line cites 1868 and quotes the 1871 text, noting the words are the same.
[fixed] A&S Table 9.5 covers n = 0 to 8; n = 10 dropped from the verifier's table check and the page names the orders checked (1, 2, 5).
[fixed] The verifier's prose checks matched a number anywhere on the page, so swapped figures passed (41/0). Each figure is now matched inside its own sentence, the dek is checked in the visible dek and in the meta description, the sign behind "if anything, is the louder" is checked, and the model's dimensions are tied to the prose. The same swap now fails 3 checks. "Every figure ... compared with this page's text" reworded to say what is checked.
[fixed] 1904 quote now includes "I think that"; "he was blunter" is now "he said it outright".
[fixed] "quoted the Astronomer Royal's view" is now "summarised".
[fixed] Dimock's words quoted whole ("Height of Whispering Gallery about 100 feet, and same diameter"); the 112 ft is said to be the drum's.
[fixed] "an ear 2.6 wavelengths in" is now "2.6 wavelengths further in" (the wall ear is itself 0.4 wavelengths in).
[fixed] Raman and Sutherland's quotation is now marked as written up the next year (the 1922 paper).
[declined] "as a mirror gathers light" in the dek: re-read. It paraphrases Airy's convergence, and Rayleigh's own 1904 summary of Airy uses exactly the mirror image; the dek does not put the words in Airy's mouth.
[fixed] 2026-10-07: the wave's figures were the coarse grid's and not converged. The page now gives the exact solution's and the finest grid's, names the coarse grid's beside them, says how delicate the exact figures are (the wall moved an eightieth of a wavelength: 15.6 to 18.6 dB a quarter round), and carries a dated Corrected line; the verifier gained grid refinement (--grids) and the exact mode sum, and checks each new sentence and the Corrected line.
[fixed] 2026-10-07, in the draft correction before publication, found by the fix-checker: the refined grids kept the sponge's damping per step (the soft figure drifted to 0.9 dB; scaled, it stays 0.4); "the exact solution cannot be given a soft wall" (open space is that solution); "every finer grid comes closer" (false at k = 2 for facing); Weyl's law stated against the zero count rather than the mode count; "the check repeats both" (it reruns the grids only on request).
[declined] 2026-10-07, fix-checker: the lath's thickness is not scaled with k (2 cells at every k). Measured by the fix-checker: k = 2 with 4 cells gives −16.1 against −16.4, k = 4 with 8 cells −18.57 against −18.53; under the page's precision.

The assay office: what a record is, and every layer re-read so far.