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 11.1 dB more than an ear 2.6 wavelengths further in, and a narrow lath against the wall cuts it by 12.9 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.

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.

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.

The standard runs, which the check below repeats, come out like this. A quarter of the way round, with a hard wall, the ear against the wall hears the whisper 11.1 dB louder than the ear 2.6 wavelengths into the room; half way round, 5.5 dB louder. Make the wall soft, so that whatever reaches it is swallowed, and the same two ears differ by only 0.4 dB (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 8.1 dB louder at the wall than the soft one, a quarter round.

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 12.9 dB. A screen four wavelengths wide standing across the straight line between whisperer and listener changes it by 0.2 dB. 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 0.9 dB in this model, 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 9.4 dB more than the ear a quarter round behind. But the soft-walled room shows a similar gap, 11.0 dB, so 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”.

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.

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 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. 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.