Whispering Gallery (3D)

What the 3D scene shows

A Blender-built domed gallery (marble floor, windows, balustrade, columns) with two people: the whisperer at S and the listener at L. The rippling film over the floor is the real 2-D FDTD pressure field, height and colour both following the pressure. White lines are rays from S, the yellow line is the tracer ray and the yellow ring is its caustic circle. The gallery is scaled to the radius slider; the physics code is identical to the 2D page.

What you are looking at

In a round room a whisper spoken close to the wall can be heard clearly on the far side of the room, while someone standing near the centre hears almost nothing. Lord Rayleigh explained it in 1878 for the gallery of St Paul's Cathedral: sound that leaves the speaker at a shallow angle to the wall is bent back by every reflection and stays hugging the curved wall. Instead of spreading out in all directions, it is confined to a thin layer along the wall.

Equations used

• Wave model (FDTD). The air pressure p and particle velocity v on a staggered grid obey dv/dt = -(1/rho) grad p and dp/dt = -rho c^2 div v. They are advanced with the leapfrog scheme at a Courant number c dt / dx = 0.5 (stable below 1/sqrt 2), with a grid of 192 x 192 cells over 2.2 R.
• Walls. Cells outside the circle are solid. The normal velocity at a wall face is v = p / Z with Z/(rho c) = (1 + r)/(1 - r) and r = sqrt(1 - alpha); alpha = 0 is a rigid wall. The curved wall is a staircase of cells, which is the main numerical limitation.
• Source. A Gaussian-modulated sine burst g(t) = sin(2 pi f (t - t0)) exp(-((t - t0)/sigma)^2) is added to the pressure of one cell.
• Rays and the caustic. A ray hitting the wall at grazing angle alpha reflects with the law of reflection. All its chords are tangent to the circle of radius r_c = R cos(alpha), the caustic. Sound never enters the disc inside it, which is why the centre of a gallery is quiet for grazing rays.
• Layer width. The wave-hugging layer is of order (lambda^2 R)^(1/3) thick: a higher frequency (smaller lambda) hugs the wall more tightly.
• Spreading. In this 2-D model free sound spreads as 1/sqrt(distance); along the wall the peak decays more slowly, which graph 1 compares.

What every control does

• Room radius sets the size (St Paul's gallery has R about 17 m). Wall absorption is the fraction of energy lost at each reflection: curtains kill the effect. Central screen is an absorbing disc in the middle that blocks the direct sound but not the wall-hugging sound.
• Pulse frequency and length set the wavelength (343 m/s divided by f). The grid needs about 8 points per wavelength; the readout warns when it is too coarse. Source radius puts the speaker near the wall (the gallery) or at the centre (no whispering effect). Listener angle moves the listener around the wall.
• Ray fan, number of rays, bounces draw geometric rays from the source. Tracer ray and grazing angle show one chord path and its caustic circle.
• Speed is solver steps per frame (the animation is slowed down: the real pulse crosses the room in about 0.1 s); colour gain only changes the colours.

Honest simplifications

This is a 2-D model (a horizontal slice of a gallery with a tall, uniform wall), so spreading follows 1/sqrt(r), not the 1/r of 3-D. The real St Paul's whispering gallery also has a dome and a balustrade that matter. The staircase wall scatters a little at short wavelengths, and numerical dispersion changes the speed by a few per cent. Pressure units are arbitrary (the source strength is not calibrated).

Checks run in the page

(1) Traced chords at ten grazing angles stay outside R cos(alpha) (the caustic) to rounding error. (2) An open-grid FDTD pulse measured by cross-correlation of two probes reproduces the sound speed. (3) A lossless rigid circular room conserves acoustic energy over about three laps of the wall.