Why a wave simulation

Sound in a whispering gallery clings to the curved wall and creeps round it. Lord Rayleigh described the effect at St Paul’s over a century ago; Raman and Sutherland (1922) showed the level round the gallery is not uniform, because the waves interfere. Both are wave effects, so Sofia solved the full 3D wave equation for the gallery rather than tracing rays.

The error it found in itself

In the 2D test against the exact solution, the sound along the wall arrived slightly late. Sofia refined the grid twice (6, 12 and 18 points per wavelength), saw the delay shrink in step with the cell size, and traced it to the stepped wall of the grid: grazing waves travel a little slower over it. At this resolution that is about 1.5 ms on a 0.3 s lap, 0.5%, well inside the precision of the measured arrival times. A smooth-wall boundary would remove it.

How it was checked

Reference

Echo arrival times 0.19, 0.29, 0.59 and 0.89 s against the measured ~0.2, 0.3, 0.6 and 0.9 s (Foteinou, Stevens & Murphy, Acoustics 7(3):49, 2025). The solver beforehand: in 2D against the exact solution for a rigid circle; in 3D against the BRAS RS8 curved-reflector measurements (TU Berlin), focal level within 1 dB from 500 Hz to 4 kHz

Caveat

An idealised gallery: the measured 33.64 m diameter, a vertical rigid wall (the real one leans into the dome) and an estimated 1.8 m walkway. The grid’s stepped wall delays the wave along it by about 1.5 ms per 0.3 s lap, sized against the exact 2D solution. The measured difference between a whisper aimed along the wall and one aimed at the listener is not reproduced. The inside sketch in the opening is approximate. Cathedral model: PeeJaa on Sketchfab (Free Standard), from Google 3D Tiles; imagery © Google. BRAS RS8: Schwanbeck et al., DAGA 2026, CC BY-SA 4.0.