Whispering Gallery Acoustics Calculator

Circular Whispering-Gallery Sound Paths

In certain domed cathedrals, mausoleums, and museum rotundas, a whisper placed close to a curved wall can remain audible at a distant point around the enclosure. These whispering-gallery effects arise when shallow reflections guide part of the sound near the boundary instead of allowing it to disperse freely through the room. This calculator uses a deliberately simple circular-gallery model: it estimates the wall-following arc, the direct chord, propagation time, and a user-supplied wall-loss estimate. It does not predict intelligibility or recreate the full three-dimensional acoustics of a particular building.

For this circular whispering-gallery calculation, let the wall radius be $R$, the source-to-listener angle be $\theta$, and the sound speed be $c$. The boundary path is the arc length $s=R\theta$, with $\theta$ in radians. Its travel time is $t=\frac{s}{c}$. The straight-line separation is the chord $d=2R\sin \left(\frac{\theta }{2}\right)$. Comparing these distances shows the extra route length assumed by the wall-following model.

Whispering-Gallery Attenuation Along the Wall

For a circular whispering-gallery path, the calculator treats wall attenuation as a constant loss in decibels per meter. With attenuation coefficient $\alpha$ and arc distance $s$, the reported loss is $L=\alpha s$ decibels. The corresponding amplitude ratio is $A={10}^{-\frac{L}{20}}$. Lower entered values of $\alpha$ leave more modeled amplitude after the sound has traveled around the wall; higher values make that amplitude decline more quickly.

The calculator also displays the ratio $G=\frac{s}{d}$. Despite its display label, this value is a geometric arc-to-chord ratio, not an acoustic amplification measurement. It indicates how much longer the modeled boundary route is than the direct straight-line distance. Circular guidance, reflection losses, air absorption, source directionality, and listener position can all affect actual loudness in ways this ratio does not represent.

Using the Whispering Gallery Acoustics Calculator

To model a circular whispering-gallery route, enter the gallery radius, the angular separation between speaker and listener, a wall attenuation value, and the sound speed to use. Clicking “Compute Acoustics” returns arc distance, chord distance, travel time, the arc-to-chord ratio, attenuation, and remaining amplitude ratio. The angle is converted from degrees to radians before the arc calculation. The copy button copies the displayed results table for use in notes or comparisons.

Whispering Gallery Architectural Context

Whispering effects have been noticed in circular and domed spaces because curved hard surfaces can repeatedly redirect sound. A real structure may have a dome, balustrade, cornice, or other boundary geometry rather than a perfect vertical circular wall. This calculator is therefore most useful for comparing circular layouts or exploring how radius, separation angle, and assumed loss alter a first-pass estimate. Measurements made in the actual space remain necessary when a design decision depends on audibility or privacy.

Sample Circular Whispering-Gallery Scenario

Consider the calculator’s default circular gallery: radius 20 m, a 180° separation, sound speed 343 m/s, and wall attenuation of 0.05 dB/m. The modeled arc is $\pi R$, approximately 62.8 m, while the chord is 40 m. The arrival time is about 0.183 s. The wall-loss model gives approximately 3.14 dB of attenuation and an amplitude ratio near 0.70. The displayed arc-to-chord ratio is about 1.57, meaning the assumed wall route is roughly one and a half times the direct distance.

For the same 20 m radius and 343 m/s sound speed, this circular whispering-gallery table shows that the modeled arc and its travel time rise directly with the entered angle. The calculator uses the entered angular route: an angle greater than 180° represents travel around the longer portion of the circle rather than automatically replacing it with the shorter arc.

Elliptical Galleries Versus Circular Whispering Paths

Elliptical whispering-gallery behavior differs from the circular-wall approximation calculated here. In an ideal ellipse, rays from one focus can reflect toward the other focus; that geometry depends on focal positions and reflection paths. This calculator instead applies circular arc and chord relationships, so it should not be used to estimate focus-to-focus behavior in an elliptical room without a separate reflection model.

Whispering Gallery Acoustics Equations

For the circular whispering-gallery model, the calculator evaluates the following equations:

$\theta =\frac{\pi \phi }{180}$, $s=R\theta$, $d=2R\sin \left(\frac{\theta }{2}\right)$, $t=\frac{s}{c}$, $L=\alpha s$, $A={10}^{-\frac{L}{20}}$.

Here $\phi$ is the entered angle in degrees, $\theta$ is the same angle in radians, $s$ is arc distance, and $d$ is chord distance. The JavaScript applies these equations directly to the four form inputs and reports the resulting quantities with displayed rounding.

Applications for Circular Whispering-Gallery Estimates

This circular whispering-gallery calculator can support classroom geometry demonstrations, preliminary exhibit planning, and comparisons between alternative radii or speaker-listener positions. It can also help illustrate why a longer wall-following path produces a longer arrival delay and more loss when the assumed dB-per-meter value remains fixed. For architectural work, treat the results as a screening calculation rather than a substitute for acoustic measurement, simulation, or professional design analysis.

Limitations of the Circular Whispering-Gallery Model

This whispering-gallery calculator represents sound travel as a two-dimensional circular arc. Actual galleries may have domes, varying wall profiles, openings, railings, surface joints, people, and other features that change reflection and scattering. Its single attenuation coefficient also combines effects that in practice vary with frequency, incidence angle, and material condition. The reported amplitude ratio reflects only the entered distance-loss model; it does not account for source level, background noise, reverberation, diffraction, or the listener’s hearing threshold.

Further Exploration of Whispering-Gallery Acoustics

To explore circular whispering-gallery behavior further, vary one input at a time and compare the reported arc, delay, and amplitude ratio. Radius and angle increase the arc distance, while a larger attenuation coefficient increases the modeled decibel loss over that distance. A more detailed study would require a defined three-dimensional geometry and frequency-dependent surface data, along with measurements or an acoustic simulation appropriate to the space.