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How do elliptical whispering galleries carry sound from one focus to another?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

In analytic geometry, an ellipse is defined as a closed curve where the sum of the distances from any point on the curve to two fixed points, called foci, remains constant.1

The phenomenon of a whispering gallery relies on a fundamental geometric property of this shape: any sound wave originating at one focus and reflecting off the boundary of the ellipse will be directed precisely toward the other focus. Because sound waves follow the same law of reflection as light rays, the elliptical walls act as a reflector that converges the sound energy at the second focal point.12

In architectural structures like domes that approximate an elliptical shape, this focusing effect allows a person whispering at one focus to be clearly heard by someone positioned at the other, even across a large distance. While this is a fascinating application of geometry, such shapes are typically avoided in concert hall design to prevent the unintended focusing of sound and the creation of distracting echoes.12