2D Whispering Gallery

gEllipse

This animated gif file is inspired the Whispering Gallery Exhibit at the Chicago Museum of Science and Industry (CMSI). There are numerous examples of whispering galleries around the world. The one at CMSI is in the form of a three dimensional ellipsoid of rotation, and it has the property that spherical waves (your whisper), located at one focal point of the ellipsoid are reflected off the wall of the gallery in such a way as to refocus the wave at the other focal point. A listener at the other focal point experiences no dissipation in amplitude of the acoustic wave, so your whisper is heard loud and clear. (Things are actually somewhat more complicated in reality because you need some sort of acoustic imaging surface near the focus to compensate for the fact that your whisper does not generate a spherical wave.)

We can model this behavior in two dimensions by considering an ellipse. Suppose that at some instant a collection of “tennis balls” traveling at constant speed emanate radially from one focus. These balls will bounce off the elliptical wall at the same angle as the angle of incidence. We see that the balls all meet at the other focus at the same time. This is a consequence of the defining property of an ellipse that the sum of the distances of a point on the ellipse to its foci is constant. The two-dimensional equivalent of a spherical wave front is a circular wave front. We may think of the tennis balls as riding on one of these wave fronts. Watching these balls gives us an idea of how a wave front reflects off the wall as it travels. Note that at any instant of time, the wave front consists either of a circle or two circular arcs. Each circular arc has one of the foci as its center. At first, the balls emerge from a focus in the form of a circular wave front. As a ball bounces off the wall of the chamber, it switches from one arc to the other. After all balls have bounced off the wall, they form a circular wave front collapsing onto the opposing focus. This process then repeats in the animation. The 3D case is recovered by rotating this picture about the major axis of the ellipse.

For a further discussion of the acoustics, see this page on reflecting waves.