Nearly 400 years ago, Galileo noticed a puzzling phenomenon that he described in his book, Dialogues Concerning Two New Sciences:
Figure 1. Chladni patterns, described by Galileo but named after the 19th-century physicist who first experimented with them systematically, are formed by grains of sand on a vibrating plate. The sand collects in regions that do not vibrate, called nodal lines. Historically, mathematicians' studies have treated the density of the plate as known and the nodal lines as unknown. For over a decade, the author has studied the reverse of this problem: If we know the nodal lines, what can we tell about a system's physical properties? In these images, a plate is attached to a mechanical vibrator (right). Different rates of vibration give rise to different patterns. The long, wiggly regions between nodal lines are typical of plates or membranes with a nonuniform mass distribution. Even though this plate is uniform, we still see the long, wiggly regions because the plate is pinned at the center.
Photographs by Tom Griffin, Rensselaer Polytechnic Institute, courtesy of the author.
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