Acoustic resonators with variable nonuniformity

Abstract

The standard method of altering the fundamental frequency of a one-dimensional resonator is to change the effective length. Another method is to introduce or alter the nonuniformity (variation in cross-sectional area), which can be accomplished very simply. If one end of an open-open nonuniform resonator is slapped by hand, the fundamental frequency of the open-closed system is heard. If the other end is slapped, the fundamental frequency is in general different. Furthermore, if the resonator is cut in half along its length, and one set of ends is placed on a flat surface, the fundamental frequency is heard if the halves are clapped together. Because the configuration can be altered by turning either or both halves upside down, as many as four different frequencies can be generated. The dependence of frequency upon nonuniformity can be understood and calculated perturbatively by Rayleigh’s energy method, or more generally by the theorem of adiabatic invariance. For the simplest case of piecewise-uniform resonators, improved accuracy is obtained by matching solutions of the wave equation in the uniform regions. With this theory, resonators have been accurately designed to yield musical intervals. Demonstrations of these ‘‘slappers’’ and ‘‘clappers’’ will be given.