Solid Torsional Horns

The propagation of torsional waves in tapered solid elastic rods is examined both theoretically and experimentally from the viewpoint of acoustic horn theory. Such tapered rods in torsional vibration are dubbed torsional horns. Two differential equations are derived that describe the propagation of torsional waves. One of these is an “exact” wave equation that can be readily solved only when the horn boundaries fit a separable coordinate system. The other is an approximate wave equation based on the assumption that the wavefronts are plane cross sections of the horn. This equation is very similar to Webster's plane-wave equation for compressional waves in an acoustic horn. Experimentally determined standing-wave patterns and resonance frequencies of torsional horns are compared with the solutions of the two wave equations for selected horn contours. A quantitative estimate of the error introduced by the plane-wave approximation is obtained for the exponential horn.