Transverse Vibrations of a Symmetric Tapered Reed

Abstract

The tapered reed, used in ultrasonic whistles, is a symmetric and homogeneous bar of rectangular cross section, uniform in the center and uniformly tapered to a sharp edge at each end. It is vibrating transversely under free-free conditions in the plane of the taper, and in its lowest symmetric mode; i.e., it must be mounted at the nodes. We have calculated the frequencies and positions of the nodes for such bars, combining the exact methods of Rayleigh and Kirchhoff for bars of uniform and tapered cross sections, respectively. The main problem is that of joining the two different solutions at the junction of the two regions. Results were obtained for a representative range of values of the ratio of length of tapered portion to total length. The effect of the surrounding medium has not been considered.