Buckling as a source of sound, with application to the modeling of cicada sound generation

Abstract

A basic model of a ribbed finite plate is first considered, with the plate connected to a parallel surface by a nonlinear spring. When individual ribs are placed under compression, the linearized version of the model predicts eventual exponential growth of the transverse displacement when the compressional load exceeds the buckling load. The nonlinear spring, however, stops this growth and a subsequent oscillation ensues. The anatomy of the cicada is, of course, much more complicated, and the basic model is extended to give a mathematical formulation of the model proposed by Bennet-Clark and Daws (J. Exper. Biol. 1999) for the Cyclochila australasiae (a relatively large species of cicada), with explicit elasto-mechanical parameters, a principal objective being the computational prediction of the observed far-field waveforms. Possible explanations are advanced for the anatomical cause of the nonlinear springs. Further extensions of the theory are applied to the development of mathematical models for species of cicadas (considerably smaller) commonly found in the United States.