Flexural pulse propagation in nonuniform elastic bars by geometric acoustics

A direct numerical procedure is developed for analyzing the refraction of a flexural wave which impinges on a finite region of variable mechanical impedance in an elastic bar. This region is approximated by a set of uniform impedance elements, regarded as Timoshenko beams, to which the methods of geometric acoustics are applied. The analysis is performed for an infinite harmonic wave train and is extended to the case of a pulse by means of numerical Fourier synthesis. Numerical results are obtained for several examples in which experiments were performed and the calculated pulses are in close agreement with those observed experimentally. It is found that the contribution of the continuing interaction between propagating and nonpropagating waves accumulates within a nonuniform region and can significantly affect the net pulses of transmission and reflection.