REFLECTION AND TRANSMISSION OF THE ENERGY OF HARMONIC ELASTIC WAVES IN A BENT BAR

Abstract

Harmonic elastic waves in a uniform bar with a semi-infinite straight input section, a bend with constant radius of curvature, and a semi-infinite straight output section are considered, with the effects of rotary inertia and shear deformation being neglected. For a given incident extensional or flexural wave, the relative energy fluxes of the reflected and transmitted waves of both kinds are determined as functions of radian frequency, curvature and bend angle. Explicit formulae are derived for the special case of a sharp bend, and particularly simple formulae are obtained for small bend angles. The energy flux of the reflected flexural wave due to an incident extensional wave with unit energy flux is the same as that of the reflected extensional wave due to an incident flexural wave with unit energy flux. For transmission, there is the corresponding reciprocity. For a sharp 99.9 ° bend, an incident extensional wave does not produce a transmitted wave of the same kind for any frequency. For a sharp right-angle bend, in contrast, the energy flux of the transmitted extensional wave is up to 4% of that of the incident extensional wave, depending on the frequency. Numerical results for waves with different frequencies, and for sharp bends and smooth bends with different curvatures, are presented in polar radiation diagrams for bend angles between 0° and 360 °.