The WKB approximation for analysis of wave propagation in curved rods of slowly varying diameter

The Wentzel-Kramers-Brillouin (WKB) approximation is applied for asymptotic analysis of time-harmonic dynamics of corrugated elastic rods. A hierarchy of three models, namely, the Rayleigh and Timoshenko models of a straight beam and the Timoshenko model of a curved rod is considered. In the latter two cases, the WKB approximation is applied for solving systems of two and three linear differential equations with varying coefficients, whereas the former case is concerned with a single equation of the same type. For each model, explicit formulations of wavenumbers and amplitudes are given. The equivalence between the formal derivation of the amplitude and the conservation of energy flux is demonstrated. A criterion of the validity range of the WKB approximation is proposed and its applicability is proved by inspection of eigenfrequencies of beams of finite length with clamped-clamped and clamped-free boundary conditions. It is shown that there is an appreciable overlap between the validity ranges of the Timoshenko beam/rod models and the WKB approximation.