rspa.royalsocietypublishing.org Research Cite this article: Nielsen R, Sorokin S. 2014 The WKB approximation for analysis of wave propagation in curved rods of slowly varying diameter. Proc. R. Soc. A 470: 20130718. http://dx.doi.org/10.1098/rspa.2013.0718 Received: 29 October 2013 Accepted: 25 March 2014 Subject Areas: mechanical engineering, applied mathematics Keywords: WKB approximation, asymptotic analysis, wave propagation, non-uniform rods Author for correspondence: Rasmus Nielsen e-mail: rn@m-tech.aau.dk The WKB approximation for analysis of wave propagation in curved rods of slowly varying diameter Rasmus Nielsen and Sergey Sorokin Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstraede 16, 9220 Aalborg, Denmark The Wentzel–Kramers–Brillouin (WKB) approxim- ation 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. 1. Introduction The Wentzel–Kramers–Brillouin (WKB) approximation is a well-established and recognized tool in theoretical physics in general, and in quantum mechanics in particular. Its application for solving the Schrödinger equation provides simple-structured solutions that describe the motion of particles and phenomena such as tunnelling. Examples can be readily found in classical compilations and textbooks such as Fröman & Fröman [1] or Bender & Orszag [2]. The fundamental feature of the WKB method is its ability to approximate the solutions of 2014 The Author(s) Published by the Royal Society. All rights reserved.