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

Abstract

The Wentzel-Kramers-Brillouin (WKB) approximation is applied for asymptotic analysis of timeharmonic 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 clampedfree 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.

Effects of corrugation shape on frequency band-gaps for longitudinal wave motion in a periodic elastic layer

“…43), j is named as the wavenumber of the compound wave and in Mead (1996) as the phase constant of the wave motion; real values of j correspond to propagating waves and complex values to attenuating waves. Note that the notion of the wavenumber as a feature of the compound wave (Brillouin, 1953) differs from the one implied in the asymptotic method of G. Wentzel, H. Kramers, and L. Brillouin (the WKB method) (Pierce, 1970;Nielsen and Sorokin, 2014). The WKB operates with the "local" wavenumber of a wave propagating in a non-uniform structure with this local wavenumber being a function of the spatial coordinate (Pierce, 1970;Nielsen and Sorokin, 2014).…”

“…Note that the notion of the wavenumber as a feature of the compound wave (Brillouin, 1953) differs from the one implied in the asymptotic method of G. Wentzel, H. Kramers, and L. Brillouin (the WKB method) (Pierce, 1970;Nielsen and Sorokin, 2014). The WKB operates with the "local" wavenumber of a wave propagating in a non-uniform structure with this local wavenumber being a function of the spatial coordinate (Pierce, 1970;Nielsen and Sorokin, 2014). However, for studying wave motion in periodic structures, the notion implied in the classical work (Brillouin, 1953) seems to be more convenient.…”

Eigenfrequencies and eigenmodes of a beam with periodically continuously varying spatial properties

“…On the other hand, several approximate analytical methods for determining the eigenfrequencies of non-uniform beams and rods were developed in [4,13,14]. In particular, the WKB method [15] was proposed for the analysis of spatially slowly varying beams and rods [16][17][18]. In contrast, the averaging procedure for processes in periodic structures described in [19] can be used to study spatially rapidly varying systems.…”

“…Analysis of such eigenmodes and eigenfrequencies is of particular importance for applications (e.g. [18,20,21]), though efficient tools for this are yet to be developed.…”

“…However, these analytical approaches are applicable only in the presence of a small parameter in the governing equations. So, either the amplitude of modulation should be small [4], or the variation of structural parameters should be slow [16][17][18] or rapid [19], or a certain parameter (e.g. the thickness) of the corresponding homogeneous structure should be small [14].…”