A Poisson scaling approach to backward wave propagation in a tube

A mathematical analysis of wave propagation along an elastic cylindrical tube is presented, with the aim of determining the range of Poisson’s ratio for which backward wave propagation (i.e. at negative group velocity) can occur near the ring frequency. This range includes zero Poisson’s ratio and a surrounding interval of positive and negative values, whose width depends on the thickness of the tube. The whole range of Poisson’s ratio is considered, so that the work applies to modern materials, e.g. composites. All results are presented in simple analytic form by means of a dominant balance in parameter space. The identification of this balance, which is unique, is a main new result in the paper, which makes possible a new type of shell theory based on ‘Poisson scaling’. The mathematical approach is deductive from the equations of motion, rather than being based on kinematic hypotheses. A key finding, accessible via the Poisson scaling, is that the regime of negative group velocities extends to high wavenumbers, while being confined to a narrow band of frequencies. Thus responses localized in space are possible for near-monochromatic forcing, an important fact for nonlinear theories of tube dynamics near the ring frequency. This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)’.