Characteristics of group velocities of backward waves in a hollow cylinder

Abstract

It is known that modes in axially uniform waveguides exhibit backward-propagation characteristics for which group and phase velocities have opposite signs. For elastic plates, group velocities of backward Lamb waves depend only on Poisson's ratio. This paper explores ways to achieve a large group velocity of a backward mode in hollow cylinders by changing the outer to inner radius ratio, in order that such a mode with strong backward-propagation characteristics may be used in acoustic logging tools. Dispersion spectra of guided waves in hollow cylinders of varying radii are numerically simulated to explore the existence of backward modes and to choose the clearly visible backward modes with high group velocities. Analyses of group velocity characteristics show that only a small number of low order backward modes are suitable for practical use, and the radius ratio to reach the highest group velocity corresponds to the accidental degeneracy of neighboring pure transverse and compressional modes at the wavenumber k = 0. It is also shown that large group velocities of backward waves are achievable in hollow cylinders made of commonly encountered materials, which may bring cost benefits when using acoustic devices which take advantage of backward-propagation effects.

A Numerical Study on the Excitation of Guided Waves in Rectangular Plates Using Multiple Point Sources

“…The number of wave modes is also much larger than corresponding Lamb-type wave modes. Backward waves can also be seen in Figure 2b with negative group velocities [45,46]. These waves are highly dispersive.…”

Backward waves with double zero-group-velocity points in a liquid-filled pipe

“…35 For further information on the properties of backward waves and ZGVs in a hollow pipe, the reader is referred to existing literature. 13,17,[33][34][35] In section III A, the coupling effect between the forward-propagating waves in a liquid cylinder and the backward-propagating waves in a hollow pipe is analyzed to study the generation of backward waves and ZGVs in a liquid-filled pipe.…”

“…Secondly, in a hollow pipe, it has been known that the strong repulsion between two neighboring compressional and shear waves may generate a backward wave which possesses only one ZGV point at the lowest frequency of the backward wave. 35 The existence condition and dispersion properties of backward waves and ZGVs are determined by the value of Poisson's ratio and the ratio of the inner to the outer radius of the hollow pipe. 13,17,35 Moreover, all the longitudinal modes, with the exception of the lowest one, can exhibit backward waves and ZGVs.…”

“…35 The existence condition and dispersion properties of backward waves and ZGVs are determined by the value of Poisson's ratio and the ratio of the inner to the outer radius of the hollow pipe. 13,17,35 Moreover, all the longitudinal modes, with the exception of the lowest one, can exhibit backward waves and ZGVs. 35 For further information on the properties of backward waves and ZGVs in a hollow pipe, the reader is referred to existing literature.…”

“…14 while, for hollow pipes, backward waves also depend on the ratio of the inner to the outer radius. 35 Cui et al 35 numerically studied the influence of the radius ratio on group velocities of backward waves in hollow cylinders.…”