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Compression waves in a rod

The features of longitudinal compression waves propagation

in a finite-cross-section homogeneous elastic rod

(linear modelling)

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

e-mail: selftrans@yandex.ru , selflab@mail.ru

Abstract

This paper continues a broad circle devoted to our new method to obtain exact analytical solutions for vibrant elastic systems. Here we present an example of application of this method to a practical problem seeming a simplest at the first sight. However even in this simplest problem the exact analytical solutions reveal quite serious features being unknown or disregarded before. In particular, we prove that, according to the dynamics of process, Poisson coefficient must be negative, in order the rod stretch to correspond its thinning, and vice versa. We also prove that the velocity of accompanying transversal waves propagation is equal to the velocity of longitudinal wave. The same, the velocity of accompanying longitudinal wave is equal to the velocity of transversal wave if inequal velocities of the main longitudinal and transversal waves. We establish that in frames of linear modelling the dynamical variation of a rod density has a non-harmonic periodical pattern; this essentially broadens the conventional range of linear approximation.

Classification by MSC 2000: 74B05; 74B10; 74H45; 74J15; 74J30; 74K10; 74S99

Classification by PASC 2001: 46.40.-f; 46.40.Cd; 46.40.Ff; 46.70.Hg

Keywords: Elastic distributed lines; Elastic rod vibrations; Surface waves; ODE; Poisson coefficient

1. Introduction

The problem of linear modelling of wave processes in an elastic infinite rod with finite cross-section is well-known. This is a particular case of a more general problem of a wave propagation in bounded elastic media. Conventionally, "when there is an abutting surface, the surface waves can arise. These waves, alike the gravitational waves in a liquid, were studied first in 1877 by Lord Rayleigh [1] who shoved that their action is fast damping with depth and that the velocity of their propagation is less than the velocity of waves within the body" [2, p. 23].

This conclusion was based on the consideration of the equation of deformation in an isotropic body like

where = _xx + _yy + _zz , and are Lame components, "fully determine the elastic properties of an isotropic body" [2, p. 17].

In particular, for a plane elastic wave in an unbounded isotropic medium, when the transfer w(w₁, w₂, w₃) depends only on one of Cartesian co-ordinates - e.g. x, and time t, it is supposed that in absence of mass forces "for the components of transfer vector w we yield the following equations: