V.5 No 1

17

On longitudinal excitation of elastic medium having a moving boundary

On longitudinal excitation of elastic medium having a moving boundary

S.B. Karavashkin and O.N. Karavashkina

apt. 187, bldg. 38, Prospect Gagarina, Kharkov 61140, Ukraine

Phone: +38 (057) 7370624

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

In these notes we are studying two most typical models of a wide class of problems on longitudinal excitation propagating in an ideal elastic medium having a moving ideally reflecting boundary. These problems have most broad area of applications, from mechanics to fundamentals of Relativity and electromagnetism. We showed that in both models the pattern of oscillations is described by progressive waves modulated in amplitude. With it, in case of stationary source and moving boundary, the wave propagation velocity is proportional to the speed of boundary motion, and in case of synchronous shift of the source and boundary, the conditions of modulation essentially depend on the shift velocity. Besides, we present the technique to transmit non-distorted measures of space and time between mutually moving reference systems.

Keywords: wave physics, dynamics of elastic systems, oscillations in elastic media with a moving boundary, techniques to transmit the spatial and temporal measures between mutually moving reference frames

Classification by MSC 2000: 30E25; 70G60; 70J35; 70J50; 74H45; 74J05

Classification by PASC 2001: 02.60.Lj; 06.30.Ft; 06.60.Sx; 46.25.Cc; 46.40.-f; 46.40.Cd

1. Introduction

We wrote these notes seeing some colleagues studying the wave propagation in elastic medium with moving boundary. Basing on our original method to find exact analytical solutions for models of elastic systems, we analyse here two main problems of this class, to make more precise the pattern of oscillation processes in the systems with moving boundary. Of course, this will be a brief analysis, it will not touch many aspects related to non-ideal boundary, line impedance, presence of dissipation of excitation being typical for spatial models. Our method makes us able to deepen into these aspects, though we think, meanwhile such level is sufficient to understand basic aspects. Hopefully, presented results will give the colleagues more sound position in their study and help them to improve their views on the phenomenology of processes that take place in elastic systems with a moving boundary.

2. Model with a stationary source and moving boundary

To clear the problem, let us consider, how propagates a longitudinal compression wave along the 1 D elastic line having the density and tension T_f , as shown in Fig. 1.

Fig. 1. Infinite elastic distributed line, along which the boundary moves with the velocity v in positive direction of axis x

In this scheme, the point of line excitation by the external force coincides with the zero coordinate, and strongly rigid boundary moves along the axis x from some initial point x₀ 0  with the velocity v. We will be interesting in the region of elastic line located between the excitation point and moving boundary, i.e.

To simplify our study without any loss of generality, premise the amplitude of external force to vary harmonically:

where F₀ and are the amplitude and frequency of external force variation, and j is the complex unity.

In accordance with [1, p. 28], the equation for momentary shift x of the elastic line elements at the arbitrary point of studied interval is

Noting the boundary being absolutely rigid and moving and the wave having an inversion at the boundary, the equation of the wave reflected from the boundary is