Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 359481, 14 pages
doi:10.1155/2008/359481
Research Article
Vanishing Waves on Closed Intervals and Propagating Short-Range Phenomena
1Faculty of Applied Sciences, Politechnica University, 061071 Bucharest, Romania
2Modeling and Simulation Department, ITT Industries, Washington, DC 20024, USA
Received 29 May 2008; Accepted 24 June 2008
Academic Editor: Carlo Cattani
Copyright © 2008 Ghiocel Toma and Flavia Doboga. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This study presents mathematical aspects of wave equation
considered on closed space intervals. It is shown that a solution of this
equation can be represented by a certain superposition of traveling waves
with null values for the amplitude and for the time derivatives of the resulting
wave in the endpoints of this interval. Supplementary aspects
connected with the possible existence of initial conditions for a secondorder
differential system describing the amplitude of these localized oscillations
are also studied, and requirements necessary for establishing
a certain propagation direction for the wave (rejecting the possibility of
reverse radiation) are also presented. Then it is shown that these aspects
can be extended to a set of adjacent closed space intervals, by considering
that a certain traveling wave propagating from an endpoint to the
other can be defined on each space interval and a specific mathematical
law (which can be approximated by a differential equation) describes the
amplitude of these localized traveling waves as related to the space coordinates
corresponding to the middle point of the interval. Using specific
differential equations, it is shown that the existence of such propagating
law for the amplitude of localized oscillations can generate periodical
patterns and can explain fracture phenomena inside materials as well.