Peripheral zone of disturbance from a local impulse in a string mesh formed by two systems of strings with different impedances

Abstract

The work is based on recurrent relations of a new type for squares of normalized orthogonal polynomials. These relations are applied to problems of transverse oscillations of a string mesh. For a homogeneous mesh, the influence function of a local impulse is represented by Kravchuk polynomials whose discrete arguments are two-dimensional numbers of nodes and discrete time that are counted from the point and time of application of the impulse. We apply the Szegö generalization of Laguerre polynomials to find an asymptotic representation of the influence function in the case where strings of two systems have essentially different impedances. Introduction of coordinates that move together with the expanding influence front simplifies the analysis of the displacement diagram for first waves and their dynamics. We derive a linear first order partial differential equation for squares of normalized orthogonal Laguerre polynomials (modified by the authors) that gives a continual description of the process in the whole. Bibliography: 7 titles.