The solution and continualization of a wave problem for a string mesh

Abstract

The problem of the propagation of a perturbation along a string mesh, formed by two systems of parallel strings with different characteristic impedances, is considered. An exact solution of the problem is obtained using established recurrence relations for orthogonal Kravchuk and Chebyshev-Hermite polynomials, and its continuous analogue is determined using limiting representations. The partial differential equations obtained are, in a limiting sense, equivalent to the initial equations in discrete variables. The structure of a mechanical system which is a continuous analogue of the string mesh is determined. Analogues of Green's functions are constructed in discrete and continuous versions. A comparison is made with the corresponding results at a physical level of rigour. In the case of an unbounded homogeneous mesh, formed by the two families of strings with a single transit time of the perturbation between the nodes, the function of the effect of the pulse which is applied to one of the nodes is obtained. This is identical to the square of the normalized Kravchuk polynomials apart from the specification of the arguments.