A proof of the convergence of the segmentation approach used in analysis of one-dimensional linear systems

摘要

A numerical approach that involves discretization of a linear system with spatially varying properties into a sequence of a large number of segments, has been in vogue for quite some time, in order to obtain the transfer matrix for a nonuniform cross section of general shapes. In this paper, a mathematical proof of the convergence of the method is presented. A method is also presented with which one can obtain the equivalent differential equation, to whose solution the segmentation approach converges. This can be useful in ascertaining the correctness of the approach in case the segment transfer matrices have been derived using some empirical principles. The techniques have been illustrated by suitable examples.