Hyperreal waves on transfinite, terminated, distortionless and lossless, transmission lines

Abstract

A prior work (Graphs and Networks: Transfinite and Nonstandard, Chapter 8. Birkhauser: Boston, Cambridge, MA, 2004) examined the propagation of an electromagnetic wave on a transfinite transmission line—transfinite in the sense that infinitely many one-way infinite transmission lines are connected in cascade. That there are infinitely many such one-way infinite lines results in the wave propagating without ever reflecting at some discontinuity. The present work examines the case where the cascade terminates after only finitely many one-way infinite transmission lines, with the result that reflected waves are now produced at both the far end as well as at the initial end of the transfinite transmission line. The questions whether the reflected waves are infinitesimal or appreciable and whether they sum to an infinitesimal or appreciable amount are resolved for both distortionless and lossless lines. Finally, the generalization to higher ranks of transfiniteness is briefly summarized. Copyright © 2005 John Wiley & Sons, Ltd.