Exact solution of the propagation of ON-OFF signals by dispersive waves

TL;DR

This paper derives an exact integral solution for ON-OFF signal propagation in dispersive waves, revealing how signals travel with group velocity and how boundary thickness expands over time, applicable to various dispersion relations.

Contribution

It provides the first exact integral solution for ON-OFF signal propagation in dispersive media, applicable to any dispersion relation, and extends analysis to approximate solutions and complex switching patterns.

Findings

ON-OFF signals propagate with group velocity.

Boundary thickness increases proportionally to the square root of time.

Exact solutions are derived for specific dispersion relations.

Abstract

The propagation of ON-OFF signals with dispersive waves is examined in this study. An integral-form exact solution for a simple ON-OFF switching event is derived, which holds for any dispersion relation. The integral can be exactly calculated for two types of dispersion relations. Further, the analysis of these solutions shows that the ON-OFF signal propagates with the group velocity and that the boundary thickness of the signal increases with time, typically at a rate proportional to the square root of time, owing to dispersion. Additionally, an approximate solution for a general dispersion relation is derived, and a for a higher-complexity ON-OFF switching pattern is constructed.