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

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 for a higher-complexity ON-OFF switching pattern is constructed.