  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.
