A rotating cosmic string spacetime has a singularity along a timelike curve
corresponding to a one-dimensional source of angular momentum. Such spacetimes
are not globally hyperbolic: they admit closed timelike curves near the string.
This presents challenges to studying the existence of solutions to the wave
equation via conventional energy methods. In this work, we show that
semi-global forward solutions to the wave equation do nonetheless exist, but
only in a microlocal sense. The main ingredient in this existence theorem is a
propagation of singularities theorem that relates energy entering the string to
energy leaving the string. The propagation theorem is localized in the fibers
of a certain fibration of the blown-up string, but global in time, which means
that energy entering the string at one time may emerge previously.