  We develop a general theory for the existence, uniqueness, and higher
regularity of solutions to wave-type equations on Lorentzian manifolds with
timelike curves of cone-type singularities. These singularities may be of
geometric type (cone points with time-dependent cross sectional metric), of
analytic type (such as asymptotically inverse square singularities or first
order asymptotically scaling-critical singular terms), or any combination
thereof. We can treat tensorial equations without any symmetry assumptions; we
only require a condition of mode stability type for the stationary model
operators defined at each point along the curve of cone points. In symmetric
ultrastatic settings, we recover the solvability theory given by the functional
calculus for the Friedrichs extension.
