  We study the implications of time-varying wave mechanics, and show how the
standard wave equation is modified if the speed of a wave is not constant in
time. In particular, waves which experience longitudinal acceleration are shown
to have clear relativistic properties when a constant reference speed exists.
Moreover, the accelerating wave equation admits only solutions propagating
forward in time, which are continuous across material interfaces. We then
consider the special case of electromagnetic waves, finding that the
Abraham-Minkowski controversy is caused by relativistic effects, and the
momentum of light is in fact conserved between different media. Furthermore, we
show that the accelerating waves conserve energy when the wave is moving along
a geodesic and demonstrate two example solutions. We conclude with some remarks
on the role of the accelerating wave equation in the context of the arrow of
time.
