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In models of warm dark matter, there is an appreciable population of high momentum particles in the early universe, which free stream out of primordial over/under densities, thereby prohibiting the growth of structure on small length scales. The distance that a dark matter particle travels without obstruction, known as the free streaming length, depends on the particle's mass and momentum, but also on the cosmological expansion rate. In this way, measurements of the linear matter power spectrum serve to probe warm dark matter as well as the cosmological expansion history. In this work, we focus on ultra-light warm wave dark matter (WWDM) characterized by a typical comoving momentumq_*and massm. We first derive constraints on the WWDM parameter space (q_*,m) using Lyman-αforest observations due to a combination of the free-streaming effect and the white-noise effect. We next assess how the free streaming of WWDM is affected by three modified expansion histories: early matter domination, early dark energy, and very early dark energy.