We consider flows, called Wu flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of Wuflows and we show that Wu flows have purely absolutely continuous spectrum in the orthocomplement of the constant functions. As an application, we obtain that time changes of the classical horocycle flows for compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions for time changes in a regularity class slightly less than C2. This generalises recent results on time changes ofhorocycle flows.