  We consider flows, called $W^{\rm u}$ 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 $W^{\rm u}$ flows and we
show that $W^{\rm u}$ 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 $C^2$. This generalises recent results on time changes
of horocycle flows.
