  This work introduces a formulation of resolvent analysis that uses wavelet
transforms rather than Fourier transforms in time. Under this formulation,
resolvent analysis may extend to turbulent flows with non-stationary mean
states; the optimal resolvent modes are augmented with a temporal dimension and
are able to encode the time-transient trajectories that are most amplified by
the linearised Navier-Stokes equations. We first show that the wavelet- and
Fourier-based resolvent analyses give equivalent results for
statistically-stationary flow by applying them to turbulent channel flow. We
then use wavelet-based resolvent analysis to study the transient growth
mechanism in the logarithmic layer of a turbulent channel flow by windowing the
resolvent operator in time and frequency. The computed principal resolvent
response mode, i.e. the velocity field optimally amplified by the linearised
dynamics of the flow, exhibits the Orr mechanism, supporting the claim that
this mechanism is key to linear transient energy growth. We also apply this
method to non-stationary parallel shear flows such as an oscillating boundary
layer, and three-dimensional channel flow in which a sudden spanwise pressure
gradient perturbs a fully-developed turbulent channel flow. In both cases,
wavelet-based resolvent analysis yields modes that are sensitive to the
changing mean profile of the flow. For the oscillating boundary layer,
wavelet-based resolvent analysis produces oscillating principal forcing and
response modes that peak at times and wall-normal locations associated with
high turbulent activity. For the three-dimensional turbulent channel flow, the
resolvent modes gradually realign themselves with the mean flow as it deviates.
