  Traditionally, the response of a turbulent flow to modulated perturbations is
expected to be complex. We conduct direct numerical simulations of turbulent
Plane Couette flow, the shear flow between two differentially moving plates, to
reveal intriguing behaviour in response to a modulation in the velocity of one
of the plates: the response of the flow to modulations can be calculated from a
superposition of the response to each Fourier mode of the modulation. We fix
the shear Reynolds number as $Re_s=3\times10^4$, and measure the propagation of
a modulated forcing in the flow close to the resonant frequency of the system.
We find that the amplitude and phase response of the flow can be largely
captured using models based on laminar flow, regardless of the waveform used to
force the flow. Furthermore, we find that linear superposition can effectively
describe the flow's response to these perturbations as long as the modulation
is smaller than the base flow. Our findings unveil the persistence of linear
superposition in a turbulent flow under specific conditions, even when
perturbations occur at time-scales closely aligned with the system's
characteristic time-scales.
