  We study mean-field dynamo action in a background linear shear flow by
employing pulsed renewing flows with fixed kinetic helicity and nonzero
correlation time ($\tau$). We use plane shearing waves in terms of
time-dependent exact solutions to the Navier-Stokes equation as derived by
Singh \& Sridhar (2017). This allows us to self-consistently include the
anisotropic effects of shear on the stochastic flow. We determine the average
response tensor governing the evolution of mean magnetic field, and study the
properties of its eigenvalues which yield the growth rate ($\gamma$) and the
cycle period ($P_{\rm cyc}$) of the mean magnetic field. Non-axisymmetric modes
of the mean magnetic field decay as $t \to \infty$ and hence are deemed
unimportant for mean-field dynamo. Both, $\gamma$ and the wavenumber
corresponding to the fastest growing axisymmetric mode vary non-monotonically
with shear rate $S$ when $\tau$ is comparable to the eddy turnover time $T$, in
which case, we also find quenching of dynamo when shear becomes too strong.
When $\tau/T\sim{\cal O}(1)$, the cycle period ($P_{\rm cyc}$) of growing
dynamo wave scales with shear as $P_{\rm cyc} \propto |S|^{-1}$ at small shear,
and it becomes nearly independent of shear as shear becomes too strong.This
asymptotic behaviour at weak and strong shear has implications for magnetic
activity cycles of stars in recent observations. Our study thus essentially
generalizes the standard $\alpha \Omega$ (or $\alpha^2\Omega$) dynamo as also
the $\alpha$ effect is affected by shear and the modelled random flow has a
finite memory.
