  In a previous paper, Oruba, Soward & Dormy (J.Fluid Mech., vol.818, 2017,
pp.205-240) considered the primary quasi-steady geostrophic (QG) motion of a
constant density fluid of viscosity $\nu$ that occurs during linear spin-down
in a cylindrical container of radius $L$ and height $H$, rotating rapidly
(angular velocity $\Omega$) about its axis of symmetry subject to mixed rigid
and stress-free boundary conditions for the case $L=H$. Here, Direct Numerical
Simulation (DNS) at large $L= 10 H$ and Ekman number $E=\nu/H^2\Omega=10^{-3}$
reveals structured inertial wave activity on the spin-down time-scale. The
analytic study, based on $E\ll 1$, builds on the results of Greenspan & Howard
(J.Fluid Mech., vol.17, 1963, pp.385-404) for an infinite plane layer
$L\to\infty$. At large but finite distance $r^\dag$ from the symmetry axis, the
meridional (QG-)flow, that causes the QG-spin down, is blocked by the lateral
boundary $r^\dag=L$, which provides a QG-trigger for inertial waves. The true
situation in the unbounded layer is complicated further by the existence of a
secondary set of maximum frequency (MF) inertial waves (a manifestation of the
transient Ekman layer) identified by Greenspan & Howard. Their blocking at
$r^\dag=L$ provides a secondary MF-trigger for yet more inertial waves that we
consider in a sequel (Part II). Here, for the QG-trigger, we solve a linear
initial value problem by Laplace transform methods. The ensuing complicated
inertial wave structure is explained analytically on approximating our
cylindrical geometry at large radius by rectangular Cartesian geometry, valid
for $L-r^\star=O(H)$ ($L\gg H$). Other than identifying small scale structure
near $r^\star=L$, our main finding is that inertial waves radiated away from
the outer boundary (but propagating towards it) reach a distance determined by
the group velocity.
