  We consider the time-dependent response of a gravitationally-thinning
inviscid liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal
ambient pressure disturbances. The theoretical investigation employs the
hyperbolic partial differential equation developed by \cite{weinp1}. The
response of the curtain is characterized by the slot Weber number, $W_{e_0} =
\rho q V/2\sigma$, where $V$ is the speed of the curtain at the slot, $q$ is
the volumetric flow rate per unit width, $\sigma$ is the surface tension, and
$\rho$ is the fluid density. Flow disturbances travel along characteristics
with speeds relative to the curtain of $\pm \sqrt{uV/W_{e_0}}$, where $u =
\sqrt{V^2 + 2gx}$ is the curtain speed at a distance $x$ downstream from the
slot. When the flow is subcritical ($W_{e_0} < 1$), upstream traveling
disturbances near the slot affect the curtain centerline, and the slope of the
curtain centerline at the slot oscillates with an amplitude that is a function
of $W_{e_0}$. In contrast, all disturbances travel downstream in supercritical
curtains ($W_{e_0} > 1$) and the slope of the curtain at the slot is vertical.
Here, we specifically examine the curtain response under supercritical and
subcritical flow conditions near $W_{e_0} = 1$ to deduce whether there is a
substantial change in the overall shape and magnitude of the curtain responses.
Despite the local differences in the curtain solution near the slot, we find
that subcritical and supercritical curtains have similar responses for all
imposed sinusoidal frequencies.
