= Solution
For a one-layer <shallow water> flow the <Froude number> is $F=u/\sqrt{g'h}$. <Supercritical flow> has $F>1$, so both <characteristic speeds> $u\pm\sqrt{g'h}$ point downstream. <Subcritical flow> has $F<1$, allowing information to travel upstream. At <hydraulic control> the flow passes through $F=1$ at a geometry-induced critical section; that condition fixes discharge in relation to upstream head when the exit is free.
For escaping light air, reflect the usual lower-layer picture vertically: a lowered doorway roof acts as an inverted <broad-crested weir>. The lower return flow can be ignored dynamically only while it has a sufficiently large available area and small <velocity>. If the warm interface lies far below the doorway top, the cool inflow occupies a substantial dynamically active fraction of the opening; its inertia and pressure cannot be replaced by a motionless deep ambient. A two-layer exchange model is then required. If the warm layer reaches below the doorway bottom, the assumed separate lower replacement path is lost entirely.
Outside a freely controlled doorway the outgoing current becomes supercritical and then rises rapidly. That upturn violates the small-vertical-acceleration <shallow water> approximation, but it does not alter the upstream discharge as long as it does not submerge or back-pressure the control. Information cannot return through the supercritical outflow to the critical section. A blocked conservatory or a drowned doorway is different and can move the control upstream.
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