Thing is that shells don't fly 'very fast'... they are nearly universally below 200m/s near their practical ranges for indirect fire ~ only with low angle direct fire over open sites can you expect supersonic flight throughout.
I think that modern long-range howitzer shell never goes subsonic. About 320-350 m/sec. is the lowest speed for maximum range.
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There exist a specialist (but not more complex) series of counter artillery and counter mortar radar that perform this exact function. I think it possible that there is some confusion about exactly what is being discussed, but equally I can see a gun position that was static and a 'nuisance' being pinpointed over a matter of hours or days, even with the 'wrong' equipment...
Yes, but these radars use a specialized search mode, not a manual-steered narrow "pencil beam".
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Is it also possible that the Rgt had attached a 'proper' counter artillery radar element, using the ADA net to coordinate with fixed wing strike aircraft?
** And to add to the AD-system as C-Arty Radar... the 'official' Counter artillery radar was a development using elements of the Krug Radar, on an MTLB chassis...
There was a specialized artillery radar, designated as 1RL239, but it appears that it wasn't exported from USSR.