When did consonantal “v” start being transcribed as “β”?

Question

Since I learned Latin using ecclesiastical pronunciation, I have a general interest in the shift from the classical pronunciation of "v" as /w/ to /v/.

This question is more focused though: I am interested in finding out at what point Greek transcriptions of Latin names began using "β" instead of "ου" to transcribe Latin's consonantal "v".

Cursory research reveals two close examples which use a different approach:

• Josephus (AD 37-100), in Jewish Antiquities 15, transcribes "Varro" as "Οὐάρρωνος."
• Plutarch (AD 45-120), in his Life of Caesar 5, transcribes "Vetus/eri" as "Βέτερι."

Something tells me, though, that Plutarch usually transcribes in the other way. I also recall that later works from Constantinople almost always use "β."

I am aware that there will not be a sharp dividing line, but I am interested in knowing if there are any guidelines about time and place that will help me to predict which way a Greek text will transcribe this sound.

Answer 1

Sturtevant, The Pronunciation of Greek and Latin, p. 88, has a relevant passage, though he doesn't go into details:

Since Attic and Hellenistic Greek had no such sound as Classical Lat. v, namely [w], a substitution was necessary and the choice lay between ου and β. For a long time the former was preferred; but as [β] became a more and more frequent pronunciation of v, the digraph ου became less and less appropriate to represent it. This may be the whole explanation of the increasing preference for β which, by the second or third century A.D., led to its regular use as the substitute for Lat. v.