Does a phrase ending in V7/V - V end in an authentic cadence or a half cadence?
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It is half cadence, but I will try to explain it to you as simply as I can:
It will not give the listener the 'ending' feel. The ending feel will be on the V-I.
The way someone listens to the V7/V - V is like this:
But because the V has the leading tone, it sounds like it needs to be led somewhere (That somewhere is the tonic (I) ). If you don't lead the V to the I (or VI), it will leave something to be desired.
Secondary dominants were classified as "modulations" by 18th and 19th century music theorists, and the term "transient modulation" was used when the modulation only lasted for the two chords in question, but in the 20th century they were re-classified as "secondary dominants" which could appear before a chord on any degree of the scale.
Aurally, you don't really know whether this is a secondary dominant or a modulation till you hear what comes next. Since music is experienced as a time-sequence of sounds, almost everything is "understood" by the listener with hindsight. Of course the experience of a listener here is different from that of a musicologist looking at the complete score in whatever order he/she chooses.
A listener could arguably interpret the D7 as a chromatically altered supertonic seventh (Dm7) which is hardly going to create a "Uh? This is really dissonant" reaction, unless he/she was living in the age of Palestrina.
Assuming that you still feel C Major as the home key, and haven't modulated to G Major, this is a half-cadence (imperfect). Although, I see why you are asking the question; the movement V7/V to V is harmonically the same as a V7-I perfect cadence in the dominant, G Major. But it is functionally different with respect to C Major.
You nearly answered it yourself. Calling the last chord V makes it unfinished - an imperfect cadence (half). What happened to get to the V doesn't really come into the equation, although, as Bob says, that in itself may be considered a perfect cadence - but only when the part is in that key. As in the key of G - which it isn't.
If it ends on V, it's an imperfect cadence. If it's ended on I in a new key, it's a perfect cadence. (I'm using the British terms, but same difference).
One secondary dominant seventh doesn't necessarily make a modulation. But it might. What happens next? A section in G major? A return to C major? Or maybe both - a repeat back to the beginning (C major) but second time it continues in G major?
It's a lot more simple than you think. Admit the possibility of ambiguity - will it go this way, or that way? That's what makes music interesting! What if that B melody note was used as a pivot and the music jumped straight into E major? Would that make the preceding chord V in C or I in G? Does it matter?