@ringhloth
That's not the way to look at it. It's a BIG ocean and the hard part is actually for the submarines to find the convoy. The escorts already know where the convoy is and it's not like you're going to come across 100 sub's all in one go. Firstly, because if they were that concentrated, they'd be way to easy to spot, secondly, if they were that concentrated there'd be only a tiny possibility of them actually encountering anything and thirdly, if there were that many sub's in one place, only the few on the perifery of the formation would be able to attack anyway.
How about this? Hypothetically.
If you need 200 destroyers to stop 50 submarines in the North Atlantic from attacking your convoys of merchant ships.
You actually need the same 200 destroyers to stop 100 submarines from attacking that same number of convoys of merchant ships.
They'll just be a lot busier as the lines of sub's may be a little longer &/or there may be more lines of sub's in order to increase the chances of some of them catching a convoy.
OK, so say you have 8 escorts to a convoy and you cross a sub' line. You're still only going to encounter two or three sub's.
At first.
Push them down & carry on your way. You will probably get hunted-down but you'll still only be encountering an additional two or three sub's (less any you've damaged/sunk) per day. After a few days, you may be facing a dozen sub's and you'll get overwhelmed as escorts get pulled of to push a contact down and another slips through the gap.
But, after a couple of days, you should have got reinforcements of your own &/or be within range of your land-based air.
So, realistically, yeah, you're going to need more escorts, but it's not in proportion to the number of sub's out there. It's in proportion to the number of sub's you're likely to encounter once you cross their picket line. You would probably need to have enough escorts to keep them down, individually, until they run out of air. So we're looking at quite a lot of escorts, but not double.
Looking at it the other way. If you were to double the number of escorts because there are twice as many sub's
IN THE OCEAN, you'll more than double the efectiveness of your screen simply because your escorts are where they need to be and the sub's are spread-out trying to find you. Now the sub's are back-footed as you can overwhelme them by keeping them down.
IMHO
But I don't think that's the main question from this thread.
If Germany build three times as many subs, what does Britain do.
What CAN Britain do?
I see CAMs earlier & catapult ships earlier.
I see RDF getting refined earlier. Not getting the instantaneous (2-3 seconds) to get a bearing Huff-Duff on all their escorts four years early, but better RDF on small ships than they had and, maybe, in widespread application of Huff-Duff two years early along with more & better applied land-based RDF.
And I see Chamberlain saying NO to the Sudetenland and backing the Czecks. Especially after the second "most unsatisfactory" meeting.