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I might be wrong about it being under a supporting wall but there are definetely bricks sitting above it on part of its length.
There is quite a gap from the air brick to to the joist and it sits above the DPS but as Bobbo mentioned over the years there has been a build up of dirt etc meaning that when there is heavy rain it can run down and drip over that particular joist. Hard to explain without photos.
A joist under a supporting wall doesn't sound right.
Water coming though an air brick doesn't sound right either - And if it did, it should still be nowhere near the joist.
Whereabouts is the air brick in relation to your DPC / floor level?
(Sorry - no builder recommendations there, I'm afraid)