This bidding section is meant for beginners and those players who like to keep it simple. Test your knowledge.

What should South 1 and South 2 bid?

Solution

South 1 is too strong to respond 1♠, since he can do so with zero points as well: he is under obligation to bid. And because South may bid 1♠ with zero points, North will usually pass that bid: only if he has a lot of extra strength, he will make another bid. So with 15-16 HCP and some distributional points North will pass 1♠. In which case 4♠ will probably be a good contract...
South 1 should therefore jump to 2♠ to show some strength (8-11). After all he has 8 HCP and one or two distributional points (one for the doubleton in diamonds — for NS are certain to have a spade fit — and often one for the fifth spade: in case partner has four). If the partner of the doubler knows that his side holds the majority of points, he can jump. North shows at least 13 HCP, so South jumps from eight. Such a jump is not forcing but invitational (since the bid is limited, showing 8-11). The logical consequence is that North will pass with a minimum hand.
Bidding over partner's take-out double is completely different from bidding over his opening! If North had opened 1♣, South would have been right in bidding 1♠, for now that bid is forcing, since it is unlimited: 6 HCP or more (possibly a lot more!).

South 2 bids 2♣, a cuebid, a bid in the opponents' suit. This is the only forcing bid available to the partner of the doubler. Thus South informs North that the partnership will bid game. Only South doesn't know yet which game. That is what NS are going to investigate next by bidding suits and/or notrump. Both trust their partner not to pass below game. 4♠ is the most obvious game but a direct jump by South 2 over the double to 4♠ would be premature: perhaps North has only three spades, in which case 3NT may be the superior game.

Note that with South 2's distribution (4-3-4-2) but South 1's point count (8-11) the jump to 2♠ would have been correct, even when this may be a 4-3 fit. South has no choice then: after all this bidding section is meant for beginners and those players who like to keep it simple...