The aircraft itself will also carry an underwater acoustic beacon (pinging at about 40KHz) or two so that underwater wreckage can be located once the general area has been established by the ELT that should have been jettisoned on impact. Both items (the ELT and the UAB) have limited battery life -- they pump out huge signals relative to their size. By convention, the UAB is located in the same general area as the FDR and CVR longitudinally.
In the case of the Air France flight incident, it would seem that the ELT did not work or was at a significant distance from the crash site by the time the search reached the position. That the black boxes (the FDR and CVR) are expected to be recoverable at this point seems to indicate that the UAB did work (it would have been detected by a sonobuoy dropped by the search airplane, which looked to me like it was equipped with anti-submarine warfare equipment, judging by the MAD boom on the tail).
Sending a signal makes it easier to find - that's not stupid whether or not it sinks.
Sinking means that it doesn't move much. However, since it's attached to the plane, it's likely to go down with the airplane whether or it it would sink on its own, whether or not it sinks on its own is irrelevant.