Basically designed as the submarine equivalent of an interceptor, get somewhere fast and open fire.
IIRC the top speed of the alfa was above the speed of the NATO torpedoes in use at the time.
https://en.wikipedia.org/wiki/Alfa-class_submarine#Impact
Like most fears over high-performance Soviet hardware though, the threat turned out to be overblown compared to the actual numbers build / operational state.
BTW the density of the sub is also for the most part pretty much equal to the density of water.
The only times where it's going to be either greater or lesser than water is when it's doing a static dive or fast ascent.
When the sub does it's initial dive and takes on water into it's buoyancy tanks it becomes about the average density of salt sea water once that is achieved it can use it's diving planes (foreplanes) to adjust it's depth.