This episode uses many stock footage shots from the movie "The Enemy Below"
DDG-51 is actually the USS Arleigh Burke, lead ship of the class, built by Bath Iron Works in Maine. The commanding officer of a Burke class destroyer would typically be a Commander, not a Captain as shown in this episode.
Typically warships do not identify themselves by name when hailing other vessels over radio, here when hailing the Russian warship they identify themselves simply as the "USS Cayuga". Normally warships identify what country they are with and use their hull number. The proper way to hail and challenge a vessel not answering hail would be "unidentified vessel (state location and bearing of vessel) this is United States Naval warship 51, identify yourself and change your course or you may be subject to defensive action".
The images of the Cayuga firing missiles is not of an Arleigh Burke-class destroyer, which uses vertical launch tubes built into the deck. It appears to be stock footage of one of the early Ticonderoga-class guided missile cruisers, the first three ships of that class had MK-26 deck mounted missile launchers like the ones shown here.
The scene of the Cayuga firing a Harpoon missile is stock footage of the U.S.S. Badger (FF-1071), a Knox-class frigate, firing a Harpoon from her MK-16 Harpoon launcher.