Most DDs are deaf over about a 40 degree area from behind. Just like our subs. Of course, the lookouts can still see you and at night, the search lights are going. So watch the periscope. When you find one napping, sneak in from behind (Slow and Silent Running) to about 500 yds. Stick him that way. Generally, I come in at about 90 down and come to periscope depth when I'm about 1000 yds from him. Remember to mark the map for his location and heading before making the approach.
Old_tex