When I hit an escort I dive after striking and do the cat and mouse bit for about 15-20 minutes. Pay attention to the sound guy and his reports. Listen for the depth charges and you can tell about how far off they are. Soon enough you'll know when you can go to periscope depth and take a look.
Destroyers run wide circular search patterns. Usually you can line then up for a nice flank shot. Or, if they spot your scope they will charge you and you can get a good head on shot (generally what I end up doing. My crew must think I'm nuts
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