Yeah, but on the full engine power DD will run 30-35 kts. 12 kts was a deadline for ASDIC usage, as I know, 'cause of self-cavitation. On this speed you can hear submarine diesel, especially if wind have right direction. Trained ear can separate self-noise (boiler machines in DD) from diesel tact. In heavy fog all sounds are a little bit different, however... As from my experience, I could hear a tugboat passing near my liner-ship (small baltic cruise) in a fog. I don't know the distance, but it was few hundreds of meters - liner had length about 150-170m I think, and I still was able to see his bow from aft in this fog.
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