IRL the following would happen:
They'd look and listen for signs of a kill. Debris is the most obvious symptom. There was a sub, for example, that was sank in March '42 off the coast of the US. The DD crew found a human heart and lungs (!). Clothing, wood, and supplies/provisions might float to the surface. An extremely large oil slick is another symptom (don't remember if diesel leaves an oil slick).
Sometimes the passive sonar operator would hear the crushing noises on their gear.
Sometimes there would be no symptoms. On many occasions, the uboat got "lucky" because the DD erroneously thought he got a kill and took off.
Much of what happened involved a decent amount of guesswork. On the flip side, the uboat skippers were wrong to the tune of about 40-50% as to how much tonnage they sunk. Frequently, the tanker they thought was going down got the fires under control and limped back to port.
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Crimguy
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