No, nothing I was talking about was related to the speed of sound in water (which is where I assume you're getting 0.66 seconds/km, since it's about right). I agree the speed of signal propagation seems unlikely to matter much in most situations.
My point is in this scenario there's no appreciable difference between "if you can hear them, they've already found you" and "if you can hear them, they'll find you in 40 seconds"
The observation that they are "detecting the searcher before the searcher detects the submarine" is academic and inconsequential when we're talking about such a negligible amount of time between detections.
But there is an appreciable difference between "if you can hear them, they've already found you", and "if you can hear them, they might never find you because you can see them looking from farther away than they can see, and they might just be outright searching the wrong area".
All my arguments went to the latter kind of problem, not the former. Your 40 seconds number seems to be based on the speed of signal propagation, not the range at which things can be detected.
Even in the event where they do find you, the arguments I gave would suggest that it would take vastly longer than the speed of signal propogation to do so, because they would have to move closer to you. Giving you substantially more notice than 40 seconds.
I was taking it for granted that knowing the location of an unmanned, expendable sonar buoy or probe before you're discovered is not tactically important.
Now that you have me thinking, it would clearly allow you to stay out of detection range if you had the good fortune for their pings to initiate within your detection range but outside theirs.
