[TorpX]

I like that you worked this out and all, but there is a way that achieves the same essential result with less maneuvering. [I'm not familiar with Newton's method so I can't say anything about that.]

The last time I was in such a situation I used the Normal Approach Course. This means setting a course normal to the bearing of the target.

Using your example:

I am running submerged at 2 kt. and detect screws bearing 120 deg. relative. I wait a little while and the bearing moves to 119 deg.

Now I order 4 knots and turn 90 deg. to port from his bearing. He will then be at 90 deg. relative.

This is a Normal Approach Course. It allows an intercept with a minimum of speed, provided an interception is possible. If the target is too fast, the target track is too far away, or it has already passed, you cannot intercept it this way.

I continue to monitor the sound. If I am gaining bearing, I can reduce speed. If I am loosing bearing, I can increase speed to 6 knots, but going higher isn't really an option, unless the target is already close. If I am still loosing bearing at 6 kt., I must either consider surface pursuit options, or let it go.

Like your method, this isn't intended to provide detailed target data (nor is it necessary), but rather to get you within attack range. You then have the standard problem of a submerged periscope approach.

The NAC lacks the fancy geometry of sophisticated sonar-only methods, but in practical terms works just as well (at least imo).