The mathematics of roving searches

[Crannogman]

Originally Posted by ColonelSandersLite

Yes, except I see big flaw in the reasoning here. Suppose we are patrolling roughly E-W in a hypothetical N-S shipping lane. We add a northward cant to our patrol as in your examples. This absolutely does decrease the speed vector of any ship moving north, thus giving better odds of detection. Shipping lanes go both ways though and it also has an inverse effect on any ship heading south, decreasing odds of detection. Supposing that you're trying to find a target you have some prior knowledge of (a radio reported convoy for instance), reducing the speed vector would surely help to actually locate them. Otherwise, I suspect that it doesn't actually help due to the inverse nature this has on finding targets going the other way. Though it might due to reasons I can't quite fully articulate at the moment.

I think you're missing the fact that figures 1 and 2 show the same thing - the sub is going due East & West in both, and the ships are traveling North-South. The change is this: in Fig1, the "camera" is hovering above the same spot on the earth. In Fig2, the "camera" is hovering above the same ship. Everything else in unchanged. The sub only appears to have a northward cant because the frame of reference is moving south. To a ship moving south, the same sub would appear to have a southward cant.
It's an exercise in relativity. The reason to use the ship's frame of reference is to visually display the area searched by the sub and the areas to which the sub is blind. Fig3 demonstrates that, as the speed of the sub increases relative to the speed of the ship, the gaps in its search pattern shrink until there is noplace to hide