Algebraic Firing Solution

[TorpX]

Final Form:


After I got the basic solution equations, I spent some time checking the math by computing the answers to a few sample problems and plotting the results on graph paper to verify the the answers are correct, and not just a heaping pile of algebraic missteps. Then, I went back to the equations and multiplied out the terms to look for ways to simplify the equations, or at least, put them into a more pleasing form. I was able to use trig reduction formulas in a few areas, but could not really make any major simplifications. I also put the trig functions in terms of the track angle and added a factor, Lg for what I call “launch lag”, as experiments indicated that this was greater than I had expected.










Residual problems:


Most of my work now is on developing a program that will do these calculations and some other auxiliary tasks to make this method a practical proposition. As I said before, the equations for t1, t2, and t3 are the core of this method. It is not immediately obvious, but the denominators of the expressions for t1, and t3 can be zero, resulting in either or both of t1 and t3 being undefined. If these cases were all “no-solution” cases, this wouldn't be a problem, but I found this not to be the case. For example, if the sub is heading north and the target south, on tracks that coincide, go = 0, t1 is undefined, though clearly a solution exists! On the other hand, if sub and target are heading in opposite directions, as before, but on parallel tracks, t1 is undefined, and no solution exists. I spent quite a bit of effort trying to sort through these cases of “parallel tracks”, in an effort to separate the wheat from the chaff. Then it occurred to me that there was a much better way.


I arranged the program to calculate t1 and t3, if they are defined. If t1 is undefined, t1 is set to zero, and passed to the calculation of t3, to check if this will yield a solution. If t3 is undefined, the alternate equation is used. Then the provisional solution is checked by adding up the positions of target and torpedo through each phase, so that the final position coordinates can be compared. If they match up, the solution is considered valid, if not, it is rejected. Easy!




Still to do:


I still have quite a lot to do on the AFS II program. The basics are in place, but a lot of details and refinements are not done. I think I have the torpedo data for the Mk 10 pinned down pretty well, but haven't started on the Mk 14 yet. I might try to compile data for one or two German torpedoes, later.


I'll be posting some sample problems in the near future, to illustrate the method.

I'm sure there are some aspects of this that aren't entirely clear, but I'm not sure how far into the weeds people want to get. If anyone has any questions, I'll do my best to answer them.

Edit: I forget to mention that all these equations are for arbitrary units; that is distance in length units, time in time units, speed in length/time units. The reason for this is, that it is much easier to check the math this way.


TorpX



P.S. If anyone can give me some tips on getting equations and formulas display in a proper manner, I'd be grateful. Using the Paint program was not my first choice, but I couldn't get the math to display well in wordpad.