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As some may have noticed, I have already mentioned that this paper doesn't really seem to be very clear but now I believe its mathematical basis is fundamentally flawed. I believe this a model used by the US Navy and cited many times in recent years so I'm assuming that I must be wrong on this, however it's such a simple thing to prove that it doesn't work...
THE PAPER IN QUESTION:
http://www.jhuapl.edu/techdigest/td1703/thomas.pdf [Broken]
The problem arises from the fact this model calculates imaginery angles, which is completely unphysical. There are two sources of imaginery components:
1)
Zi' formula on page 3 can be rewritten by taking out the common factor of [Cos(delta) + Z(z1',delta)]/2.
The other term is then 1+Cos[(2i-1)etc..] which can only vary from 0 to 2.
Z(z1',delta) can take a maximal value of 1 and it turns out that by using typical values that it is very close to 1 typically.
So in total Zi' varies from 0 to 2.
Now, in the expression for f, we have one term that is written as (1-(Zi'^2))^(0.5), so this obviously can be imaginery. Not only can it be imaginery, it must be for low i.
2) The second big problem lies in the numerator of f...
Zi' varies from 0 to 2. But Z(z1',delta) is a fixed value close to 1.
Consider i is high => Zi' is low and hence the bracket becomes negative.
The second bracket is the numerator cannot save the day since if Zi' is low for say delta=0, then we still have a positive number here and hence the numerator will be imaginery overall.
The first bracket in the numerator term inside the square root can also be imaginery.
The only get out clause to both of these problems would be hoping that the summation would result in the imaginery terms cancelling out, but I have observed this not to happen, unsurprisingly.
So I believe this paper's model is fundamentally flawed and I am somewhat annoyed if these are indeed genuine errors. Have I missed something or do others agree?
Natski
P.S. I have attached a mathematica notebook with my workings. They aren't really necessary to understand the problem here. I have changed some of the variables' symbols to make life easier.
THE PAPER IN QUESTION:
http://www.jhuapl.edu/techdigest/td1703/thomas.pdf [Broken]
The problem arises from the fact this model calculates imaginery angles, which is completely unphysical. There are two sources of imaginery components:
1)
Zi' formula on page 3 can be rewritten by taking out the common factor of [Cos(delta) + Z(z1',delta)]/2.
The other term is then 1+Cos[(2i-1)etc..] which can only vary from 0 to 2.
Z(z1',delta) can take a maximal value of 1 and it turns out that by using typical values that it is very close to 1 typically.
So in total Zi' varies from 0 to 2.
Now, in the expression for f, we have one term that is written as (1-(Zi'^2))^(0.5), so this obviously can be imaginery. Not only can it be imaginery, it must be for low i.
2) The second big problem lies in the numerator of f...
Zi' varies from 0 to 2. But Z(z1',delta) is a fixed value close to 1.
Consider i is high => Zi' is low and hence the bracket becomes negative.
The second bracket is the numerator cannot save the day since if Zi' is low for say delta=0, then we still have a positive number here and hence the numerator will be imaginery overall.
The first bracket in the numerator term inside the square root can also be imaginery.
The only get out clause to both of these problems would be hoping that the summation would result in the imaginery terms cancelling out, but I have observed this not to happen, unsurprisingly.
So I believe this paper's model is fundamentally flawed and I am somewhat annoyed if these are indeed genuine errors. Have I missed something or do others agree?
Natski
P.S. I have attached a mathematica notebook with my workings. They aren't really necessary to understand the problem here. I have changed some of the variables' symbols to make life easier.
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