Submarine collision chances

cavedave

Hello. Can you help me with this calculation?

There are 25 submarines in the Atlantic. Each has the volume of a 24.7m radius sphere. The area of the Atlantic they can go in (the top 300 meters) is 30 million cubic km.

If I model the submarines as gas particles and have the Atlantic as the box holding them the problem becomes a mean free path calculation.

I've worked out these figures here and here.

I figure it out to be about one in every 6 million days there should be a collision. If you can point out any errors in the reasoning or calculation I would really appreciate it.

Professor_Fink

cavedave wrote: »

If you can point out any errors in the reasoning or calculation I would really appreciate it.

Submarines have sonar ;-)

Capt'n Midnight

subs don't travel in a random walk
nor is the depth they travel at random, if they aren't on the surface they are likely to be deep enough to avoid all surface traffic

the boomers are likely to be moving very slowly or not at all, the Russian Typhoon class are designed to sit on the sea bed

great idea for modeling, but the model must match the real world situation
maybe model for the likely hood that trawler in the irish sea catches a sub ?

cavedave

Professor_Fink

Originally Posted by cavedave View Post
If you can point out any errors in the reasoning or calculation I would really appreciate it.
Submarines have sonar ;-)

When wandering the Atlantic don't use sonar as this gives their position away.

Capt'n Midnight

subs don't travel in a random walk
nor is the depth they travel at random, if they aren't on the surface they are likely to be deep enough to avoid all surface traffic

the boomers are likely to be moving very slowly or not at all, the Russian Typhoon class are designed to sit on the sea bed

great idea for modeling, but the model must match the real world situation
maybe model for the likely hood that trawler in the irish sea catches a sub ?

They don't randomly move. But neither do particles. Subs do try not follow set paths and i think particle movements are a reasonable approximation of this. Fair point on the surface traffic, but if you say they are between 20 and 320 meters you still have the same volume of a box.

I'm trying to model the french and british submarine crash but the trawler one is interesting.

Capt'n Midnight

http://en.wikipedia.org/wiki/Random_walk

the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler can all be modeled as random walks.

cavedave

A fair point you can model the effects of a particle traveling through a box using monte carlo methods. Here the changes in direction are caused by collisions with the sides and each other though otherwise they travel in straight lines. The path of a foraging animal is a good comparison to a wandering nuclear submarine. But i think in this case a mean free path calculation might be easier.

The problem i have is that n the number of particles to unit volume makes sense when its particles per meter. But i cannot figure out what it means when it is 25 particles of radius 24.7m in 30 000 000 kilometers cubed.

Image of the equation http://4.bp.blogspot.com/_CcLjYF5EMyU/SaXWBjGWx7I/AAAAAAAAA0k/0bonb1JGx_c/s1600-h/eq2.latex

Professor_Fink

Well, you're going to run into some major problems. Firstly, you can probably assume that every submarine from a given country is aware of other subs from that country operating within the same area. In which case, this collection of subs will have self avoiding walks. Secondly, not every position in the ocean is equiprobable, and so collision probably will be increased in some areas, and reduced in others in a way that does not conserve the probability of collision. Lastly, most subs are going to have partial information about others operating in the area, even if they are not from the same area. While active sonar may not be used under common circumstances, you can be pretty sure they are listening out for other subs and ships. Without taking factors into account, it seems highly unlikely that you can produce an even remotey accurate estimate of the collision probability.

Professor_Fink

Capt'n Midnight wrote: »

http://en.wikipedia.org/wiki/Random_walk

Subs almost certainly won't follow random walks. The captain is almost certainly following orders that require either patroling of a certain area (which prevents diffusion) or following some particular path (in which case they will be constrained to remain close to the path). Both cases seem at odds with a random walk model. The reason foraging animals can be modeled as following random walks is essentially because that's what foraging requires. You look in one position, and if there is no food, you look some place else.

cavedave

Professor_Fink Well, you're going to run into some major problems. Firstly, you can probably assume that every submarine from a given country is aware of other subs from that country operating within the same area.

that assumption is taken into account with the 25 figure. There are about 50 nuclear subs in the world. Not all will be in the atlantic. The 25ish american and british ones should know roughly where each other are same for the 20 ish russian subs. So i reckoned there would be at most 25 subs you could run into in the Atlantic.

Secondly, not every position in the ocean is equiprobable, and so collision probably will be increased in some areas, and reduced in others in a way that does not conserve the probability of collision. Lastly, most subs are going to have partial information about others operating in the area, even if they are not from the same area. While active sonar may not be used under common circumstances, you can be pretty sure they are listening out for other subs and ships. Without taking factors into account, it seems highly unlikely that you can produce an even remotey accurate estimate of the collision probability.

True. However the point of having subs that can launch nukes from anywhere that no one knows where they are means that the opposition have to assume your subs are completely randomly distributed. They are listening for other subs but again the point of having subs is that the other guy cannot know where they are. Still though you are right some information and non randomness will be present in the subs position i am just as a first approximation assuming it is not.

*In[29]:= MeanFreePath = 1 êH1.4 * 3.14 * SubSphericalRadius2 TimeBetweenCollisions = MeanFreePath * SubsPerUnitOceanL êSubVelocity
Out[29]= 4.80437 μ 1010
Out[30]= 8.57923 μ 109
8.6 Gigaseconds, that' s 279.6 years.

Capt'n Midnight

TE=Capt'n Midnight;59193442]subs don't travel in a random walk [/QUOTE]

Professor_Fink wrote: »

Subs almost certainly won't follow random walks.

If the subs did a random walk then they would eventually pass through every space.

maybe they have an arrangement of depths for different countrues to avoid collisions a bit like the way the RAF used to do with the Buccaneers