Bomb blast radius?

[RTHolmes]

An exponential decay is y~e^-x rather than y~1/x^2. If you use an inverse square law to model blast damage you will have infinite bomb damage at zero radius (the whole dividing by zero thing).

sure about that? my gut says inverse square is a good, if simplified model. I'm not too bothered by infinite "damage" for r=0, although I'm too tired atm to work out exactly why (something to do with a finite amount of energy and a sphere with surface area of 0)

[2Slow]

An easy way to do this would be to drop a single 1000 lb bomb in the middle of the high building density area of a town. Fly back over and take a screen shot of the damage with something of known length in the frame (a tank as suggested earlier). Then the tank can be used as a scale and the radius at which a 1000lb bomb's damage decays to the 250 lb damage level by measuring the distance at which the last town building is destroyed.

Good idea!  In case anyone is interested, 1 mm of measurement equals 33 scale feet.  What I have done is insert the town.bmp into a MS Word document.  I do not expand it or shrink.  This is the map I work off of.  Oh yeah, I am working off of a printed map document.

[tf15pin]

sure about that? my gut says inverse square is a good, if simplified model. I'm not too bothered by infinite "damage" for r=0, although I'm too tired atm to work out exactly why (something to do with a finite amount of energy and a sphere with surface area of 0)

Energy falls off exponentially in a dampened medium in every case I have ever come across (bullets, electrons, electromagnetic radiation, water waves...) So if damage is related to energy it will also follow an exponential decay. We are essentially describing a shock-wave which would be modeled as a propagating dirac-delta function. (read:"moving high pressure front") In one of your earlier responses you called it an exponential decrease but then gave an inverse square relation for the equation. Now if you want to talk about the amount of shrapnel moving through the surface of some given sphere that would follow an inverse square law.

An argument against an inverse square law being appropriate is the zero radius problem, if you cant predict how much damage the bomb does exactly where it lands then how can you predict it anywhere else with any confidence? If you tried to do anything with it you would also find that working with an exponential decay mathematically is much easier than working with an inverse square law anyway. So if simplicity of the model alone were to be used as a measure of the usefulness of  a model I would go with the exponential decay.


Finite energy in a sphere with zero surface area? Sounds like we are starting to talk about black holes now. How did you find out about the black hole bomb that the bish nation is developing?!?!  

[tf15pin]

An exponential decay is y~e^-x rather than y~1/x^2. If you use an inverse square law to model blast damage you will have infinite bomb damage at zero radius (the whole dividing by zero thing).

For this case let our damage function be D(r)=D0 e^[-c r]: where D is the damage at radius r, D0 is the damage at radius 0 (the max damage of the bomb), c is some coefficient of decay that we have to determine from dropping a bomb and fitting the curve to our observations, and r is the radius.

The first thing that has to be done is to determine c. To do this we will take a 1000 lb bomb and drop it near something that takes a known amount of damage to destroy (town building = 250lbs if I remember right). So we make the drop and find that the maximum distance a 1000 lb bomb can land and take out a building is 100'. We now have all we need to know! We throw some algebra at it to give us c = -1/r Ln[D/D0], plug in the numbers, c =  -1/100 Ln[250/1000] = 7.21*10^-3 (1/ft). Then we plug our new found c into our general formula and have a function of damage at a given radius for 1000 lb bombs. If things are modeled consistently in the game this function will hold for all bomb types. So for this example we would have
Damage = Bomb weight e^[(7.21*10^-3) r].  

I made a nice plot of the function but I cant figure out how to post and image on here so the function and Idea will have to stand alone.

REMEMBER THIS IS AN EXAMPLE AND SOMEONE NEEDS TO MAKE A MEASUREMENT SO THE CURVE CAN BE PROPERLY FIT. So you just need a known bomb weight destroying an object of known hardness at a known maximum distance and then it can be fit.

I missed a set of parenthesis around the 1/100 when I ran the numbers for the "c" calculation through my calculator so the correct value for c is 0.0139 1/ft.

[RTHolmes]

Energy falls off exponentially in a dampened medium in every case I have ever come across (bullets, electrons, electromagnetic radiation, water waves...) So if damage is related to energy it will also follow an exponential decay.

decay/damping is certainly a factor but this is surely more of a propagation problem. I was thinking in terms of the amount of energy in the shock front hitting a fixed size object at varying distance, or equivalently the overpressure. I did forget that shock fronts arent elastic sound waves (whose energy does decrease at 1/r²) but are entirely different (single front, traveling in one direction and much faster than sound). a little sniffing around the net reveals that the shock wave energy starts to reduce with distance at 1/r³ (Ive woken up a little since dinner but why its a cubic deal is beyond me at this point) and the rate gradually, continuously drops to 1/r² as the shock wave slows and eventually becomes a sound wave.

I still dont have a problem with r=0, expand the gas in a sphere with 0 volume and you will have infinite overpressure.

interesting problem

[Ping]

 
Think I just entered a foreign language class.

[RTHolmes]

this page has some good stuff on the physics of explosions.

there is an example there for a 1000lb bomb. the critical distance (ie. where the shock wave becomes a sound wave and the overpressure degenerates at 1/r²) is about 2000m.

so the overpressure degenerates at a rate of 1/r³ at r=0 down to a rate of 1/r² at r=2000m. so for our town buildings splash, the short distance from the explosion means that 1/r³ is a decent approximation

[tf15pin]

In the real world over pressure may be the main mechanism for imparting damage to soft structures like non-reinforced town buildings. In the game the damage a structure can withstand is not measured in force/area of overpressure. It is measured in equivalent pounds of explosives. It takes 2750lbs of damage to take down a hangar not 14 psi of overpressure. If things in game were damaged considering over pressure alone, you could drop a 100lb bomb and destroy anything in the game just by hitting it directly since, as you say, it has infinite over pressure at r=0. In my experience a single 100 lb bomb does not kill much even with direct hits.


Until a r^-3 model can predict how much damage the bomb will inflict on a structure how useful is it?

If you can make it work so that Damage = (appropriate constants)/r^3 with appropriate units and all while satisfying the r=0 boundary condition I would be thoroughly impressed.

[4deck]

Think I just entered a foreign language class.

I just scrached my arse too.

[RTHolmes]

well the shock front propagates from the explosive/air boundary outwards, if you can find an existing bomb with r=0 I will be impressed

did you check the link?

[tf15pin]

The bomb does not have radius equal to zero but r=0 is the origin for the explosion, so the casing air interface right before the casing ruptures and the shock wave begins propagating is r=0.

I read the link, it has a lot of good information about shock waves but no information about fragmentation or heat generation so it is missing a large portion of what causes damage when a bomb explodes.

The energy of a shock wave will dissipate much more rapidly than the energy of any bomb fragments so if we are looking for the single damaging effect that has the largest radius it would be the shrapnel anyway. 

Q.E.D.

[2Slow]

Insert Quote
Quote from: tf15pin on August 18, 2010, 11:57:21 AM

An easy way to do this would be to drop a single 1000 lb bomb in the middle of the high building density area of a town. Fly back over and take a screen shot of the damage with something of known length in the frame (a tank as suggested earlier). Then the tank can be used as a scale and the radius at which a 1000lb bomb's damage decays to the 250 lb damage level by measuring the distance at which the last town building is destroyed.

Good idea!  In case anyone is interested, 1 mm of measurement equals 33 scale feet.  What I have done is insert the town.bmp into a MS Word document.  I do not expand it or shrink.  This is the map I work off of.  Oh yeah, I am working off of a printed map document.

Based on my efforts at the bombing range (offline town bombing) I have a damage radius for 1000 pound bombs.  If one measures from the inside wall to inside wall of destroyed buildings then I have it at 82.5feet.  From outside edge to outside edge then the radius is 115.5 feet.  2.5mm or 3.5mm (length of 5 or 7mm divided by 2) measurements on that map as described above.

B17/B24 formation would have a destruction swath of 437' or 13.44mm on the map.

[RTHolmes]

1000lb town building at 41.25' nice

you didnt happen to test 500lbers did you?

The bomb does not have radius equal to zero but r=0 is the origin for the explosion,

yup (in the centre of our spherical explosive charge).

... so the casing air interface right before the casing ruptures and the shock wave begins propagating is r=0.

nope. the explosive/air boundary at which the shock front starts to propagate through the air is at r=d/2, where d=diameter of the explosive charge.

overpressure instantly collapses buildings, not frag or heat. and I assume our bombs are GP bombs where the casing is just thick enough to deliver the explosive charge to target, designed for overpressure, not frag like a nade or cluster bomblet, or heat like an incendiary.

[2Slow]

1000lb town building at 41.25' nice

you didnt happen to test 500lbers did you?


yup (in the centre of our spherical explosive charge).

nope. the explosive/air boundary at which the shock front starts to propagate through the air is at r=d/2, where d=diameter of the explosive charge.

overpressure instantly collapses buildings, not frag or heat. and I assume our bombs are GP bombs where the casing is just thick enough to deliver the explosive charge to target, designed for overpressure, not frag like a nade or cluster bomblet, or heat like an incendiary.

I confused some numbers and words.  I have corrected below:

Based on my efforts at the bombing range (offline town bombing) I have a damage radius for 1000 pound bombs.  If one measures from the inside wall to inside wall of destroyed buildings then I have blast diameter 165.5' (5mm on scaled map) when measured from the inside facing wall to the other inside facing wall.  This gives one a radius of 2.5mm (83') for inside to inside.  From outside edge to outside edge then the blast diameter is 231' (7mm on scaled map.)    This gives one a radius of 3.5mm (115.5') for outside to outside.

B17/B24 formation would have a destruction swath of 437' or 13.44mm on the map.

I have not tested a 500 pound bomb sortie.  Want to bet it turns out to be 50% of the 1000 lb.  250 lb would be 50% of the 500 lb.  100 lb would be a 10% of a 1000 lb bomb.  Way to simple so probably not correct.   

[RTHolmes]

if the 1000lber can just drop a building at r=~25m (82'), a 500lber should do it at ~21m (68') using 1/r³. of course we dont know how the blast damage is actually modelled in AH so who knows.