Mr. William Hamby III, Ph.E.
1394 Del Sol Lane
San Diego, Ca. 92154
2004 July 13, Copywrite
The reader must realize that this paper is only an overview of a very complex phenomena. To appreciate more exact details I highly recommend a book by Mr. G.F. KINNEY titled, "Explosive Shocks in Air".
Unfortunately, differences in the human condition and or human perception has created a world environment where the unjustified, flagrant disrespect for human life has resulted in the use of explosives for the sole purpose of murder and meyhem has become somewhat commonplace.
Fortunately, many decent, moral, law abiding peaple are on a constant hunt to prevent and capture these destructive individuals or organizations. Some work in law enforcement, others in anti-terrorism task forces, others in security, survailance or technology.
This dissertation is written from the technological perspective, by the author and a combination of work from the United States Department of Defence, United States Department of the Army (Bombing Analysis of the Murrah Federal Building Oklahoma, U.S.A. 1995 April 19), and a incredible book "Explosive Shocks in Air", by Mr. Kinney. The purpose of this paper is to instruct the reader how to quickly and easily estimate the size of a bomb blast and its unique chararistic "signature" or pattern for the purpose of perhaps identifing and capturing the persons, and or the organizations that are responcible.
One primary element that should clearly instill in the reader is how much damage a very small quantity of explosive can do to persons and structures, while similarly they are very statisticaly survivalable given a modest distance of separation from the epicenter. Some feel that economic lack of resources and or inequality, greed and language differences will aways provide a basis for war of some form or another worldwide at any given time. However, long term, I believe the most effective form of security shall be prevention; and the long term education of all the children of the world to address differences and grievances with each other with respect and in a more productive and unifieing way.
This paper is dedicated to those who have suffered as a result of explosives.
All objects moving in air, impart energy and therfore, transmit an energy to the air. This energy generates SOUND waves, and sometimes SHOCK waves, and very rarely, HYPERSONIC (shaped charge) and PLASMA waves.
It is very easy for a human, or even a dog to recognize the sound of a loved ones voice, as opposed to confusing it with a cricket, or a jet. This is elemental signature or class identification. A more subtle signature identification would be noticing a loved one is just starting to get a cold or sore throat. The widespead use of spectrum analysis and F.F.T. analyzers has simplified the generic parameters and descriptions of classes of objects, but still the human mind and ear are extremely adept at this.
The size, shape, velocity, temperature, frequency and harmonic content of the object determines its "signature" and somewhat, its identity or object class.
Air density = d = 1.18 kilograms / cubic meter
c= speed of sound = 343 meters per second
ATM = Atmosphere = standard air pressure = 101325 newtons / square meter
Most sound is measured in watts or decibels, by definition and measurement
the begining of human hearing is 0 decibels (db).
The 2 primary definitions of an acoustic decibel are:
For most ordinary low power sounds under about 120 db, db(W) and db(P) are approximately the same, however above about 120 db the db(W) typically becomes higher than db(P) and it becomes important to state which type reading was made.
Although not evident a human will percieve just 1 watt of acoustic power as "extremely loud". In general, ordinary bombs, SHOCK waves and sonic booms generate 10 EXP+8 to 10 EXP+10 WATT seconds of energy.
Sound pressure (p) can also be called "overpressure" and is most conviently expressed as:
Most chemicals and objects are "stable". They live out their life or reality in the same stable, unchanging (boring) condition or state, second by second, minute by minute, year by year. Slowly, peaple get wrinkles or rocks turn into sand.
Chemically they are becoming more "entropic" i.e. entropy. Entropy is a process of "increasing disorder" or "increasing randomness". A rock turning to sand is a nice example of a SLOW RATE of entropy. A bomb explosion is an example of a FAST RATE of entropy. Say; 1000 pound rock turns to sand at a rate of 1 pound per 1000 years (boring, if you dont believe me try watching it). Humans degenerate at the approximate rate of 0.859 percent per year, or 2 to the power of 81 D.N.A. replications per 80 year lifetime.
Explosives upon detonation increase entropy perhaps 1,000,000 times per millionth of a second! This also changes its volume by about the same amount. As an example, 1 gram of T.N.T. (tri-nitro-touloulene,C7H5(NO2)3) about a half a thimble full as a solid becomes about 600 liters of gas (refridgerator size). When this happens it makes a SHOCK wave and pushes out anything in its way, worse still it does it so extremely fast, sometimes 1 mile per second or even faster. Thats why bomb explosions cause nearby objects to fly outwardly huge distances, these flying objects are called "schrapnel" and is how a granade works. A high percentage of human injurys from bomb explosions are from the schrapnel. If the SHOCK wave is strong enough it can also injure or kill directly (no schrapnel), or make the person a large piece of schrapnel.
Generally speaking most explosives are "conventional" that is to say they are "non-nuclear", in a strict sence they are "chemical" bombs. Irrespective of the intial source of energy either chemical or nuclear, the effect on humans and air molecules is the same. Scientifically, it is convienient to understand that the SHOCK waves they generate ARE EXACTLY THE SAME, once they are adjusted to the same distance and or size, force level. In other words
This is valid rule until approximately 247db (6,575 P.S.I.) the chemical T.N.T. internal limit, and or 1000 atmospheres (14,720 P.S.I.)"overpressure" (the volumetric radius limit of a spherical charge of T.N.T.). This rule is NOT valid for PLASMA, HYPERSONIC waves or INSIDE a nuclear bomb.
The primary characteristics of a bomb explosion are:
1.chemical or nuclear type
2.yield amount expressed in terms of wieght equivilent T.N.T.
3.The airburst altitude: air hieght, surface or underground blast
Plasma is usually when the temperature of something is so high the electrons are no longer are "happy". This causes an internal breakdown in the basic strenth or structure of the material, i.e. steel comes to its "triple point", essentially a gas. Superheated plasma can be generated in the laboratory as electricity, arc welding or a shock wave. Plasma is also created by lightning or several meters very near the center of an nuclear explosion.
An explosive "shaped charge" is an ordinary explosive which has been made into a special shape, usually a deep cone like a champange glass. Shaped cones concentrate and accelerate the SHOCK wave and raise the SHOCK temperature extremely violently, SHOCK cones can easily exceed 4000'Farenheit, so the SHOCK wave pressure blows and "melts" and through its target. Some shaped charges can go through 19 inches of steel or 10 feet of solid concrete!
The chemical types can be any number of chemicals however the universal measurement standard is T.N.T.. However there are many more than listed here: lead picrate, lead picolinate, sucrose octinitrate, sodium perchorate, Plastique, C4, R.D.X., picric acid, P.E.T.N., amytol, and the weakest is gunpowder. Typically chemist's rate explosive to the energy yield of T.N.T. i.e. 1 = T.N.T., 0.10 = gunpowder by wieght for equal bomb strength (i.e. it requires 10 times as much gunpowder by wieght to equal the energy of T.N.T.. Nuclear bombs are also calibrated against T.N.T. however it is usually tons (2000 pounds or metric tons 2204.9 pounds). For large nuclear bombs kiloton (1000 ton) or even MT (megaton = 1,000,000 tons) yield. Sometimes the detonation velocity is important as in particular for construction site rock shattering, higher is better. For mining and dirt removal (heave yield) a slow velocity is prefered, a range of detontion velocities is from 300 feet per second(gunpowder) to 2 miles per second for some "high explosives". Detonation velocity (DET. V) is only a partial contributor to the total energy of an explosive. Oxegen combustion rate, total nitrogen content per unit mass and chemical efficientcy are other factors.
This paper also recognizes T.N.T. as the reference standard yield, either chemical or nuclear.
Geologists have determined much about the nature of the earth and one important concept is that the earths crust is very consistent in terms of measurement, dencity, resistance to force ect.. This has greatly revolutionized earthquake science and volcanology. Similarly it has now begun to revolutionize bomb and SHOCK wave theory. Since the earth is so consistent worldwide, once reference bomb yield explosions have been measured and calibrated, there is a very high degree of accuracy when making estimates about energy yields from earthquakes, volcanoes or as in this paper bomb wieght yield estimation.
Several separate sources have calibrated and measured the crater sizes from a large number of reference bomb explosions both nuclear and chemical and have proved to be extremely consistent and accurate. Until now however there has not been a simple way to correlate bomb yield with the crater size or total ejected volume. Another benefit from this method is that even though some bombs occur in building structures, cars or buses those materials are basically "transparent" to the explosive SHOCK. The SHOCK wave does not lose much force and energy from hitting them and continues through them quite effortlessly. The earth however has a massive resistance factor and the SHOCK wave loses a huge amount of energy and force in the creation of a crater. So basically no matter where the bomb was, as long as it was somewhat close to the ground in ALTITUDE("Surface Burst"), it is possible to cross correlate all bomb explosions of all sizes and determine the reference yield.
Obviously, the larger the yield the larger the crater, but actually it is a cubic base equation (to the power of 3). Later I will show the many equations I have created to express either the yield, distance or crater size to any size explosion. To reduce the size of this dissertation I have not included all the reductions and proofs to generate the equations listed here, however I have evaluated dozens of explosions from many different aspects and have concluded an
In order to simplify and tremendously shorten this paper, I will briefly say that SOUND waves(X air) are "happy" air molecules and that SHOCK waves(Y air) are "angry" or even "cranky" air molecules.
The SHOCK velocity is a capital U. The AIR PARTICLE velocity is a lower case u. THEY ARE DIFFERENT THINGS!
In bomb mathematics every variable has a subscript X or Y. The
This is the shocked air.
Simplistically, this is due to the fact that SHOCK waves generally have a higher energy, speed, temperature and quicker release of energy than SOUND waves. SHOCK waves also lose overpressure extremely rapidly with increased distance. SOUND waves lose overpressure less rapidly and is proportional to distance squared. More simply SOUND loses 6.02 db every doubling of distance from the source. SOUND waves travel by air molecules "bumping" into one another, one nieghbor molecule bumps the next nieghbor. This collision speed has a limit, and this speed is the
If the object in question exceeds the speed of sound it is said to be "SUPERSONIC" and it has broken the sound barrier. Supersonic objects generate a "SONIC BOOM". Sonic booms are tremendously loud and powerful. The supersonic speed is approximately 343 meters per second (767 miles per hour), airplane pilots call this "MACH 1". Mach 2 is 2 times the spped of sound, mach 3 is 3 times the speed of sound , ect. HYPERSONIC waves are when the object is going faster than MACH 5, however the air does nothing new or spectacular however continues to overheat.
SHOCK waves travel at the
They are usually extremely fast i.e. MACH 1.5 to MACH 4
The SHOCK wave itself is very, very thin "sheet" of energy. Mathematically it can be reresented by:
Interestingly, to approximate the instantaneous power expressed in WATTS per square meter; divide the Mach velocity by the shock thickness and multiply by the overpressure squared / air density * speed of sound.
Mach(x)= Mach 2 = U = SHOCK velocity = 686 meters per second
u = air particle velocity = 429 meters per second
Q = 2.955 EXP+5 newtons = 203.4 db(Q)
p = overpressure = 205.0 db = 3.546 EXP+5 newtons / square meter = 3.5 atmospheres =3.5 ATM
8.42 EXP+17 WATTS per square meter!
As clearly evident: theory, reality and human comprehention are not always overlapping.
In a more powerful blast the WIND pressure (Q) could exceed p by a factor of up to 5.76 times.
In SOUND and SHOCK waves there is an actual air particle movement i.e. a displacement and a velocity. However in sound each individual air molecule (particle) is shaking back and forth VERY slowly, about 0.001 inch per second. In a SHOCK wave the air particle velocity is very high, and can actually exceed 1000 miles per hour, and is only one directional , outward. The
This relationship is physically dependant in several ways;
Both have elements of predictability and much in common. A simple concept to visualize is that of the air particle velocity. Most extremely loud SOUNDS like a rock concert or a scream rarely generate higher than an
Mathematically and in reality, it is possible to show that there are overlapping conditions in which you have both a SOUND and a SHOCK (and the generation of an air particle velocity). A human being can just begin to perceive a blast wind particle velocity of about 1 foot per socond at 128 db. At 177 db the air itself begins to show signs of non-linearity due to the blast wind particle velocity. At 190 db the non-linearity is to large and SHOCK wave formulas must be used.
This SHOCK wave pressure does not shake back and forth like a loudspeaker playing music (SOUND), it is just a ONE WAY, outward pushing, positive over pressure force above the
So a very destructive 2 P.S.I. overpressure SHOCK wave makes a total of 16.72 P.S.I. absolute pressure. Only the initial peak values are used.
SOUND waves are usually measured in decibels. (peak or R.M.S.)
This is a rule of convention only, because its very inconvient to measure SOUND using a P.S.I. meter, and no one has converted SHOCK waves to decibels until now, by this author.
The reason I chose to express SHOCK wave pressure as decibels is that once you do it, it makes all other bomb calculations very easy, quick and straight forward. In the long run it greatly simplifies all work calculations. Futhermore, SHOCK wave size and intencity typically generate and degenerate logarithmicly (cubicly i.e. to the power of 3), and decibels are already logarithmic so SHOCK wave and DECIBELS readily transpose and substitute back and forth with each other once you learn this papers simple equations.
Furthermore, if a noise source is louder than 194.09 db or faster than the speed of sound most of its fundamental characteristics become mathematically non-linear. Then the air no longer behaves as sound but more like shock waves or sonic booms, which for very complicated reasons are hard to estimate. Bomb explosions are tricky to measure or estimate because they get significantly more violent the closer you get (to the power of 3) so sometimes crater size is used to estimate bomb energy with the general formula: the cube root of bomb energy change equals crater size change.
Allthough it is not intuitively clear, SHOCK fronts degenerate into SOUND waves, with an increasing distance from the source. Therefore; measured overpressure degenerates initially at a cubic rate then at a "slower" rate, i.e from the third power to the second power continuosly. The exponents are 3.0000 then 2.99, 2.98, 2.97,..., 2.02, 2.01, so on and finally 2.0000 for pure SOUND as the distance from the source is increased.
This is a major reason why bomb yield simplification has been difficult goal to achieve. Nearby, the SHOCK wave decibel drop can be high as 18.06 db every distance doubling (cubicly), then fading off to 11 to 9 db loss every distance doubling , until its just a loud SOUND (elastic wave) and loses 6.0206 db every distance doubling, just like SOUND at a very far away distance, sometimes miles in the case of a nuclear bomb. This wave is decaying at the same rate as SOUND (-6.02db / double distance), although it may not have frequencies in it like sound. It may just have a peak value and then diminish.
I have created a term called the
More simply it is the far away critical distance at which an additional distance doubling will create a 6.02 db loss in peak overpressure. Using data or measurements or my formulas this critical distance can be determined. Then, from this critical distance it's decibel level is recalculated by distance meter "HALF-INGS" until it is a theoretical distance of 1 meter from the source.
As an example a medium bomb 0.57 tons = 208.97db(NP) will have a critical distance of say 2048 meters(about 1.5 miles). So at 2048 meters a pressure meter says 0.040 P.S.I.(142.75db) then we know that at 4096 meters the pressure level will be half this or 0.020 P.S.I. (136.73 = 142.75 - 6.02db). To further show this, the pressure readings at 1024 meters are 0.10252 P.S.I. in other words at 1024 meters there is a 2.563 times increase in the pressure, not just a 2.000 increase that we would expect from SOUND so at 1024 meters it is still behaving as a SHOCK wave. At a very far 8192 meters the measured pressure level is 0.007 P.S.I. which is lower than the 0.01 P.S.I. we would expect, even as a SOUND wave, this is because of additional air friction and heat and geography or weather in the SOUND wave. This is called the "FAR FIELD". Far field measurements are scientifically useless, inaccurate and hard to predict. Extensive data shows the critical distance D is approximately 3000 to 4000 meters for a 6.5 ton yield.
Mathematically critical distance can repesented as;
D = critical distance
6.02 < 20 log (pressure at 0.5 D / pressure at D) = to NEAR = SHOCK zone
6.02 < 20 log (pressure at 0.5 D / pressure at D) = to FAR = FAR zone
i.e. 2 meters , 4 meters, 8 meters, 16 meters, 32 meters,..., example; if: J = 9 then critical distance is 512 meters.
1024 meters = 2 EXP(10) meters
This GAINS a specific db value of db(G);
This is now added to the actual peak overpressure gauge readings at the specific critical distance = db(P). So;
The critical distance in theory is exact; in reality data suggests it is a range ZONE that varies +- 25 percent, i.e. 0.75 D to 1.25 D.
This method has proved to be quite accurate and reproducible in both directions, provided the range span is not to extreme; (db(G) > 100 db)) or D > 65000 meters (40 miles), i.e. (J > 16)
The db(NP) level is extremely accurate but underates the true energy and destructive power of a bomb because it does not include the blast wind pressure (Q).
For any db level higher than 207.46 db(P) the blast wind pressure (Q) is higher than shock wave overpressure (P) or (NP).
It is more accurate to include the blast wind db(Q) when rating the destructive energy of higher power bombs.
Some high power SHOCK waves are even beyond the description of rigid mathematics, i.e. "functionaly discontinous". Furthermore, many SHOCK waves superheat the air, and must be described using gas chemistry. This paper will try to remain as simple as possible and not involve advanced gas chemistry or mathematics.
all formulas are log base 10
db(N) = distance normalized air TOTAL ENERGY sound pressure plus wind to a theoretical distance of 1 meter
db(NP) = distance NORMALIZED peak shock front OVERPRESSURE to a theoretical distance of 1 meter pressure
db(Q) = peak DYNAMIC pressure i.e. blast WIND after shock wave pressure front passes by at a certain location
db(PR) = REFLECTED shock overpressure is the momentum force a wall feels when hit by a shock overpressure
6.5 tons T.N.T.= 216.0 db(NP)+-0.3DB (NP = normalized PEAK SHOCK OVERPRESSURE not including blast wind Q pressure which usually is much higher), and a 43.68 foot crater, critical distance is approximately 4000 meters.
W = WIEGHT OF T.N.T. = YIELD
or very convieniently;
peak SHOCK OVERPRESSURE only, does not include blast wind DYNAMIC pressure = db(Q), i.e. db(P) less than 207.46
DB(P) CHANGE = 20 * LOG (change in crater WIDTH)
CRATER WIDTH CHANGE = (change in explosive WIEGHT) raised to the power of 0.3333
DB(P) CHANGE= -20 * LOG (change in DISTANCE to the power of 3)
(DISTANCE FROM BOMB1/DISTANCE FROM BOMB2) = (W1/W2) TO THE POWER OF 0.33333
D1 = D2 * W TO THE 0.33333 POWER
DB(NP) = 210.6 + 6.67 * LOG (TONS T.N.T.)
BLAST WIND VELOCITY = u = OVERPRESSURE/((AIR DENCITY = 1.18) * C), BELOW 177 db(P)
BLAST WIND VELOCITY = u = (5*OVERPRESSURE/7*ATM) * (C/SQUARE ROOT(1+6*OVERPRESSURE/7ATM))
MACH NUMBER = M = (SHOCK VELOCITY)/C = SQUARE ROOT(1+(6*OVERPRESSURE/7*ATM))
OVERPRESSURE = p =(SHOCK ATM)-(ATM) = ATM (7*(M SQUARED - 1)/6)
BLAST WIND PRESSURE = Q = 2.5 * OVERPRESSURE SQUARED/(7*ATM + OVERPRESSURE)
Q under 194 db(P) APPROXIMATES to Q = (atmosperes squared)/2.801
Q between 194 db(P) to 234 db(P) must be CALCULATED from formulas above
Q between 234 db(P) to 240 db(P) approximates to Q = 2.30 * overpressure
Q between 240 db(P) to 250 db(P) approximates to Q = 2.40 * overpressure
Q between 250 db(P) to 260 db(P) approximates to Q = 2.49 * overpressure
Q over 260 db(P) IS BASICALLY Q maximum, Q = 2.5 * overpressure
db(PR) REFLECTED SHOCK OVERPRESSURE = 2 * OVERPRESSURE + 2.4 * Q = 2 * OVERPRESSURE(7*ATM + 4*OVERPRESSURE/7*ATM + OVERPRESSURE) = overpressure units may be newton /square meter or ATM for quicker, simpler results
At very close range so db(P) is more than 207.46, varies +-3.0 db because of db(Q):
DB(P,1 ton t.n.t.) = 265 - 20 * LOG (meters DISTANCE to the power of 3)
1 foot crater width = 0.156 pounds of T.N.T.
10 foot crater = 156 pounds
Crater WIDTH = 7.2 * Crater DEPTH, approximately
+6.0206 DB(P) CHANGE = +8.0017 TIMES MORE EXPLOSIVE WIEGHT
+6.67 DB(P) CHANGE = +10.000 TIMES MORE EXPLOSIVE WIEGHT
+10.000 DB(P) CHANGE = +31.5682 TIMES MORE EXPLOSIVE WIEGHT
The is also a dependant relationship between the total ejected volume of crater dirt and the explosive yield.
The (NP) level is extremely accurate but underates the true energy and destructive power because it does not include the blast wind pressure, recall that any db level higher than 207.46 db (P) the blast wind pressure (Q) is higher than shock wave overpressure (P) or (NP).
With some practice useage I believe most engineers will find these formulas quite handy and useful.
Mr. G.F. KINNEY author of "Explosive Shocks in Air"
"Empirical Scaling Laws for Truck Bomb Explosions Based on Siesmic and Acoustic Data"; Mr. Koper, Mr. Wallace, Mr. Reinke, Mr. Leverette. Bulletin of the Seismological Society of American, Vol. 92, No. 2, pp 527-542, 2002 March.