Proof of Taylor-Sedov blast wave equations including all constants that previously needed numerical integration, and a simple blast arrival time law

It's amazing how poor the Taylor-Sedov shock wave mathematical physics is; they use the very "wooden" (i.e. inflexible!) machine or computer-like approach of solving the equations of motion with the use of zero physical understanding. Hence, they both fail to get a complete simple analytical solution. As a result, they can only derive crude features of the solution analytically, and have to rely on numerical methods (stepwise integration) to determine constants. There is a simple way around this (called the use of actual mechanistic understanding or physical logic), and I've written up this proof in a paper as an extract from a draft internet book in the PDF paper linked here (the maths in this paper have been carefully checked, although some of the introductory words that explain its purpose may need editing for style, etc.). I've just submitted this paper to an online physics archive. Instead of solving the differential equations of motion, you can instead build up a mathematical proof that leans on physical understanding to guide the mathematical proof to a full analytical solution without any need for numerical solutions, that can easily be extended to predict blast arrival times for any distance:



Above: blast arrival times as a function of distance, as measured at various Jangle and Castle series nuclear surface burst tests, from Operation Castle weapon test report WT-934, 1959. The curve corresponds to the blast arrival time equation in the theoretical proof paper.

Weapon test report WT-934 is an excellent summary of weapons effects observed at very high yield (far bigger than current stockpiled USSR and US MIRVed missile warheads) nuclear tests.

Another nuclear weapon test report I recommend to all who are interested in the effects of nuclear terrorism is Jangle report WT-414 (PDF linked here), which summarizes the radiation dose rate as a function of time and other effects measurements made at the first ever Nevada surface burst, 1.2 kt Sugar and also the first ever shallow underground shot, 1.2 kt Uncle at 17 feet underground.

This was the test of the 1950s earth penetrator warhead design, which maximises ground shock destruction to hardened underground targets and minimises above ground collateral damage like thermal radiation and blast; even the fallout that escapes into the atmosphere is constrained to very close-in distances, since it consists almost entirely of large particles. Despite the poor scanned quality of the declassified photocopy, it has beautiful graphs of curves showing how the pulse of initial radiation (at many upwind and downwind locations in each test) decays to a minimum as the cloud rises above the ground (air shielding) and then rises to a second pulse as the fallout (or base surge in the underground shot) settles out. However, it does not include very good versions of the fallout patterns, which were later improved by combining ground level surveys with aircraft surveys corrected for the decrease in radiation level from ground deposited fallout altitude.

For the final versions of the Sugar and Uncle fallout patterns, see Albert D. Anderson, U. S. Naval Radiological Defense Laboratory, A Theory of Close-In Fallout from Land-Surface Nuclear Burst, Journal of the Atmospheric Sciences, vol. 18, Issue 4 (August 1961), pp. 431–442 (online PDF version linked here), Triffet's prepared statement to the June 1959 Congressional Hearings Biological and Environmental Effects of Nuclear War (PDF linked here), and DASA-1251 (the Pacific tests volume of DASA-1251 which contains map scale errors exaggerating atoll area fallout by large factors for some of the 1952-4 Pacific tests, unhelpfully chooses to ignore the fully reconciled 1956 fallout patterns from WT-1317, instead using very poor quality unreconciled data for ocean and land, but is at least a fairly good compendium of some Nevada test fallout patterns, although there is a better documented 1.65 kt surface burst Small Boy test fallout pattern in Freiling's 1970 ACS book).

J. Robert Oppenheimer's attempts to suppress Feynman's path integral formulation of quantum mechanics



Above: Professor Freeman Dyson on how Oppenheimer tried to suppress Feynman. Feynman was respected by Oppenheimer as a junior Los Alamos physicist. For example, Oppenheimer put Feynman in charge of showing journalist William L. Laurence the Los Alamos project, and also placed Feynman in charge of the "computer" - punched card sorter - calculations of the plutonium implosion bomb design, when the Stanley Frankel who was originally in charge of computations, wasted months calculating tables of logarithms! But as soon as Feynman came up with a radical reformulation of quantum mechanics, Oppenheimer and almost all the other famous scientists opposed him. Oppenheimer, described as a "bigoted old fool" by Dyson in the video above, also opposed Teller's hydrogen bomb, and when Teller successfully managed to diminish Oppenheimer's influence, he lost friends who admired Oppenheimer. There is a strong communist style dictatorial culture in physics even today. Just as in the Soviet Union, where only an elite few were members of the Communist Party and suppressed all dissent and innovation with unethical means, the same occurs in physics. Much of the problem stems back to the merger of teaching with research at universities in the nineteenth century. Teaching old stuff has nothing to do with innovative research, so the resulting mix up has had some tragic results in science. Oppenheimer and Bohr in particular opposed Feynman in the arrogant belief that he was ignorant of old stuff, rather than trying to understand his innovation which overthrew the old stuff; they didn't want radical progress and they easily made up false claims about innovators being ignorant of status quo, instead of assessing innovations. Similarly, in the Soviet Union, a central planning committee composed of a select handful could not do the job of organizing science and society, as seen in President Reagan's 31 May 1988 speech on the virtues of freedom versus communism at Moscow State University (he was in Moscow to negotiate arms control treaties from a position of strength not from a position of weakness; see the previous post, linked here):

"The key is freedom - freedom of thought, freedom of information, freedom of communication. The renowned scientist, scholar, and founding father of this university, Mikhail Lomonosov, knew that. 'It is common knowledge', he said, 'that the achievements of science are considerable and rapid, particularly once the yoke of slavery is cast off and replaced by the freedom of philosophy.'

"Some people, even in my own country, look at the riot of experiment that is the free market and see only waste. What of all the entrepreneurs that fail? Well, many do, particularly the successful ones; often several times. ... And that's why it's so hard for government planners, no matter how sophisticated, to ever substitute for millions of individuals working night and day to make their dreams come true. The fact is, bureaucracies are a problem around the world."

Update: Manchester University Home Office Scientific Advisory Branch civil defence researcher Dr Smith has now defended her PhD thesis and gone to the Cabinet Office Strategic Unit at Westminister, according to the news release linked here.

"The role of the [Strategy] Unit is to provide a cross-departmental perspective on the major challenges facing the UK, to work with departments in developing their key policies, and to provide strategic advice and support to the Prime Minister/No 10." - http://www.cabinetoffice.gov.uk/strategy.aspx

She very kindly replied to an email I sent last year about her 32 pages long article, "Architects of Armageddon: the Home Office Scientific Advisers' Branch and civil defence in Britain, 1945-68" published in BJHS, 2009. What Dr Smith did in that article is to introduce the Scientific Advisory Branch research which lies behind the Protect and Survive civil defence advice in Britain during the Cold War. I lived fairly near the National Archives (then the Public Records Office) in Kew, and spent some of my spare time from July 1990 onwards visiting to read the scientific research reports of the Home Office (HO catalogue class) and other departments like Aldermaston and Porton Down (e.g., AIR, ES and DEFE catalogue classes) as they were released under the 30 year rule. Lord William Penney, head of the Atomic Weapons Research Establishment (now the Atomic Weapons Establishment) in the 1950s, had I believe lost his wife in the Blitz and was extremely keen on civil defence research, so the whole point of the first British nuclear weapon test at Monte Bello in 1952 was to simulate a clandestine terrorist attack on Britain by a ship smuggled offshore inside the hull of a ship. That would get past air defenses and radar, and the radioactive base surge and contamination would cause immense radiation problems.

I was disappointed that Dr Smith did not go into that test, but she does go into the research done in Britain to produce nuclear weapons protection. As discussed in the last post, the Morrison shelter was developed because in November 1940 (at the height of the Blitz) the national Shelter Census showed that 60% of Londoners weren't sheltering at night due to the time taken to reach cold damp outdoor shelters. The blackout made travel at night dangerous, while the cold and damp made sheltering in usually flooded Anderson shelters (ground water innundated most of them) very uncomfortable, since London was bombed 57 consecutive nights in order for the Nazis to avoid the more efficient daytime air defences. Most people "ducked and covered" under tables indoors during the Blitz. The Ministry of Home Security investigated the casualty rates and discovered that many people survived inder strong tables when their homes were totally blown up, and therefore decided to design and issue half a million indoor Morrison shelters (Minister Sir John Anderson was replaced by Herbert Morrison when his shelter policy failed). The punch line is that the Morrison shelter was invented by Sir John Baker (shown proving the value of Morrison shelter when your house is blown up in the video below) and his assistant (both at university and in private practice) Edward Leader-Williams, and Leader-Williams in 1955 drafted the first government manual on how people can use such inner-refuge techniques against nuclear weapons effects! Hence Morrison shelter design which saved so many lives in World War II led to Protect and Survive.

I'm going to briefly extract below some of the key civil defence parts from the previous post (which is very lengthy since it analyses in depth how Reagan's strategy of "negotiation from a position of strength" led to pressure on Gorbachev which forced him to accept the INF treaty and also radical changes inside the Soviet Union, which precipitated the fall of communism; contrasted to Prime Minister Chamberlain's 1930s policy of "negotiation from a position of weakness" which led to worthless treaties with Hitler, who was encouraged to be as aggressive as he liked):




Above: the new Blitz-experience-based Shelter at Home handbook, published in June 1941, marked a shift of civil defence policy away from cold, damp, flooded outdoor shelters toward the more popular home Morrison protected bed shelter. The British Government under Prime Minister Chamberlain had failed to properly fund civil defence research against high explosives in good time before World War II, resulting in idealistic solutions which were not properly tested for practical effectiveness before being deployed in panic after the September 1938 Munich crisis (when Prime Minister Chamberlain was intimidated in his second meeting with Hitler). The panic civil defence countermeasures were outdoor trenches in public parks and the "Anderson" shelter, a corrugated steel arch buried in the ground and covered with earth.

Most of the Nazi bombing of Britain occurred during the Blitz (between 7 September 1940 and 10 May 1941), when the U.K. Government's Shelter Census of central London in November 1940 found that 60% of the public were sleeping in their own homes during air raids, instead of getting up and dressed to go to a shelter upon the attack warning siren. Only 4% used the Underground system shelters, 9% used other public air raid shelters, and 27% used domestic Anderson shelters (Morrison indoor shelters were not even introduced until March 1941). The 60% who did not go out to any kind of shelter during air raids:

"stayed in their homes, sleeping downstairs, under stairs, under tables, in cupboards."

(For this census, see the "Home Shelters" tab at the internet site linked here, but beware that it falsely states that Morrison shelters were available, which was completely incorrect in November 1940.)

"... distribution of 'Andersons' had begun before their testing had been completed. At the opening of 1939 'load tests' had shown that 'Andersons' were strong enough to bear the weight of any debris falling on them from the type of house for which they were intended. But it was not until some months later [Sectional Steel Shelters, Cmd. 6055, July 1939] that a series of 'explosion tests' proved conclusively [that they] could withstand without damage a 500 lb. [227 kg] high explosive bomb falling at least fifty feet away [equivalent to a 12 kt Hiroshima nuclear bomb some 50(12,000/0.227)^1/3 = 1,880 feet away: thus, Anderson shelters would have survived undamaged at ground zero after the air burst that high over Hiroshima] ... It was established at the same time that they would protect their occupants against blast from a bomb of this size bursting in the open at a distance of thirty feet or more. But this soundness of the 'Andersons' from a structural soundpoint, it soon became clear, was counterbalanced by an important practical defect, namely liability to flooding."

- Terence H. O'Brien, Civil Defence, H.M. Stationery Office, London, 1955, p. 196.

There were very good reasons for the failure of 60% of the public to utilize outdoor shelters: Britain has a cool, wet climate with a high water table, so any below ground structure rapidly became damp and cold during the winter, and flooded by rain. Before World War II it was believed that Nazi bombing would be in the daytime for reasons of accuracy, like World War I bombing. In fact, the Blitz was nighttime bombing, when people were trying to sleep, because anti-aircraft guns and fighter aircraft found it much harder to shoot down bombers in the dark at nighttime, despite searchlights and early radar sets. London was bombed 57 consecutive nights. Most people simply did not have time, upon hearing the air raid warning siren, to get dressed and go out to a cold, damp or flooded public or back yard Anderson shelter, which in winter were often dark to allow some people to try to sleep and uncomfortable compared to a home bed (London's underground rail communal shelters being an exception to the rule). Attempts to evacuate millions of women and children proved a failure, since most evacuees returned home after a few months of the outbreak of war, when the predicted air raids had still not occurred.

See Richard M. Titmuss, Problems of Social Policy, H.M. Stationery Office, London, 1950, online HTML version linked here, and Terence H. O'Brien, Civil Defence, H.M. Stationery Office, London, 1955, online PDF linked here. O'Brien at pp. 325-7 points out that the Government plan was to evacuate 4,000,000 women and kids before the outbreak of war in early September 1939, but unknown to the Government fewer than half of those decided to leave, and furthermore:

"By Christmas more than one-half of the 1,500,000 mothers and children concerned had returned home; in the London and Liverpool areas about two-thirds of the evacuated children had returned. (The first count taken in January 1940 disclosed that about 900,000 had returned.) ... this evacuation scheme had, as Mr Titmuss says, 'largely failed to achieve its object of removing for the duration of the war most of the mothers and children in the target areas'."

Consequently, there was a shift based on these experiences away from the large failures of the "outdoor" shelter and evacuation policy, towards providing better protection within the home itself. Sir John Fleetwood Baker and his assistant Edward Leader-Williams at the Ministry of Home Security developed an indoor shelter which could absorb the energy of the falling debris from the collapse of a normal house. There is a really brilliant scientific proof film (in the linked Cambridge University engineering faculty internet site, right click on the video link to choose to save the 7 MB mpg video file) showing precisely the mechanism by which the Morrison shelter deflects slightly in order to absorb the kinetic energy of a falling house by plastic deformation; Baker simply puts his own pocket watch inside a tiny model Morrison shelter within a model house, and then slams down a 10 pound load to represent the debris of the collapsing house, and his watch remains safe!

These Morrison table shelters were named after the Minister of Home Security (Herbert Morrison) and were introduced in March 1941. More than 500,000 were issued by November 1941, and they simply consisted of a strong dinner table containing a mattress for sleeping. They were 6' 6" long x 4' wide x 2' 9" high with a top consisting of 1/8" solid steel plate, with welded wire mesh sides and a metal lath floor. One wire side lifted up, allowing people to crawl inside the structure, where there was sleeping space for several people. These were placed in a ground floor (or basement) "refuge room", a technique revived for blast, thermal flash and fallout radiation shielding by the U.K. Government in its 1980 civil defence manual against nuclear attack, Protect and Survive. Edward Leader-Williams, assistant to Morrison shelter designer Sir John Baker during the experiments, worked in the U.K. Home Office Scientific Advisory Branch until 1965, and in 1955 initiated the basic Protect and Survive "inner refuge" research against nuclear war.

"In one examination of 44 severely damaged houses it was found that three people had been killed, 13 seriously injured, and 16 slightly injured out of a total of 136 people who had occupied Morrison shelters; thus 120 out of 136 escaped from severely bomb-damaged houses without serious injury. Furthermore it was discovered that the fatalities had occurred in a house which had suffered a direct hit, and some of the severely injured were in shelters sited incorrectly within the houses." - Wikipedia

The 22 May 1940 booklet Your Home as an Air Raid Shelter had already marked a change in policy as the discomfort and flooding of outdoor Anderson shelters became clear. As a result of the experience gained during the Blitz bombing, it was revised and greatly improved in June 1941 to create the new handbook (featuring the indoor Morrison shelter), Shelter at Home, which states:

“people have often been rescued from demolished houses because they had taken shelter under an ordinary table ... strong enough to bear the weight of the falling bedroom floor.”

The discovery of this table "duck and cover" effectiveness in air raids led to a revolutionary shelter design; the indoor Morrison table shelter of 1941. (For publication dates of these booklets, see T. H. O’Brien, Civil Defence, H.M. Stationery Office, 1955, pages 371 and 529.) It is the forerunner to the “inner core refuge” adopted for protection against thermal flash, blasted flying debris and fallout radiation in a nuclear war in the 1980 booklet Protect and Survive.


Above: the facts about the life-saving ability of the Morrison table shelter during aerial bombing in World War II Britain: it protects against the collapse of buildings regardless of whether that collapse is caused by TNT, a hurricane, an earthquake, or a nuclear bomb. A U.K. Government press release from November 1941, Morrison Shelters in Recent Air Raids, states:

“A report of Ministry of Home Security experts on 39 cases of bombing incidents in different parts of Britain covering all those for which full particulars are available in which Morrison shelters were involved shows how well they have stood up to severe tests of heavy bombing.

“All the incidents were serious. Many of the incidents involved direct hits on the houses concerned, a risk against which it was never claimed these shelters would afford protection. In all of them the houses in which shelters were placed were within the radius of damage by bombs; in 24 there was complete demolition of the house on the shelter.

“A hundred and nineteen people were sheltering in these ‘Morrisons’ and only four were killed. So that 115 out of 119 people were saved. Of these only 7 were seriously injured and 14 slightly injured while 94 escaped uninjured. The majority were able to leave their shelters unaided.”

The top set of instructions for building the Morrison shelter and using it as a table between air-raids are taken from the instruction manual for building the Morrison shelter, How to put up your Morrison “Table” Shelter, issued by the Ministry of Home Security, H.M. Stationery Office, March 1941 (National Archives document reference HO 186/580), which states:

“The walls of most houses give good shelter from blast and splinters from a bomb falling nearby. The bomb, however, may also bring down part of the house, and additional protection from the fall of walls, floors and ceilings is therefore very essential. This is what the indoor shelter has been designed to give. Where to put it up, which floor? Ground floor if you have no basement. Basement, if you have one. ... Protect windows of the shelter room with fabric netting or cellulose film stuck to the glass (as recommended in Your Home as an Air Raid Shelter). The sides of your table shelter will not keep out small glass splinters.”

According to the article, "Air Raid Precautions" in Nature, vol. 146, p. 125, 27 July 1940, "more than 700,000 copies" of Your Home as an Air Raid Shelter, were sold by the end of July 1940.

Your Home as an Air Raid Shelter

Above: The U.K. Government film, Your Home as an Air-Raid Shelter was issued in 1940 to accompany a manual of the same title, giving improved information based on bombing experience.

“The public outcry about conditions in the largest public shelters, often without sanitation or even lighting, and the appalling inadequacy of the over-loaded and ill-equipped rest centres for the bombed-out led to immediate improvements, but cost Sir John Anderson his job. ... His successor as Home Secretary, Herbert Morrison ...

“The growing reluctance of many people to go out of doors led the new Home Secretary to look again at the need for an indoor shelter… The result was the Morrison shelter, which resembled a large steel table … During the day it could be used as a table and at night it could, with a slight squeeze, accommodate two adults and two small children, lying down. The first were delivered in March 1941 and by the end of the war about 1,100,000 were in use, including a few two-tier models for larger families. Morrisons were supplied free to people earning up to £350 a year and were on sale at about £7 to people earning more. … the Morrison proved the most successful shelter of the war, particularly during the ‘hit and run’ and flying-bomb raids when a family had only a few seconds to get under cover. It was also a good deal easier to erect than an Anderson, and while most people remember their nights in the Anderson with horror, memories of the Morrison shelter are usually good-humoured.

“... A government leaflet, Shelter at Home, pointed out that ‘people have often been rescued from demolished houses because they had taken shelter under an ordinary table... strong enough to bear the weight of the falling bedroom floor’. I frequently worked beneath the solid oak tables in the school library during ‘imminent danger periods’ and, particularly before the arrival of the Morrison, families became accomplished at squeezing beneath the dining table during interrupted meals. ... Although the casualties were mercifully far fewer than expected, the damage to property was far greater. From September 1940 to May 1941 in London alone 1,150,000 houses were damaged ...”

- Norman Longmate, How we Lived Then - A history of everyday life during the Second World War, Pimlico, 1971.





Above: in the major British Cold War civil defence drive of 1981, the British Home Office shamelessly promoted the tried and tested World War II Morrison shelter as the "Type 2" in its 1981 manual, Domestic Nuclear Shelters, and included the World War II Anderson shelter which the Home Office had successfully tested at the first British nuclear weapon test, Hurricane in 1952 at Monte Bello (see U.K. National Archives document reference DEFE 16/933), as the "Type 3". Their "Type 1" was simply Cresson H. Kearny's earth and door covered trench shelter from the U.S. Oak Ridge National Laboratory reports on blast tests of the shelters, plus his Expedient Shelter Handbook and his Nuclear War Survival Skills, which in turn were based on American blast and fallout radiation shielding tests of the Soviet Union civil defense manual shelters. In part, therefore, Britain's ability to credibly site cruise missiles and neutron weapons in the 1980s against the Soviet expansionism and massive SS-20 threat, was derived from translations and tests of research published in Russian in the Brezhnev era Soviet civil defense manual! Kearny in 1987 improved the improvised fallout radiation "core shelters" by developing the method of stacking water-filled plastic bag lined boxes on a table to provide shielding where earth or sand was not available.

For data on tests of successful improvised shelters against chemical weapons, see the 2001 Oak Ridge National Laboratory report, ORNL/TM-2001/154, linked here; similar chemical warfare protective actions to those described and defended empirically in this report were to be included in a revised edition of Protect and Survive in the late 1980s which was never published after the Cold War ended. Note that the idea of sealing up an inner refuge room with duct tape against poison gas - based on 1930s Porton Down experiments - dates right back to the 1938 British Government civil defence handbook distributed to all 14,000,000 households after the Munich crisis in September 1938, The Protection of Your Home Against Air-Raids.

Update: my shock wave proof paper is now online at http://vixra.org/abs/1003.0259