Classical Physics


Analytical Proof of the Taylor Equation Including Taylor's Constant Sγ Which Previously Required Numerical Integration, with Applications

Authors: Nigel B. Cook

British mathematician Sir Geoffrey I. Taylor in secret work for British civil defence in 1941 (declassified in 1950 and published in the Proceedings of the Royal Society, vol. 201A, pp. 159-186), derived the strong shock solution equation, namely distance, (equation) , where (equation) is the ambient (pre-shock) atmospheric density, t is time after explosion, E is the energy released and Sg is Taylor's calculated function of g, requiring a complex step-wise numerical integration. We present a proof of the equation (equation), implying that Taylor's so-called constant (equation), not requiring any complex integration. This is useful for close-in shock waves from nuclear explosions and supernovae explosions. We further obtain the general arrival time of the shock wave (equation), by noting two asymptotic solutions; namely, at very great distances, the blast decays into a sound wave so the arrival time t approaches the ratio of distance to sound velocity (equation), while at very close-in distances the strong shock equation previously derived becomes accurate, and there is also an easily included effect at intermediate distances due to the expansion of the hot air in reducing shock front arrival times. The errors of method made by Taylor for nuclear test explosions in air were also made by Russian mathematician Leonid I. Sedov who applied similar cumbersome numerical integrations in a 1946 paper (published in the Journal of Applied Mathematics and Mechanics, vol. 10, pp. 241-50).

Comments: 3 pages, see paper for equations in abstract

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Submission history

[v1] 28 Mar 2010

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