## Analytical Proof of the Taylor Equation Including Taylor's Constant S_{γ} 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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