# What is the exact value of the constant in the similarity solution for blast waves?

Recently I used the Rankine-Hugoniot equation to reason that the limit of the speed of shock waves for extremely strong shocks is $\bigl(\frac{6P}{5r_0}\bigr)^{1/2}$ where $P$ is the pressure of the shock and $r_0$ the regular density of the air. But, in contrast, I used the mathematical form of the similarity solution to deduce that the speed of the shock is $\frac{2}{5}\bigl(\frac{P}{r_0}\bigr)^{0.5}$. Why the inconsistency? And what is the real value of the constant?

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• Can you explain which Rankine-Hugoniot equation and which similarity solution you used? – user10851 Mar 16 '16 at 19:41
• I used all of the Rankine-Hugoniot equations to deduce a familiar fornula for the ratio of densites (behind and ahead of the shock) as a function of the pressures ratio (see wikipedia article). According to this formula, r_2/r_1 approaches (\gamma + 1) /(\gamma - 1) as P_2/P_1 goes to infinity. For air \gamma = 7/5 and that gives the results r_2/r_1 = (12/5)/(2/5) = 6 and u_s = ((6P)/(5r_1))^(1/2). As to your second question, i used the similarity solution R(t) = C* (Et^2/r_0)^(1/5). – user2554 Mar 17 '16 at 11:25
• – honeste_vivere Mar 17 '16 at 12:56
• I saw this question already, but i didnt find there the result for the constant C. Maybe it can be implied from the answer there by reading "beneath the lines" but i didnt succeed in doing so. So my question is simple- what is the value of C when written as numerical value?? If you know the answer please simply write it as a number. – user2554 Mar 17 '16 at 15:00
• @user2554 - That constant is a scaling factor that depends upon a specific problem. It is not something you find analytically, you find it numerically. – honeste_vivere Mar 23 '16 at 13:00