Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/194

follows. The object's kinetic energy is equated to the work required for lifting the gravity center of the object to a height proportional to the size of the object,

$E\sim {\frac {K^{2}}{M}}\sim MgR.$

(XXII-2)

Into $K$ of Eq. (XXII-1) we substitute the expression of the object's mass $M$ by the characteristic dimension $R$ and the object's density $\varphi$, and get

$M\sim \rho R^{3};\quad {\frac {p_{0}^{2}}{c_{0}^{2}}}{\frac {(R^{3})^{2}}{\rho R^{3}}}\sim \rho R^{3}gR;\quad {\frac {c_{0}^{2}}{p_{0}^{2}}}\rho ^{2}gR={\text{idem}}.$

(XXII-3)

The sign idem adopted in similarity theory signifies that similarity will take place if the term on the left remains constant. For all explosions in the air under normal conditions $c_{0}=$const, $p_{0}=$const, the criterion is simplified and $\rho ^{2}gR=$idem.

This criterion also includes exact simulation - the change in acceleration $g$ is inversely proportional to the size $R$ (centrifugal simulation). But on the basis of the approximations made earlier we obtained a criterion which also admits another solution: the change in density is inversely proportional to the root of its dimensions. This method was proposed by Khariton and this author [105]. This method allows for a sufficiently wide change in the scale. By substituting a material with density 2 (stone) with a material with density 11 (lead) it becomes possible to reduce $R$ by a factor of 30, which corresponds to the reduction of the charge by a factor of 27,000, i.e., it is possible to simulate the explosion of 1 ton of explosive by the explosion of 40 g of the same substance.

In Khariton's many experiments, the edgewise standing bricks turned out to be convenient indices for the distance at which the momentum of a shock wave drops to a specific value.

It is obvious that centrifugal simulation is necessary in more complex cases in which, along with a rigid structure, the soil also plays a role. The approximate simulation by changing density, as proposed by Khariton and this author, is considerably narrower in scope and the advantage of this method is only the simplicity of experimentation.