Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/191

By reducing the dimensions of the charge, the beam and the distance between them by a factor of $n$, the oscillation period of the beam will decrease by a factor of $n$, and the frequency will increase by a factor of $n$. This can readily be verified with the aid of elasticity theory for any specific practical method of securing the beam.

The mass has decreased by a factor of $u^{3}$, at a similar point the shock wave momentum per unit of surface has decreased by a factor of $n$ on account of a decrease of the shock wave width and a decrease in shock wave action time at a constant peak pressure, and the surface receiving the pressure has decreased by a factor of $n^{2}$. Thus linear velocity reached by the beam as a result of the effect of pressure momentum will be independent of the size of the beam. The amplitude of the oscillations will be of the order of the product of velocity $\times$ period, i.e., it will be proportional to the size of the beam. Hence we see that the relative deformation and density of elastic energy proportional to the square of initial velocity are identical in the model and in actuality. The result will also be identical, namely, the presence or the absence of destruction. Let us note that similarity will not be violated by friction which depends on velocity and on the load in the case when the load is also assigned in the fundamental shock wave action, since velocity and pressure are the same in similar systems.

A less trivial case is the one frequently encountered in structural mechanics. It is the case in which the stability of the structure and the effort required for its destruction depend on the structure's weight. The simplest instance of this kind is the sandy area without cohesion. Another instance is a stack of bricks, the solidity of which depends on the weight of the bricks and on the friction produced by the pressure of one brick on the other. Khariton emphasizes that such a type of stability very frequently determines the resistance of a structure to destruction. The stack of bricks represents one extreme example in which the weight determines internal cohesion. A solid steel box, which is easier to topple as a whole than to destroy, is another extreme example in which the explosion works against the force of gravity.