on which depends the absence of similarity in the case of a change in scale, differs appreciably from unity. Thus, if we write this criterion as a ratio of characteristic length to the size of the charge , then for a charge weighing 1 kg we get . The ratio of static soil pressure to blast pressure yields, at a crater depth of several meters, a quantity of the order of -. Thus, the criterion in the most varied formulations turns out to be sharply different from unity. This means that we are dealing here with the case in which not all the quantities are of the same order. It is obvious that we find ourselves in the domain of extreme or critical laws, in a domain, that is, in which we may expect self-simulation in the same way as self-simulation arises in hydrodynamics at very high or very low, Reynolds Numbers.
We now have to find the physical nature of this self-simulation.
Let us give a closer look to the toppling of a wall (see Fig. 56). At the beginning of the preceding chapter we brought it up as an instance for a process which lasts considerably longer than the action of the shock wave (in this case the time ratio yields another criterion which sharply differs from unity), i.e., a process in which the decisive role is played by the general wave momentum. We divided the process into two stages: 1) the action of the wave on the object which determines its momentum, and 2) the motion of the object by inertia, which overcomes the force of gravity, and we readily find the conditions for similarity.
In fact, the object's momentum , equal to the force momentum, (for a geometrically similar change of the system, in which the dimensions of the object and the distance between the object and the charge change proportionally to the dimension of charge ) is proportional to
| (XXII-1) |
where is the area on which the wave acts, is the pressure momentum per unit of surface. The momentum of the object sufficient for its toppling will be determined as
