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April, 1930] DIMENSIONAL ANALYSIS 381
so that
Kd = L*T = Z//2.
It may be expected, then, that a body moving through a dense medium will be affected in a natural way on the screen only when the damping constant of the system is reduced in proportion to the five-halves power of the length reduction factor. It should be noted, however, that in reducing the length dimension by L, we have reduced the area of the model by L2 and hence the liquid itself need have its damping constant with respect to any moving system reduced by only Z//2, for the damping constant of any system in general is proportional to the area of the moving member.
I would like to indicate at this point that the placing of waves in the same category as falling bodies, as was done in the paper by Mr. Ball, is not justified. So far as spray and other falling particles of water are concerned, the parabolic relation of length and time is probably justified, but in the case of the waves themselves, the theory of falling bodies no longer holds. In this case the velocity of the waves is constant, and the length-time relation is a linear one. I will illustrate by an example. Suppose we have a pool of water 100 feet in length, and suppose we drop a pebble in at one end. It will take t seconds for the ripples, or waves, to reach the other end. Now suppose it is desired to make a picture of such an action in miniature and choose a pool only ten feet long. It will now actually take one tenth of t seconds for the waves to traverse the pool, so in order that it appear as though they traversed it in t seconds, it will be necessary to crank ten times as fast, so that the time ratio is the same as the length ratio. A failure to realize this fact probably accounts for some unconvincing ocean scenes which we have seen done in miniature.
The problem which now presents itself is, how are we to picture in miniature a ship blowing up in a rough sea. If we scale our time intervals to accommodate the falling debris, we have not timed properly for a good illusion in the case of the waves. It may, however, be shown that the depth of the water used is a factor in the determination of the velocity of propagation of waves, and since our model ship is probably floating on a model ocean it is not improbable that we may so design the tank as to give the desired effect. The problem of surface wave motion is not readily adaptable to simple and direct mathematics, and will not be treated here. This