Industrial Research 



281 



with respect to which wind tunnel experimentation is clifTieult 

 and flight testing very dangerous. It has been the subject of a 

 number of mathematical investigations, the results of which 

 have reached a sufficiently advanced stage that they are now 

 being used to predict the critical speeds and flutter frequencies 

 of aircraft while still in the design stage. Even more important, 

 the mathematical investigation of this problem jioints the way 

 to modifications of design which will insure that lluttor cannot 

 occur in the usable speed range. 



Telephony provides a second example: 



(to) The equipment in an automatic telephone exchange must 

 be capable of connecting any calling subscriber with any called 

 subscriber. It consists of several stages of switches, each of 

 which can be caused to make connection with a number of trunks 

 which lead in turn to switches in the next succeeding stage. 

 Enough switches must be provided so that only a very small 

 proportion of subscribers' calls will fail to be served immediately. 

 Since the demands made by the subscribers fluctuate from 

 moment to moment, the number of switches required depends 

 in part upon the height to which the crests occasionally rise in 

 this fluctuating load. It is also influenced, however, by the way 

 the trunks are arranged, by the order in which the switches 

 choose them, and by many other factors. Experimental ap- 

 praisal of the effect of these various factors is impossible, both 

 because it would be very costly, and because it would be exceed- 

 ingly slow. Mathematically, however, they have been studied 

 by the theory of a priori probability,' which is used not only in 

 determining how much apparatus to install in a working exchange, 

 but also in comparing the relative merits of alternative arrange- 

 ments while in the development stage. 



Sixth: Mathematics is frequently useful in devising 

 so-called crucial experiments to distinguish once for all 

 between rival theories. A famous example in the field 

 of physics was the study of the refraction of starlight 

 near the sun's disk, which afforded a means of deciding 

 between Newtonian and relativistic mechanics. In 

 this case, mathematical investigation showed that the 

 result to be expected was different according to the 

 two theories, and astronomical observations confirmed 

 the prediction of relativistic mechanics. In the indus- 

 trial field, an example of this kind comes to me from 

 Dr. Joseph A. Sharpe, Chief Physicist in the Geophysi- 

 cal Laboratory of the Stanolind Oil and Gas Companj': 



(n) As an example of the second sort of use of analysis there is 

 the case of our study of "ground-roll," the large amplitude, 

 low frequency surficial wave which caused so much grief in the 

 early days of seismic reflection prospecting when filters were not 

 used as extensively as at present. We hope to use our study of 

 this wave motion as an aid to a better understanding of the 

 properties of the surficial layers of soil and their effects on the 

 reflected waves in which we are primarily interested. 



Two views on the ground-roll are current, although neither is 

 based on very much observation, and this of an uncontrolled 

 sort. One view states that the ground-roll is an elastic wave. 

 Analysis predicts that this wave w ill have a certain velocity in 

 relation to the velocities of other waves, that it will have a certain 

 direction of particle motion and relation of maximum horizontal 

 to maximum vertical component of displacement, that it will 

 attenuate with distance according to a certain law, that it will 



' Nol statistics, which is a pofteriori probability. This is one of the few cases in 

 industry where the a priori theory finds application. 



attenuate with depth in a certain way, and that its velocity will 

 follow a certain dispersion law. The second view maintains 

 that the "ground-roll" is a wave in a viscous fluid, and analysis 

 predicts a behavior which is similar in certain cases, and different 

 in others, to that of the elastic wave. Having the predictions of 

 the analysis at hand, we are enabled to devise a group of obser- 

 vations, and the special equipment for their i)rosocution, which 

 will provide crucial tests of the two hypotheses. 



Seventh: Mathematics also frequently performs a 

 negative service, but one which is sometLnics of very 

 great importance, in forestalling the search for the 

 impossible; for many desirable objectives in industry 

 are as unattainable as perpetual motion machines, and 

 frequently the only way to recognize the fact is by 

 means of mathematical argument. 



(o) A certain type of electric wave filter which is usually 

 referred to as an "ideal" filter would be very useful if it could be 

 produced. However, it has been shown mathematically that 

 such a structure would respond to a signal before the signal 

 reached it; in other words, that it would have the gift of prophecy. 

 Since this is absurd, it follows that no such filter can be built, 

 and consequently no one tries to build it. 



Still another example from the field of communica- 

 tion deals with the design of feedback amplifiers. 



(p) In practice, any amplifier is intended to handle signals in 

 a given frequency band. For various reasons, it is preferable not 

 to have it amplify disturbances outside this band, and hence its 

 gain characteristic is made to drop off as rapidly as possible out- 

 side the limits of the useful band. It has been shown theoreti- 

 cally, however, that the gain cannot decrease at more than a 

 certain rate, which can easily be computed, without causing the 

 amplifier to become unstable. As a matter of fact, the allowable 

 rate at which the gain may fall is often surprisingly low, and a 

 great deal of design effort would be wasted in the attempt to 

 obtain an impossible degree of discrimination if the theoretical 

 limitations were unknown. 



Eighth: Finally, mathematics frequently plays an 

 important part in reducing complicated theoretical 

 results and complicated methods of calculation to 

 readily available working form. So many and so varied 

 are the services falling in this category that it is diffi- 

 cult to illustrate them by means of examples. We 

 arbitrarily restrict ourselves to two, chosen primarily 

 for the sake of variety. The first comes from Mr. 

 Hibbard : 



(?) In aircraft design the metal skin, though thin, contributes 

 a larger part of the structural strength. Nevertheless, such thin 

 metallic plates will buckle or wrinkle after a certain critical load is 

 exceeded. Beyond this point the usual structural theories can- 

 liot be applied directly, and it is therefore necessary to introduce 

 new methods of attack to predict the ultimate strength of the 

 structure. These stiffened plates are difficult to deal with theo- 

 retically, but by interpreting the effect of the stiffeners as equiva- 

 lent to an increase in plate thickness or a decrease in plate width, 

 the calculations can be brought within useful bounds. 



The reduction of electric transducers to equivalent 

 T or n configurations, the interpretation of the elastic 

 reaction of air upon a microphone as equivalent to an 

 increase in the mass of its diaphragm, the postulation of 



