366 BELL SYSTEM TECHNICAL JOURNAL 



a procedure which may be successful at intermediate separations becomes 

 inadequate. Relativistic mechanics breaks down and Lorentz invariance 

 may lose its significance. This is in agreement with the experimental result 

 that, in some nuclear reactions, the energy balance, as calculated from 

 the relativistic relations, is not satisfied. Also the difficulty which has been 

 encountered in calculating nuclear phenomena by the techniques of wave 

 mechanics suggests that the extremely non-linear condition is approached 

 for the separation of the particles within a nucleus. This viewpoint suggests 

 that an understanding of the nucleus might make possible an experimental 

 determination of velocity relative to the ether. 



The reactions between wave patterns of appreciable amplitude may also 

 be viewed from a somewhat different angle. We may think of the various 

 wave patterns as being the analogs of the various modes of motion of, say, 

 an elastic plate. For very small amplitudes they have negligible effect on 

 one another. For larger amplitudes, where Hooke's law does not hold, the 

 force may be represented as a power series of the displacement. The first 

 power term represents the linear stiffness. If the frequencies of two modes 

 which are in oscillation are wi and 0)2 , the higher power terms represent 

 forces of frequencies ;;zcoi ± iicoo where m and n are integers or zero. These 

 forces set all the modes into forced oscillation at the frequencies of the 

 various forces, in amounts which depend on the impedance of the particular 

 mode for the particular frequency. When the frequency of the force coin- 

 cides with the resonant frequency of one of the natural modes, the forced 

 oscillations may be large. Thus the variation in stiffness with displacement 

 provides a coupling whereby energy may be transferred from one or more 

 modes, that is wave patterns, to other modes. But in this transfer the energy 

 always appears associated with a new frequency which is related to those of 

 the modes from which it came in accorance with the familiar formula of 

 quantum theory. 



The theory of such energy transformations with change of frequenc}^ has 

 been worked out in considerable detail for vacuum tube and other variable 

 resistance modulators, and the results show little in common with the quan- 

 tum theory beyond the relations connecting the frequencies. Wlien, however, 

 the variation is not in a resistance but in a stiffness, as occurs in the ether 

 case, the situation is cjuite different. This problem has been explored both 

 theoretically" and experimentally. It is found tliat an oscillation of one 

 frequency in one mode may provide the energy to support sustamed oscil- 

 lations of two other lower frequencies in two other dissipative modes. For 

 this to occur the frequencies involved must be related through the quantum 

 formula. Also the amplitude of the generating oscillation must exceed a 



'■• R. V. L. Harllcv, Bell Svs. Tech. Jour., 15, 424, 1936. 



"' L. W. Husscy :in<i !-. R. Wralliall, Bell Sys. Tech. Jour., 15, 441, 1936. 



