356 BELL SYSTEM TECHNICAL JOCRXAL 



tions which increase with decreasing radius. These reflections will effectively 

 take the place of the assumed reacti\'e impedance of the generator, and so 

 the latter may be discarded. The fact that the retlections take place from 

 a somewhat diffuse inner boundary prevents the amplitude from building 

 up to an infinite value at the center as it would with a linear medium. 



However, the reflected wave includes components of triple and higher 

 frequencies and, due to the non-linearity, other frequency components will 

 be generated. If the entire pattern is to be stable, all of these must satisfy 

 the boundary conditions. Their magnitudes relative to the fundamental, for 

 a particular mode of oscillation, will depend on the amplitude and fre- 

 quency of the fundamental, as well as on the constants of the medium. 

 Hence the amplitude as well as the frequency of a stable pattern of a par- 

 ticular mode should be uniquely determined. Particles of different prop- 

 erties would then be expected to consist of patterns involving different 

 modes of oscillation. 



Returning to the lack of complete reflection at the outer boundary and 

 the change it might be expected to make in the pattern with time, this 

 might be an important factor for a single particle alone in the universe. 

 Actually, however, a very large number of particles are present. If we con- 

 sider a point at a considerable distance from any one particle, a point in a 

 vacuum, the resultant of the disturbances produced there by all the patterns 

 will be very large compared with that due to any one. But the effect on a 

 particular pattern of its own loss by radiation will be determined by this 

 small component, and so will be small compared with the effect exerted on 

 it by the combined small fields of its neighbors. This combined field due 

 to a large number of patterns, randomly placed, and moving at random, will 

 constitute a randomly varying electromagnetic field in a vacuum, such as 

 has recently been postulated for other reasons. If, now, the center of a 

 pattern be placed at the point in question, this random field may occasion- 

 ally take on so large a value as to disturb the equilibrium conditions of 

 the pattern. 



It may be argued that, in spite of the merging of a given pattern in that 

 of the random group, the group as a whole will suffer a progressive loss of 

 energy through incomplete reflection. Were this to occur the total loss of 

 energy would not be evenly distributed among the j)articles. As discussed 

 below the particles would exchange energy through the mechanism of the 

 non-linearities, continually forming less stable grouj) i)at terns of greater 

 energy, which in turn suffer transitions to more stable patterns of lower 

 energy. A small continuous decrease in total energy would manifest itself 

 as an increase in the rate of transitions downward in energy comj^ared to 

 those upward. 



Associated with a standing wave pattern such as that described above 



