380 BELL SYSTEM TECHNICAL JOURNAL 



correspondingly attenuated. The arrival of the reflected waves at the gen- 

 erator adds a reactive component to the impedance of the medium, as seen 

 from the generator, which reduces the power delivered to the medium. 

 Meanwhile energy is being stored as standing waves in the medium and 

 the rate of flow of energy in the wavefront is decreasing. The energy in 

 successive shells of equal radial thickness decreases with increasing r, in- 

 stead of being uniform as it would be in the absence of reflection. In the 

 limit it approaches zero, but as the rate of decrease depends on the curva- 

 ture, the rate of approach also approaches zero. As the rate at which energy 

 is stored and that at which it is carried outward at the wavefront both 

 approach zero, the resistance which the medium offers to the generator 

 approaches zero, and its impedance approaches a pure reactance. 



The total energy stored in the medium depends on how the over-all at- 

 tenuation of the main wave is related to its amplitude. If there were no 

 attenuation, the impedance would remain a pure resistance, the energy in 

 successive shells would all be the same, and the total energy would increase 

 linearly with r, and so with the time, and approach infinity. If the attenua- 

 tion were independent of r, the total energy would approach a finite value. 

 The present case is intermediate between these, the attenuation being finite 

 but approaching zero with increasing r. If we assume it to vary as some 

 power of the amplitude of the velocity, then W. R. Bennett has shown that 

 if this power is less than the first the total energy approaches a finite value. 

 If it is equal to the first, the energy approaches infinity as log r, and if it is 

 greater than this, the power approaches infinity more rapidly. Until more is 

 known as to the actual variation of amplitude with distance, nothing 

 definite can be said about the limit of the total energy. 



APPENDIX: EQUATIONS OF THE KELVIN ETHER 



We are concerned with the wave properties of the model for wavelengths 

 long enough compared with the lattice constant so that it may be regarded 

 as a continuous medium. Its density is equal to the average mass of the 

 gyrostats per unit volume. Its elastic properties are to be derived from the 

 resultant of the responses of the individual gyrostats. 



We shall therefore begin by considering the behavior of a single element, 

 which is shown schematically in Fig. 1. Here the outer ring of the gimbal, 

 which is rigidly connected with the lattice, lies in the .v y plane. The axis 

 about which the inner ring rotates is in the .v direction, and the spin axis C 

 of the rotor is in the 2 direction. We wish to examine the effect of a small 

 angular displacement <p of the lattice, that is, of the outer ring. If it is about 

 X or z, it will, because of the frictionless bearings, make no change in the 

 rotor. If it is about y it will produce an equal displacement of the spin axis 



