MATTER, A MODE OF MOTION 353 



These equations become identical with those of Maxwell for free space, 



- T - 1 



if we replace q by £, -• by H, po by f and - by /x. Then p^q corresponds to 



D and — 2</j to B where ^ is the angular displacement of an element of the 

 medium. Or the roles of the electric and magnetic quantities may be inter- 

 changed. 



For present purposes, however, we are more interested in finite displace- 

 ments. The relations which then apply are discussed in detail in the com- 

 panion proper. It is there shown that changes of two kinds appear in (1) 

 and (2), with corresponding changes in the transmission properties of the 

 medium. The simple linear relations are to be replaced by non-linear ones, 

 which cause distortion of a wave but no reflection. In addition, a qualitative 

 difference appears in the nature of the elasticity, as was pointed out by 

 Kelvin. The restoring torque is no longer proportional to the angular dis- 

 placement alone. When the axis of a gyrostat is displaced it begins rotating 

 toward the axis of the displacement, thereby decreasing the component of 

 its spin which is normal to that axis. Thus the restoring torque for a con- 

 stant angular displacement decreases with time. The restoring torque is 

 therefore a function of the time as well as of the displacement. Because of 

 this time dependence, a disturbance of finite amplitude generates waves 

 which propagate both backward and forward. 



Vox a plane progressive sine wave it is found that the reflected waves 

 interfere destructively. However, if a central generator starts sending out 

 a diverging sinusoidal disturbance, a part of the energy is reflected inward 

 as a wave of the same frequency as the generator and another smaller part 

 as waves the frequencies of which are odd multiples of that frequency. This 

 reflection attenuates the outgoing wave. If the incoming wave is reflected 

 rather than absorbed at the generator, it tends to set up a standing wave 

 pattern. As time goes on, the impedance of the medium as seen from the 

 generator becomes more reactive and less power is drawn from the generator. 

 Due to the attenuation, the energy in spherical shells of a given thickness 

 decreases with increasing radius, so that it and the power transmitted at the 

 wave front approach zero as /' approaches infinity. This falling off is some- 

 what similar to that suffered by a wave the frequency of which lies in the 

 stop band of a filter, but with one important difference. There the attenua- 

 tion is independent of the distance. But here, since the attenuation is a 



