G58 REPORT— 1900. 



beam than tliere -was in tlie incident beam, altliougli its leugtli is shorter on 

 account of the Dopplev effect. This requires that the' undulations must oppose a 

 resistance to the advancing wall, and that the mechanical work required to push 

 on the wall is directly transformed into undulatory energfy. In fact, let us con- 

 sider the mechanism of the reflexion. Suppose the displacement in a directly 

 incident wave-train, with velocity of propagation c, to be ^ = a cos {mx - met) ; that 

 in the reflected train will be ^' = «'cos {}n'x + m'ci), where a', vi are deter- 

 termined by the condition that the total displacement is annulled at the advanc- 

 ing reflector, because no disturbance penetrates beyond it ; therefore when ,r= vt,- 



where v is its velocity, | + ^' = o. Thus we must have a = - a, and m' = m ^ ~ ", in 



c + v 

 agreement with the usual statement of the Doppler effect when v is small compared 

 with c. Observe, in fact, that the direct and reflected wave-trains have a system 

 of nodes which travel with velocity v, and that the moving reflector coincides 

 with one of them. Now the velocities d^ldt and d^'jdf, in these two trains are not 

 equal. Their mean squares, on which the kinetic energy per unit length depends, 

 are as m^ to m'-. The potential energies per unit length depend on the means of 

 (d^ldx)'- and (d^'\dxY, and are of course in the same ratio. Thus the energies 

 per unit length in the direct and reflected trains are as m" to m'-, while the lengths 

 of the trains are as m' to m ; hence their total energies are as m to 711' ; in other 

 words the reflected train has received an accretion of energy equal to 1 —7)i'jm of 

 the incident energy, which can only have come from mechanical work spent in push- 

 ing on the reflector with its velocity v. The opposing pressure is thus in numerical 



(wi'x c 

 1 1- of the density of the incident energ-y, which 

 m / « -^ °-" 



(*- 7j- 



works out to be — , — ;, of the intensity of the total undulatory energy, direct and 



C T V~ 



reflected, that is in front of the reflector. 



"When V is small compared with c, this agrees with Maxwell's law for the pressure 

 of radiation. This case is also theoretically interesting, because in the application 

 to ajther-waves | is the displacement of the njther elements whoso velocity d^jdt 

 represents the magnetic force ; so that here we have an actual case in which this 

 vector ^, hitherto introduced only in the tlieoretical dynamics of electron-tlieory, 

 is essential to a bare statement of the i'acts. Another remark here arises. It has 

 been held that a beam of light is an irreversible agent, because the radiant pres- 

 sure at the front of the beam has nothing to work an-ainst, and its worlc is there- 

 fore degraded. But suppose it had a reflector moving with its own velocity c to 

 work against ; our result shows that the pressure vanishes and no work is done. 

 Thus that objection to the thermodynamic treatment of a single ray is not well 

 founded. 



This generalisation of the theory of radiant pressure to all kinds of undulatory 

 motion is based on the conserv.ation of the enei'gy. It remains to consider the 

 mechanical origin of the pressure. In the special case of an unlimited stretched 

 cord carrying transverse waves the advancing reflector may be a lamina, through 

 a small hole in which the cord passes without friction : the cord is straight on one 

 side of the lamina, and inclined on the other side on account of the vibration ; and 

 it is easil}"^ shown that the resultant of the tensions on the two sides provides a 

 force acting on the lamina which, when averaged, agrees with the general formula. 

 In the case of an extended medium with advancing transverse waves, which are 

 reflected directly, the origin of the pressure is not so obvious, because there is not 

 an obvious mechanism for a reflector which would sweep the waves in front of it 

 as it advances. In the rethereal ease we can, however, on the basis of electron- 

 theor}', imagine a constitution for a reflector which will turn back the radiation on 

 the same principle as a metallic mirror totally reflects Hertzian waves, and thus 

 obtain an idea of how the force acts. 



The case of direct incidence has here been treated for simplicity ; that of oblique 

 incidence easily follows ; the expression for tlie pressure is reduced in the ratio of 

 tlie square of the cosine of the angle of incidence. If we averago up, after 

 Boltzmaun, for the natural radiation in an enclosure, which is incident equally at 



