given 



Waves at the Surface of a Moving Mirror. 509 



disturbance does not travel into the reflector, and must 

 therefore be annulled at its surface ; thus when w = vt, we 

 must have f+£' = identically. This gives a'=—a, and 

 m! (c — v) = m (c + v). The amplitude of the reflected dis- 

 turbance is therefore equal to that of the incident one; while 



the wave-length is altered in the ratio , which is approxi- 



2v . c+ . v . 



mately 1 -, where v/c is small, and is thus in agreement 



c 



with the usual statement of the Doppler effect. The energy 

 in the wave-train being half potential and half kinetic, it is 



by the integration of pi j~ J along the train, where p 



represents density. In the reflected train it is therefore 

 augmented, when equal lengths are compared, in the ratio 



i ) ; but the length of the train is diminished by the 



V*-*/ . . <—v 



reflection in the ratio ; hence on the whole the energy 



c + v ^ J 



transmitted per unit time is increased by the reflection in 



the ratio - — -. This increase per unit time can arise only 

 c — v 



from work done by the advancing reflector against pressure 



exerted by radiation. That pressure, per unit surface, must 



therefore be equal to the fraction of the energy in a 



1 c — v 



length (c + v) of the incident wave-train; thus it is the 



*> 2 



p- 1 j)" 



fraction -s s of the total density of energy in front of the 



c 2 -{-v 2 J aJ 



reflector, belonging to both incident and reflected trains. 

 When v is small compared with c, this makes the pressure 

 equal to the density of vibrational energy, in accordance 

 with Maxwell's electro-dynamic formula (Elect, and Mag., 

 1871). 



" In the case of light waves we can, however, imagine an 

 ideal material body constituted of very small molecules, that 

 would sweep them in front of it with the same perfection as 

 a metallic mirror actually reflects the longer Hertzian waves. 

 The pressure will then be identified physically, as in the case 

 of the latter waves, with the mechanical forces acting on 

 the screening oscillatory electric current-sheet which is in- 

 duced on the surface of the reflector. The displacement 

 represented above by J, which is annulled at the reflector, 



