Waves at the Surface of a Moving Mirror. 511 



-considerations, it is easily proved that H = E/c, so that the 

 energy per unit volume is equal to 2E 2 /87rc 2 . If the waves 

 are sinusoidal, o£ amplitude a, the average value of E 2 over 

 a wave-length is equal to a 2 /2; so that the average energy 

 per unit volume due to a train of plane electromagnetic 

 waves is independent of the wave-length and frequency of 

 the waves, and depends only on their amplitude, its value 

 being a 2 /87rc 2 . This result admits, 1 imagine, of no dispute; 

 then, if we accept the law that light exerts a pressure on a 

 mirror from which it is reflected, and therefore that work is 

 done in moving the mirror, it follows that the amplitude 

 must be altered by reflexion from a moving mirror. It is 

 -easily proved that this alteration of amplitude is produced, if 

 we rigorously apply the principles of the electromagnetic 

 theory to the problem in hand. 



4. To fix our ideas, let it be supposed that a perfectly 

 reflecting (or perfectly conducting) plane mirror moves from 

 West to East with a velocity % the surface of the mirror 

 remaining parallel to the meridian. Let x be measured from 

 a fixed point, in the direction from West to East, and let time 

 be measured from the instant when the mirror passes through 

 this fixed point, so that at the time t the position of the 

 mirror is given by x=vt. Let the incident plane waves 

 travel from East to West, their electric fields being vertical. 

 Let Ei represent the variable force that would be exerted on 

 -a stationary electric charge of unit magnitude by the incident 

 waves, the positive direction of E x being vertically upwards. 

 Then the equation of the incident wave-train may be written 



E 2 = a cos m (x + ct), (1) 



where a is the amplitude, while m = 2ir\ and \ denotes the 

 wave-length of the incident waves, and c is the velocity of 

 light. Let the equation to the reflected wave-train be 



E, = a' cos m' (x — ct), (2) 



where a' denotes the amplitude, while m' = 2ir/\' } and X' 

 denotes the wave-length of the reflected waves. The positive 

 direction of E 2 is also measured vertically upwards. 



In any plane parallel to the meridian, and at a distance x 

 from the origin, the force (E) that would be exerted on a 

 stationary unit positive charge is given by 



E=E 1 +E 2 =acos m (x + ct) +a'cos m' (x — ct). 



But if the charge is moving, the force that will be exerted on 

 it by the waves is no longer represented by this equation. 



