Waves at the Surface of a Moving Mirror. 523 



Writing vt for x in (13) and (14), and substituting the 

 values of E x and E 2 in (38), we obtain 



a (c cos 6 + v) cos m{c + v cos 0)t 4- y sin 0} 

 + a'(c cos O'—v) cos m{ — c+« cos d')t+y sin 0'} =0. (39) 

 This equation is satisfied for all values of y and t if 

 a(c cos + v) +a'(ccos O'—v) = 0, 

 m(c + v cos 0)=?n'(c— vcos #'), 

 ?7i sin 6= —m' sin 0'. 



(40) 



Then c + v cos 



c — V cos 0' 



sin0 



sin0' 



which leads to the condition 





tan 5- — 



6—1' 



• tan - 



c + v 2 " 



just as in §8. The alteration in the wave-length is identical 

 with that given by (23), and 



, ccos0 + ?< (c 2 -I- v 9 ) + 2cv cos //I1X 

 a' = — a— -a*— = -a- 4 5 , • (41; 



from (28) and (29). Hence the angle of reflexion, the 

 amplitude, and the wave-length of the reflected waves are 

 identical with the corresponding quantities when the electric 

 field of the incident waves is perpendicular to the plane of 

 incidence. 



13. The screening current induced on the surface of the 

 mirror can be determined in two ways. 



The component of the electric force which terminates 

 normally on the mirror must end there on an electric charge, 

 which travels along the surface of the mirror with the speed 

 at which the corresponding wave trails along the surface. 

 Thus the normal component E x sin due to the incident 

 wave-train ends on a charge (— E 1 sin 0/47rc 2 ) per unit 

 area, which travels from North to South with the velocity 

 { (<?/ sin 0) + v cot 0}. The corresponding superficial current, 

 in the direction from South to North, is equal to 



4& sin 6 (site + VC0te ) "iJ^ + 'oos*) per cm. 



The normal component E 2 sin 0' due to the reflected wave- 

 train ends on a charge ( — E 2 sin 0' /47rc 2 ) per unit area, and 

 travels with a velocity, measured from North to South, equal 



