Waves at the Surface of a Moving Mirror. 519 



When v = 0, 0' — — 6,' which is the ordinary law of reflexion. 

 It is apparent from (25) that when v is finite and positive 

 (i. e. when the mirror advances to meet the incident waves) 

 the angle of reflexion is less than the angle of incidence. 

 When v = c, 0' = O, so that the waves would be reflected 

 normally for all angles of incidence. If we remember that 

 the sign of 0' is opposite to that of 0, it becomes clear that 

 (O' + 0) is equal to the difference between the angles of 

 incidence and reflexion. Also, (0—6') is the sum of the 

 angles of incidence and reflexion. Then equation (24) shows 

 that (0 + 0') 12 will have its maximum value when 



(0—0')/2=^, that is, when the sum of the angles of 



incidence and reflexion is equal to it. This condition cannot 

 be realized, since cannot exceed w/2, and 0' is always less 

 than ; but we see that the discrepancy between the angles 

 of incidence and reflexion increases with the angle of 

 incidence right up to grazing incidence. 



9. The value of the ratio a' /a must now be determined as 

 an explicit function of v and 0. 



2 _ \c + ty 



6 



COS0'=- ■ " = - ^^~ V/ 2 a 



1+tan'f l+( c —)tan'? 



2 \c + v/ 2 



(c + r) 2 cos 2 9 — (c — v) 2 sin 2 » 

 (c + v) 2 cos 2 ^+(c— v) 2 sin 2 | 



(c 2 + v 2 ) cos + 2cv 

 c 2 + v 2 + 2cv cos 6 



Using this value of cos#', we find that 



c — vcos0'= o , , o a ' • • (26) 



c a + I?- 4- 2cv cos v y 



Similarly n , (c 2 — v 2 ) (c cos + v) 



J c cos — i> = i-= 5^— -^ , 



c 2 + ?•- + 2ct- cos 



c cos 0' — v c — rcos #' sin 0' 



ccos0+v ~~ c + vcosd sinO 



Then, from (22) and (26) 



, c 2 + r 2 -f 2c u cos 



(27) 

 (28) 



(29) 



