Waves at the Surface of a Moving Mirror. 513, 



Writing vt for # in (1) and (2), and substituting the resulting 

 values Ej and E 2 in (3), we obtain 



(c + v) a cos m (y + c)t-{- (c — v) a' cos m' (u — c) £ = 0. , (4) 

 This equation is satisfied for all values of t if 



(c + w)a + (c-u)a / = 0, .... (5) 



m (c + v) = m' (c — v) (6) 



Therefore, by reflexion from the moving mirror, the ampli- 

 tude is increased in the ratio (c-\-v)/(c — v). If v = c, the 

 amplitude of the reflected waves becomes infinitely great. 

 The wave-length is altered in accordance with the Doppler 

 principle. 



5. The average pressure exerted on each unit of area of 

 the mirror is now easily obtained. The energy carried up to 

 unit area of the mirror by the incident waves in a second 

 is equal to the energy comprised in a volume (c-\-v) of tbe 

 incident wave-train. The energy carried away from the 

 same area in a second is equal to the energy comprised in 

 a volume (c— v) of the reflected wave-train. The difference 

 between the energy carried away and the energy carried up 

 to the unit area of the mirror in a second is equal to the 

 work done by that area in moving against the pressure p 

 through the distance v. 



••• J»- »{«'■<—>-*(• + •> } = C { (S)V»)"(«+v) } 



_ 2a 2 v Ic + v\ 

 ~&rrc 2 V^v)' 

 2a 2 /c + r\ n . 



Notice that p becomes infinite when v = c. Sir Joseph 

 Larmor concludes* that the pressure decreases as v increases, 

 becoming equal to zero when v=c. This error has arisen 



from expressing the pressure as the fraction ^ — ~ 2 of the 



energy per unit volume of both incident and reflected wave- 

 trains, and forgetting that when this fraction becomes equal 

 to zero through v being equal to c, the energy per unit 

 volume of the reflected wave-train becomes infinitely great. 



* " The formula above obtained is of general application, and shows 

 that for high values of v the pressure must fall off." Article ' Radiation/ 

 § 2, EncyclopcBdia JSrittanica, vol. xjtii. p. 787. 



Phil. Mag. S. 6. Vol. 28. No. 166. Oct. 1914. 2 L 



