Waves at the Surface of a Moving Mirror. 517 



Oblique Reflexion — Electric Field Perpendicular to 

 Plane of Incidence. 



§ 7. Let the motion of the mirror be defined as in § 4, 

 and let the plane of incidence be horizontal, the incident 

 waves travelling up to the mirror from somewhere in the 

 quadrant between East and North. Let 6 be the angle of 

 incidence, while 6' is the angle of reflexion ; both these 

 angles are measured from the axis of x round towards the 

 North. If y is measured from the origin in the direction 

 from South to North, the equation to the incident waves may 

 be written in the form 



Et = a cos {m(x cos 6 + J sin 6 + ct)} y . . (13) 



where E T denotes the electric field at(#, y) at the time t, 

 the positive direction of E x being vertically upwards. 



The equation to the waves reflected from the moving 

 mirror may be written 



E 2 = a' cos {m!(x cos O'+y sin 6' — ct)}. . (14) 



The magnetic field due to either train of waves will be 

 parallel to the intersection of the wave-front and the plane of 

 incidence; let the sign of the magnetic field be chosen so 

 that for normal incidence (0 = 0' = 0) the positive direction 

 of the magnetic force is from South to North. 



At any point (x, y) the resultant force that would be 

 exerted on a unit positive charge in a stationary position is 

 equal to (Ej + Ea). At the same point, the force that would 

 be exerted on a unit N pole in a stationary position has the 

 component (E x cos — E 2 cos 0')/c from South to North, and 

 the component (E x sin 0— E 2 sin 0')/c from East to West. 



At the surface of the mirror, the force that would be 

 exerted on a unit positive charge moving with the mirror 

 must vanish. Thus 



E 1 +E 2 +-(E 1 cos0--E a cos0')=O' 



.\ "E l (c + vcos0) + E a (c-vcos0') = O. . . (15) 



Further, the component of the magnetic field normal to 

 the mirror must vanish at the surface of the mirror. Since 

 the value of this component of the magnetic field is not 

 affected by motion from West to East, 



EiSin^~E^sin^=0 (16) 



