Study of Huygenss Secondary • Waves. 211 



We shall now apply the solution (a) of the general 

 differential equation to the following problems : it is under- 

 stood that in each case the reflecting surface or transmitting 

 aperture is of dimensions large compared with the wave- 

 length. 



Reflexion of plane aerial waves at an infinite rigid plane, 

 whose position is given by a?=0. 



Let the plane waves be incident at an angle i on the posi- 

 tive of the plane, and the velocity-potential of the incident 

 waves be the real part of 



» d'(Vt+x cos i — y sin z). 



Superpose upon this, the value of the integral 



taken over the whole of the infinite plane, Y and Z being 

 the y and z coordinates at any point on the plane. 



The value of the integral at any point in the space on the 

 positive side of the infinite plane is 



a pdc(Vt - x cos i—y sin 



and on the negative side 



_ * pik(Vt+x cos i—y sin i) 



The resultant disturbance on the positive side is 



, _ a r d-(Vt+x cos i—y sin i) . gtk(Vt —x cos i—y sin i)-| 



J* = Ack cos i e J < Yt -y s[n « [«*■ * os ' - ,-■*■ cos *], 



which when .v — 0, is equal to zero. 

 On the negative side 



6 = and ^=0. 

 dx 



The necessary conditions are thus satisfied. The normal 

 velocity at the plane is zero on both sides of it, and the region 

 on the negative side of the plane is entirely screened from 

 disturbance. The same is true even if the reflecting surface 

 occupies only part of the plane x = 0, for at points close to 

 the plane, on either side of it, the result of the integration (d) 

 is practically the same as if it were extended over infinity. 



In the case last mentioned of a finite reflecting surface, the 

 effect of the reflected waves at points not near the reflecting 



P2 



