III-73 



Suppose that we irradiate the surface instead with the retarded incident wave 



_,(T)/ v_ ik(ax+Yz)-iui)t + iT , . im 



Pine ^""'^''^ - Pine ^ ^ Pine ^^'^''^^ ("1-52) 



In other words, p.^^ is identical at the two space-time points P = (x,z,t) and 

 Pi = (x+n5, z, t+ — - — ). The incident wave will produce a scattered wave which 

 has also been retarded by T: 



„(i") / ,v , . -iUJt iUJt 



Psc ^^'^'^^ " Psc^"^'^^^ ^ (III-53) 



Instead of shifting the time axis, we could consider a translation of the x-axis by 

 an integral number of periods n§ . This translates the surface into itself. The 

 corresponding incident wave is 



„(n?) _ ^ ^ik(ax+an5 + vz)-iujt , , ikan ? 



Pine -Pinc^ =Pinc^^'^'^^^ <"I-54) 



and causes a scattered wave 



•'sf* = Psc<'^"?'^>^"'"" ■ W'-SS) 



However, we note that p. " could equally well be regarded as a retarded ver- 

 sion of p.j^^, retarded by an amount T = kan§ /uo . The surface has remained 



the same, since S(x+n§ ) = S(x), and so we expect that the scattered waves 

 will be the same, i.e., 



Psf*-Pj"'"''"' («■-") 



Arthur B.littleJnt. 



S-7001-0307 



