SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 199 



Let us now consider the problem of waves incident from some source (or sources) 

 and reflected from the surface S. Let u denote any Cartesian component of force in 

 the incident wave, and let the point P be outside S. Then (7'l) applies, and so 



|Y 3t>_ 9w 



J \ ov fiv 



because u has no singularity inside S. 



If u' detiotes the corresponding component of force in the reflected wave, we have 

 from (7'12) 



v I 



because u' has no singularity outside S (and n' will correspond to a divergent wave 

 at infinity). 



By addition we have the result 



f \(u + u'} ^ -v | (u + u') 



J I OV OV 



where v is taken to be e~'" R /R. 



Now M + t/ = w gives the corresponding component of force in the complete wave ; 

 and this accordingly satisfies certain known relations at the surface S (the exact 

 form depending on the physical properties of S). It must, however, be clearly 



^\ 



understood that we cannot usually obtain both w and -^ by any simple methods, any 



cv 



more than the analogous problems of electrostatics can be solved by a mere appeal 

 to GREEN'S Theorem. 



However, we can obtain an approximate solution, suitable to the problem of short 

 wave-lengths, by assuming that near the reflecting surface, the character of u' can be 

 determined from that of u by the rules of elementary geometrical optics. Thus we 

 treat the reflected wave as derived from the incident by simple reflexion in the 

 tangent-plane at the point of incidence. Making this hypothesis it is an easy matter 



f\ 



to construct both w and -^ when the form of u is given. 



OV 



It will be noticed that we shall have w = 0, -^- = at all points within the 



ov 



geometrical shadow ; and so the final integral (7 '13) extends only over the illuminated 

 side of the surface S. 



Suppose now that we consider electric waves incident on a simple convex 

 conducting surface ; and take an origin O on the surface such that OP is the 

 reflected ray (in the sense of geometrical optics). Take the plane of incidence as the_ 

 plane of yz, and the normal at O as the axis of z. 



VOL, ccxx, A. 2 F 



