1903.] Electric Waves round a Conducting Obstacle. 



251 



" The Bending of Electric Waves round a Conducting Obstacle." 

 By H. M. Macdoxald, F.B.S., Fellow of Clare College, Cam- 

 bridge. Eeceived January 21, — Read January 29, 1903. 



1. The mathematical theory of the formation of a shadow when waves 

 impinge on an obstacle rests on an application of Huygens' principle, 

 which may be stated in the form that, if a closed surface S be drawn, 

 enclosing all the sources of the waves, the circumstances that obtain at 

 any point outside this surface at a definite time can be expressed in 

 terms of the state of affairs at the surface S at previous times. For 

 waves of sound, the usual analytical expression involves a knowledge 

 •of both the velocity potential of the motion and the velocity normal to 

 the surface S at each point of it for all time;* for electric waves, 

 which may be taken to include waves of light, it requires a knowledge 

 ■of both the electric and magnetic forces tangential to the surface S for 

 all time.f In the application to the theory of shadows, % the surface S 

 that is chosen coincides in part with the surface of the obstacle, the 

 remaining part being chosen so as to simplify the calculation as much 

 as possible ; for example, in the problem of the passage of waves of 

 light through an aperture in a plane screen, the surface S is taken to 

 be the plane of the screen. It is then assumed that the part of the 

 surface S which coincides with the surface of the obstacle contributes 

 nothing, and it follows that, when the wave-length is small, a shadow 

 is formed, whose boundary is determined by the extreme incident rays 

 that meet the surface of the obstacle. The assumption thus made is 

 •equivalent to assuming that the obstacle is perfectly absorbing, and for 

 waves of light incident on opaque bodies this is known to be approxi- 

 mately true, with possible exception for the case of opaque bodies whose 

 surfaces are polished. For waves of sound incident on an approxi- 

 mately rigid obstacle, and for electric waves incident on an approxi- 

 mately perfectly conducting body, this theory does not apply : as, in 

 the first case, the condensation does not vanish at the surface of the 

 obstacle ; and, in the second case, the tangential magnetic force does not 

 vanish at the surface. 



In what follows the behaviour of electric waves incident on a per- 

 fectly conducting body will be discussed, and the conditions necessary 

 for the formation of a shadow in this case will incidentally appear. 

 The results for waves of sound incident on a rigid obstacle are very 

 similar. 



2. It will be sufficient to consider a comparatively simple case, 



* Lord Rayleigh, ' Theory of Sound,' vol. 2, § 293. 

 t Macdonald, ' Electric "Waves,' § 14. 



X Cf. Stokes, ' Canib. Phil. Trans.,' vol. 9, 1849, p. 1 ; Lorenz, ' Pogg. Ann.,' 

 p. Ill, 1860; Kirchhoff, 'Berlin Sitzungsberichte,' vol. 2, 1882. p. 641. 



