40 



Lord Eayleigh. On the [May 1, 



" On the Bending of Waves round a Spherical Obstacle." By 

 LORD EAYLEIGH, F.RS. Eeceived May 1 Read May 2S, 

 1903. 



In the 'Proceedings' for January 21, 1903, Mr. H. M. Macdonald 

 discusses the effect of a reflecting spherical obstacle upon electrical and 

 aerial waves for the case where the radius of the sphere is large 

 compared with the wave-length (X) of the vibrations. The remarkable 

 success of Marconi in signalling across the Atlantic suggests a more 

 decided bending or diffraction of the waves round the protuberant 

 earth than had been expected, and it imparts a great interest to the 

 theoretical problem. Mr. Macdonald's results, if they can be accepted, 

 .certainly explain Marconi's success ; but they appear to me to be open 

 to objection. 



If C be the source of sound, P a point upon the sphere whose centre 

 is at 0, ^ the velocity-potential at P due to the source (in the absence 

 of the sphere), x the angle subtended by OC, Mr. Macdonald finds for 

 the actual potential at P, 



</> = *i(l-cos X ) ........................ (1), 



so that there is no true shadow near the surface of the sphere. If 

 C be infinitely distant, and p denote (as usual) the cosine of the angle 

 between OP and OC, 



</> = ^(l+p) ........................... (2). 



That the sound should vanish at the point opposite, and be quadrupled 

 at the point immediately under the source is what would be expected ; 

 but that (however large the sphere) the shadow should be so imperfect 

 at, for example, p = - J, is indeed startling. 



The first objection that I have to offer is that nothing of this sort 

 is observed in the case of light. The relation of wave-length to 

 diameter of obstacle is about the same in Marconi's phenomenon as 

 when visible light impinges upon a sphere 1 inch (2'54 cm.) in diameter. 

 So far as I am aware no creeping of light into the dark hemisphere 

 through any sensible angle is observed under these conditions even 

 though the sphere is highly polished.* 



But I shall doubtless be asked whether I have any complaint against 

 the mathematical argument which leads up to (2). 



As in Theory of Sound, 334, the question relates to the ratio 

 between a certain function of c (the radius) and its differential 



* It may be remarked that at the centre of the shadow thrown at some distance 

 (say 1 metre) behind, there is a bright spot similar to that seen when a disc is 

 substituted for the sphere. This effect is observed with a magnifying lens. If the 

 eye, situated at the centre of the shadow, be focused upon the sphere, the edge of 

 the obstacle is seen bounded by a very narrow ring of light. 



