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XIV. A Diffraction Problem, and an Asymptotic Theorem in 

 BesseVs Series. By R. Hargreaves, M.A.* 



THE first diffract! oij problem to which exact methods 

 have been applied with success, is the problem in two 

 dimensions solved by Spmmeri'eld. Its solution is here 

 presented in a form which is, I think, in a sufficient degree 

 simpler and more convenient than the original, to justify an 

 independent statement of the arguments. I add also a 

 solution of the problem in three dimensions, which arises 

 when the plane of the incident wave is not parallel to the 

 edge of the barrier. The solution appears first in the form 

 of a definite integral, and a direct algebraical transformation 

 is made to a series of Bessel's functions and Trigonometrical 

 functions. When the latter form is got independently the 

 crux of the problem lies in the asymptotic value of the 

 series. 



§ 1. The coordinates in the plane being (xy), the barrier 

 occupies the half of the XZ plane for which x is positive. 



The condition at the barrier may be -=— = (i. e. ^ = 0), or 



~l- =0 ; the first corresponding to zero pressure, the second 



to zero velocity in the acoustical problem. In constructing 

 the functions it is convenient to take an incident wave 

 cos k(Yt + y) ; the transition to oblique incidence is imme- 

 diate and presents no difficulties. This form of incident 

 wave involves two asymptotic conditions. For x infinite 

 and negative, y finite, -v/r must approach the limit cos&( Yt +y) . 

 For x infinite and positive, y finite, the asymptotic value 

 must be zero for y negative, cos k(Yt + y)+ cos k(Yt---y) 

 for y positive according as we are dealing with the barrier 



condition -^ = or ^— =0. 



The solution is based on the function 



N V^ C V y f 1" 7/TT v") AU 



4>0,y)=o-7=- cos \i+ k (yt+y-^\—r- ■ (l 



L \ IT Jo K * J \U 



which for r + y very great approaches the value 

 icosk(Yt+y), 



The physical conception suggesting the form of function 

 is that a wave of type Yt+y must be converted to one of 

 type V£ — r, which will correspond to divergence from the 

 * Communicated by the Author. 



