by a Screen bounded by a Straight Edge. 423 



not for any reflected waves, i. e. it is correct for a screen 

 which absorbs all the energy of the waves falling on it. 



In the cases more usually dealt with allowance must be 

 made for the reflected waves. 



To retain symmetry in the notation let the axis of y\ be in 

 the direction of propagation of the incident waves and the 

 axis of y 2 in the direction of propagation of waves reflected 

 according to the laws of geometrical optics. The reflecting 

 plane is given by y ± =y 2 . 



Write 



/i=-j f (et-yi — fi sec* *)<*«, \ 



(12) 



with a similar definition for/^ 



In the case in which the boundary condition to be satisfied 

 on either face of the diffracting half-plane is V=0 the 

 required solution is 



Y — -^r [ —j\—yjr 2 +f 2 in the region A, where the' 



ordinary reflected waves occur, 

 V==/\— / a in the region B, the geometrical )>(13) 

 shadow, 

 and V = ^i— J\— f 2 in the remaining region 0. J 



The conditions of continuity of V and its differential co- 

 efficients are satisfied, the values of V on the boundaries 

 between the regions A and 0, B and C being yfri—/ l — 2^2 

 and 1^1-/2 respectively. 



The lormula (11) is equivalent to one found as a special 

 case in my paper * on " Diffraction by a Wedge and Kindred 

 Problems." The weakness in the present demonstration lies 

 in the vagueness of the argument which leads to the trial of 

 the integral (3) proposed in the first instance. As a matter 

 of fact, in the integrals which serve for the solution of the 

 problem of diffraction by a reflecting wedge, the factor corre- 

 sponding with the U of equation (3) is not merely a function 

 of w, it depends on the azimuth. 



The proof that/ of equation (9) satisfied the fundamental 

 differential equation was based on the condition that the waves 

 had been passing for only a finite time. If this condition 

 is removed then (u — l) 1/2 i//(c£ — y — u%) does not vanish as 

 £->c© . From the physical point of view the limitation can 



* Proc. London Math. Soc. ser. 2, vol. xvi. at the foot of p. 106, 



tan a sin ^ being written for sinh ~ . 



