250 DYNAMICAL THEORY OF SOUND 



that the problem has mathematically more than one solution ; 

 either of the above formulae will lead to an exact result, and 

 we might even use a combination of the two, in any arbitrary 

 proportions. This resolution of a historic controversy is due to 

 Lord Rayleigh. 



As a verification of (3), suppose that the value of <j> at x = 

 is that due to a train of plane waves <f> = e~ ikx . Let OT denote 

 the distance of BS from the orthogonal projection of the point 

 P on the plane x 0, so that r 5 = # 2 4- w 2 . For the aggregate 

 of elements 88 forming a certain annulus of the plane we 

 may write 27ror&nr = 27rr&r. We have also dr/dx = a;/r. The 

 formula (3) therefore gives 



r 9 fe~ ikr \ 

 -x ^ e )dr = e- ik *. ...(4) 



In the case of waves transmitted by an aperture in a plane 

 screen (x = 0), we have, in (1), 3</>/3n = except over the area of 

 the aperture. If, further, the dimensions of the aperture S are 

 small compared with X, then at a point P whose distance r is 

 large compared with X, the function e~ ikr /r will have sensibly 

 the same value for all the elements of S, and we may write 



(5) 



where the first factor represents the total flux through S. 

 Under these circumstances the aperture acts like a simple 

 source, as in 82. 



It is understood of course that the expression d<f)/dn in 

 (1) or (5) represents the normal component of the velocity, as 

 modified by the action of the screen. When as in the case just 

 considered the aperture is relatively small, the distribution of 

 normal velocity over it will differ considerably from that due to 

 the primary waves alone. This distribution can be ascertained 

 approximately, in the case of plane waves incident directly on 

 a circular opening, from the electrical analogy of 82. The 

 lines of flow have the same configuration as the lines of force 



