through Apertures in Plane Screens. 263 



and the remaining condition is also satisfied if 



dslr m /diV = ikj dy]r p /dx=—ik. . . . (21) 



Thus ^f m is to be such as to make d^jr m /dx = ik ; and, as in the 

 proof of (17), it is easy to show that in (19) 



V m =+J2", %=~f r /^, ■ ■ ■ (22) 



where tfr m , ty p are the (equal) surface-values at dS. 



When all the dimensions of S are small in comparison with 

 the wave-length, (19) in its application to points at a sufficient 

 distance from S assumes the form 



+»= ,; ^jJV s > • • • • (23) 



and it only remains to find what is the value of fi yjr p d$ 

 which corresponds to dty P ld% = — ik. 



Now this correspondence is ultimately the same as if we 

 were dealing with an absolutely incompressible fluid. If we 

 imagine a rigid and infinitely thin plate (having the form of 

 the aperture) to move normally through unlimited fluid with 

 velocity u, the condition is satisfied that over the remainder 

 of the plane the velocity-potential ^ vanishes. In this case 

 the values of yjr at corresponding points upon the two sides 

 are opposite ; but if we limit our attention to the positive 

 side, the conditions are the same as in the present problem. 

 The kinetic energy of the motion is proportional to w 2 , and 

 we will suppose that twice the energy upon one side is hu 2 . 

 By Green's theorem this is equal to — \\ yjr . dyfr/dn . d$, or 

 — u\\yjrdB; so that \\yfrdS=—hu. In the present appli- 

 cation a= —ik, so that the corresponding value of ' utypdS is 

 ihk, Thus (23) becomes 



hBxe ~ ikr 



+*= SSS" (25) 



The same algebraic expression gives yfr m , if the minus sign be 

 omitted ; for as x itself changes sign in passing from one 

 side to the other, the values of ^jr m and ty p at corresponding 

 points are then equal. 



The value of h can be determined in certain cases. For a 

 circle* of radius c 



*=^-; (2(5) 



* Lamb's ' Hydrodynamics/ § 105, 



