266 Lord Rayleigh on the Passage of Waves 



aperture in an opaque screen, as given in (14), trie former 

 varying as the cube of the linear dimension, and the latter 

 as the first power simply. 



Reflecting Plate. — <£ = 0. 



For the sake of completeness it may be well to indicate 

 the solution of a fourth problem defined by the above heading. 

 This has an affinity with the first problem, analogous to that 

 of the third with the second. The form of % is the same as 

 in (30), and those for i/r TO , ty P the same as in (7). These 

 make dyjr m /dx, d^ v jdx vanish on the aperture, and so do not 

 disturb the continuity of dfyjdx. But in order that the con- 

 tinuity of (/> may also be maintained, we must have ^ m = ^ p , 

 and not as in (9) ^ m = — ^p. On the plate itself we must 

 have 



Accordingly yfr m is the same as in (12), while yjr p in (12) 

 must have its sign reversed. The realized solution is 



»,-»»= cos (nt-^-M 009 {nt - kr) . . (34) 



Two-dimensional Vibrations. 



In the class of problems before us the velocity-potential of 

 a point- source, viz. e~ ikr /r, is replaced by that of a linear 

 source ; and this in general is much more complicated. If 

 we denote it by D(kr), the expressions are* 



^H^^-ii+iw • • •> 



+ -p"°i — 2^42 2 2 2 .4 2 .6 2 ' ' ' ' " ^ ' 



where 7 is Euler's constant ('5772 . . .), and 



S OT = l+i + i+ • • • +!/»• 

 Of these the first is " semiconvergent," and is applicable 

 when kr is large ; the second is fully convergent and gives 

 the form of the function when kr is small. 



Since the complete analytical theory is rather complicated, 

 it may be convenient to give a comparatively simple deriva- 



* See for example ' Theory of Sound/ § 341. 



