264 Lord Rayleigh on the Passage of Waves 



so that for a circular aperture the realized solution is 



<£ m = 2 sin nt sin hx 



Rttc t 

 + ^jj- -2 cos '(nt—'kr). . . . (28) 



It will be remarked that while in the first problem the 

 wave (ty) divergent from the aperture is proportional to the 

 first power of the linear dimension, in the present case the 

 amplitude is very much less, being proportional to the cube 

 of that quantity. 



The solution for an elliptic aperture is deducible from the 

 general theory of the motion of an ellipsoid (a, b, c) through 

 incompressible fluid*, by supposing a = 0, while b and c re- 

 main finite and unequal ; but the general expression does 

 not appear to have been worked out. When the eccentricity 

 of the residual ellipse is small, I find that 



/ ( = |(MI(1-A« 4 ), (29) 



showing that the effect of moderate ellipticity is very small 

 when the area is given. 



From the solutions already obtained it is possible to derive 

 others by differentiation. If, for example, we take the value 

 of <f) in the first problem and differentiate it with respect to 

 x, we obtain a function which satisfies (2), which includes 

 plane waves and their reflexion on the negative side, and 

 which satisfies over the wall the condition of evanescence. 

 It would seem at first sight as if this could be no other than 

 the solution of the second problem, but the manner in which 

 the linear dimension of the aperture enters suffices to show 

 that it is not so. The fact is that although the proposed 

 function vanishes over the plane part of the wall, it becomes 

 infinite at the edge, and thus includes the action of sources 

 there distributed. A similar remark applies to the solutions 

 that might be obtained by differentiation of the second solu- 

 tion with respect to y or z, the coordinates measured parallel 

 to the plane of the screen. 



Reflecting Plate. — d<l)/dn = 0. 



We now pass to the consideration of allied problems in 

 which the transparent and opaque parts of the screen are 

 interchanged, Under the above-written boundary condition 

 * Lamb's t Hydrodynamics/ § 111. 



