22<6 The late Lord Rayleigh on Resonant 



waves of simple form in the interior of the channels occupies 

 a space which is small relatively to the wave-length, and 

 then the connexion between the condition of things outside 

 and inside admits of simple expression. 



On the outside, where the dissipation is neglected, the 

 velocity potential (<p) of the plane waves, incident and 

 reflected in the plane of xy, at angle 0, is subject to 



d 2 $ldt 2 = a\d^ldx 2 + d: 2 $ldif), . . . (1) 



or if <p cc e mt , where n is real, 



d*$/da? + d?<f>/dit* + ]P$=0 9 .... (2) 



k being equal to n/a. The solution of (1) appropriate to 

 our purpose is 



J> __ e %(nt+1cy sin 8) f J^gikx cos 8 _j_ Igg—ikx cos 8\ /g\ 



the first term representing the incident wave travelling 

 towards — #, and the second the reflected wave. From (3) 

 we obtain for the velocity u parallel to #, and the con- 

 densation 5, when A' = 0, 



M=^=^^+^ 8in0 )^cos^(A-B), ... (4) 



(XX 



1 dcf) in 



a dt a 



a*=-^^=-™««*+* rin «(A4-B), . . (5) 



so that u .B- A , n , 



as = 00sd BTA (6 > 



For the motion inside a channel we introduce in (1) on 

 the left a term hdcf>/dt, h being positive, to represent the 

 dissipation. Thus, if <£ be still proportional to e int , we have 

 in place of (2) 



d 2 (f>ldx 2 + d 2 (f>/dy 2 + d 2 <l>/dz 2 + ]e' 2 cl> = 0, . . (7) 

 where k' 2 is now complex, being given by 



J,'2 ==k 2_ inh/a 2 ( 8 ) 



If we write k' = k 1 — ik 2 , where & 1? k 2 are real and positive, 

 we have 



k ] 2 -k 2 2 = P } kjc^nh/a* (9) 



At a very short distance from the mouth of the channel 

 d 2 (j>/dy 2 , d 2 (p/dz 2 in (7) may be neglected, and thus 



<£ = ^{A'cos*^ + B'sin&'#}. ... (10) 



