Mmtchester Memoirs, Vol. xlii. (1898), No. 9. 9 



actual distribution of pressure on the inner surface, we 

 should have had, in place of (5), 



dii, w 

 ax a 



dii w pa 

 dx a ~ JiB 



) ' ' ' (29X 



and thence, on the supposition that u, w,p vary as e 



iinx 



h-n'^hB hE , , 



/ = ^2 ^£' = -^W . . (30). 



Combined with (18) and (24) this leads to 



^2= — i^:— . . . (31). 



I + iKaj/iE 



This is the formula obtained by Korteweg, on the assump- 

 tion just stated. It may easily be shown that the right 

 hand side of (31) is an upper limit to the smaller root of 

 (27). In the particular case previously considered, it gives 



^=759^0- 



The assumption referred to may be justified (as a first 

 approxim.ation) by an appeal to a fundamental theorem 

 in Vibrations,* In the cases at present in view, the 

 motion of the tube-wall may be regarded as a forced 

 vibration whose period is considerably longer than that 

 of the free vibration of the same type (i.e., of the same 

 wave-length), so that the deformations have practically the 

 "equilibrium values" corresponding to the instantaneous 

 distribution of force. 



It remains to notice the various modes of radial 

 vibration of the system. These are found by putting 

 u = o, dwldx = o, in (5). If the time-factor be e'''\ we find 

 from (5) and (23), 



c ap, /Jva) 1 B ' 



''V^IT7a77{^)f='^^ • • (32), 



where 



y = in/c, .... (ss). 



• See Lord Rayleigh's Sound, § 100. 



