278 Lord Rayleigh on Waves. 



The frequencies of vibration in the three gravest modes, being 

 inversely as the square root of the corresponding pendulum- 

 lengths, are in the ratio 



1 : 1-29 : 1*44. 



Professor Guthrie's observations give for the value of these 

 ratios 



1:1-31: 1-48. 



Possibly too low a frequency is attributed to the gravest vibra- 

 tion from the effect of insufficient depth. 



When the complete theory of the free vibrations of any sys- 

 tem is thoroughly known, it is in general easy to investigate 

 the effect of periodic forces. If u u u 2 , &c. are the normal 



2^ 2tt 

 functions, and — ; — , &c. the periods of the corresponding 



111 ^2 



27T 



free vibrations, the effect of forces whose period is — can be 



expressed in terms of the effect produced by similar forces of 

 infinite period, which last can be calculated statically. Thus, 

 if the solution of the problem according to the equilibrium 

 theory is 



A^i cos pt + A 2 u 2 cos pt + . . . , 



the true solution as modified by the inertia of the system will 

 be 



A x n\ ., A 2 nl 



i-Mi COS pt + ?-ll 2 COS pt+ 



n*—p 2 n 2 2 —p 2 



Let us calculate in this way the motion in a circular cylin- 

 drical basin due to a small horizontal force acting uniformly 

 throughout the mass of liquid, but variable with the time 

 according to the harmonic law. The equilibrium value of h 

 (the elevation) is evidently 



h = r cos 6 cos pt ; 



and the only difficulty consists in expressing r by a series of 

 Bessel's functions J x . It may be proved that 



2J 1 («r) 2J x fo»0 



- (^-1)^0 + OS-l^O*) ' ' <■ ; 



where k x , k 2 , &c. are the roots of J\(/c) = 0, and the radius R 

 is taken as unity. Thus the true value of h (after the motion 

 has been going on long enough to be independent of initial 

 circumstances) is 



, _ 2n\ cospt cos 6 Ji (#!?•) nv . 



, ~'(nj-^)(«;-i)J 1 (* l ) + -' • • (VV) 



