Lord Rayleigh on Waves. 27 5 



the corresponding period for the air-vibration i 1 ^ 



a / e Kl —e~ Kl ^ 



(I) 



. = 27r-r-A:a. 



If A/ be the wave-length of plane aerial vibrations having the 

 period i J K , 



fC=27T + \ / . 



If kI be very small, the ratio of periodic times is 



T K :T / K = a:Vgl, (J) 



and is independent of tc. Hence the two problems of the vibra^ 

 tions of air and liquid are mathematically analogous whatever 

 the initial circumstances may be ; so that if the condensation 

 in the first follows the same law initially as the elevation in 

 the second, the correspondence will be preserved throughout 

 the subsequent motion, if a 2 =gl. The initial circumstances, 

 however, must be such as not to give prominence to the higher 

 components, for which kI would no longer be small. 



When /cl is not negligible, we learn from formula (I) that 

 the period increases with I until kI is moderately great, when 

 it becomes sensibly 



T K = 27T+^ (K) 



In any case the period is independent of the density of the 

 liquid. 



Some careful observations on liquid vibrations have been 

 recently made by Professor Guthrie *, with which it may be 

 interesting to compare the results of theory. Professor Guthrie 

 used troughs whose horizontal section was rectangular and cir- 

 cular. "We will take the rectangular section first. 



Confining ourselves to those modes of vibration which de- 

 pend on only one horizontal coordinate, we may take for the 

 normal functions 



nirx 

 u= cos -j—> 



L being the length of the trough, n integral, and cc being 



measured from one end. The corresponding value of k is y 



Hence, from (I), the length of the simple equivalent pendulum 

 is 



nnl nnl 



,iLiL+£Z (L) 



e L — e~ *^ 



* Phil. Mag. October and November 1875. 

 TJ2 



nir 



