126 DYNAMICAL THEOKY OF SOUND 



asymptotic to the line y = 1. The figure shews that we have 



approximately 



) .................. (8) 



where 6? = 1, 2, 3, ..., and a s is small. It follows from (2) that 

 the frequencies of the successive normal modes of symmetrical 

 type are approximately proportional to 3 2 , 7 2 , II 2 , .... For a 

 more exact computation of the roots we have 



_ __ m _ 



ia *--- 



where s = e~ sir ...................... (10) 



Hence a 8 = ten-> (&' 2a >) = ^ s e~^-^e~^+ .... (11) 



Since ? g is small, even for s = 1 (viz. & = "00898), this is easily 

 solved by successive approximation. 

 In the asymmetric modes we have 



rj = Bsinhnuc + Dsmmx, ............... (12) 



with the terminal conditions 



B sinh J ml D sin \ml 0,j 



cosh imJ-D cos JmZ = 0,J 

 whence tanh \ml = tan \ml ................... (14) 



The roots of this are given by the intersections of the curves 

 y = tan x, y = tanh X, the latter of which is asymptotic to the 

 line y = 1 ; see Fig. 44. It appears that 



JmJ = ( + J)ir-&, .................. (15) 



where 5 = 1, 2, 3, ..., and ft is small. The corresponding 

 frequencies are approximately proportional to 5 2 , 9 2 , 13 2 , .... 

 For the more exact calculation we have 



1+ tan mn. tanh 

 where ^ = e~ 2s7r ~ i7r ........ , .......... ...(17) 



Hence ft = tan- 1 (^ 2ft )= ?^ 8 -K^~ 6 ^ + ....... (18) 



Since i = "00039, the approximation is very rapid, even for 



