128 DYNAMICAL THEORY OF SOUND 



at the clamped end, and 



," = 0, ,'" = O = JZ] (2) 



for the free end. In one class of vibrations we have 



77 = A cosh mx -f D sin mx, (3) 



with the conditions 



A cosh \ml D sin \ml 0,^1 ... 



A sinh ^ml + Dcos ^ ml = 0, 1 



whence coth \ml = tan \ml (5) 



This is solved graphically by the intersections of the curves 

 y tan cc, y = coth x, the latter of which has y 1 as an 

 asymptote ; see Fig. 44. We have, approximately, 



iwiZ = (* + i)7r + a; (6) 



where s = 0, 1, 2, 3, ..., and a/ is small. This leads to 



tana/^- 2 "'', (7) 



where f f = e~ 2 "- iir , (8) 



whence a/= ^~ 2a<! '-i ?.'*~ 6 ; + , (9) 



which can easily be solved by successive approximation, except 

 in the case of the first root (s = 0). For this special methods 

 are necessary*. In the remaining type of vibrations we have 



77 = B sinh mx + Ccosmx, (10) 



with Bsinh ml + 



.(11) 

 B cosh \ ml + C sin | ml = 0, 1 



whence coth \rnl- tan \ml ................ (12) 



The intersections of the curves y = tan ,r., y = coth x are also 

 shewn in Fig. 44. The roots of (12) are given by 



imJ = (*-i)9T-&', ............... (13) 



where s = 1, 2, 3, .... Hence 



? s 6 2 ^, .................. (14) 



where f. = e"" ir , .................. (15) 



and therefore 



A' = &*' - J ?. w + ............. (16) 



* One such method will be indicated later in connection with the radial 

 vibrations of air in a spherical vessel ( 84). Another very powerful method is 

 explained in Rayleigh's treatise. 



