84 DYNAMICAL THEOKY OF SOUND 



frequencies are therefore slightly lower than in the symmetric 

 modes of an unloaded string. As b increases the frequencies 

 all diminish, the physical reason being of course the increased 

 inertia. Finally, when M is very large compared with the mass 

 pi of the string, l/b is small, and the lowest root of (5) is given 

 approximately by a? = 1/b, whence 



n = 



in agreement with 6 (4). 



31. Hanging Chain. 



The contrast with previous continuous systems is still more 

 marked in the case of the small oscillations of a uniform chain 

 hanging vertically from its upper end, which is fixed. This 

 has no immediate acoustic importance, but it is interesting 

 historically*, and is, from the standpoint of general theory, 

 instructive in various ways. 



We take the origin at the equilibrium position of the free 

 end. The tension at a height x above this point will be 

 P = gpx, the vertical motion being neglected as of the second 

 order. Hence if y denote the horizontal deflection, we have 



or 



a* 



3 / dy 

 I*-K 



(*y\ 



rat/ 1 



To ascertain the normal modes we assume that y varies as 

 cos (nt + e), and obtain 



i( x d /] + n -y = Q ................... (3) 



dx \ dxj g * 



This can be integrated by a series, but the solution assumes 

 a somewhat neater form if we introduce a new independent 

 variable in place of as. The wave-velocity on a string having 

 a uniform tension equal to that which obtains at the point x 

 would be *J(Plp) or *J(gx). Hence if r denotes the time which 



* It appears to have been the first instance in which the various normal 

 modes of a continuous system "were determined, viz. by D. Bernoulli (1732). 

 The Bessel's Function also makes its first appearance in this connection. 



