82 DYNAMICAL THEOKY OF SOUND 



on the general principle illustrated in 9*. This is probably 

 to the relative advantage of the lower modes. The effect of 

 yielding of the bridges in modifying the natural frequencies 

 of the string has been discussed by Rayleighf; it is probably 

 in practice very slight. 



Another cause which must be mentioned as affecting our 

 results to some extent is the imperfect flexibility of the string, 

 or wire. In the case of the higher normal modes the segments 

 into which the string is divided may be so short that flexural 

 couples come into play, and tend to raise the frequency by 

 increasing the potential energy of a given deformation. This 

 will be referred to later ( 50). A further point is that the 

 abrupt forms postulated in the theory of plucked or bowed 

 strings are not exactly realized, and that such investigations as 

 those of 26, 27 are to be viewed as approximations, which are 

 however quite adequate so far as the determination of the ampli- 

 tudes of the graver and more important harmonics is concerned. 



30. Vibrations of a Loaded String. 



We conclude this chapter with the discussion of one or two 

 problems which, besides being of some interest in themselves, 

 may serve to remind us again that the harmonic scale of fre- 

 quencies is after all an exceptional phenomenon, even in the 

 case of strings. 



Take first the case of a string, otherwise uniform, loaded 

 with a mass M at its centre. It is obvious that those normal 

 modes of the unloaded string which have a node at this point 

 are unaffected. Leaving these on one side, we consider only 

 those vibrations in which there is at every instant complete 

 symmetry with regard to the centre. If the lateral displacement 

 of M be {3 cos (nt + e), we have, for the first half of the string, 



(1) 



* Some interesting experiments bearing on these questions have been made 

 by Barton and Garrett, Phil. Mag. (6), vol. x., 1905. See also Barton, Text- 

 Book of Sound, London, 1908, 361. 



t Theory of Sound, 135. 



