46] 



GRADUAL PASSAGE TO LIMIT. 



169 



As n increases without limit y preserves its form, while v s approaches 

 the limit 



102) ".- 



We have therefore for the continuous string, 



y = 



STtX 



s sm ~ cos 



_ 

 fsTf-l /S \ 



(r v 7 ' i ~ ") - 



The frequencies of the different terms of the series are in the ratios 

 of the integers. Such partial vibrations are called harmonics or 

 overtones of the lowest or fundamental, for which s = 1. Since, if 

 we consider a single term of the series, the excursions of all the 

 particles are in the same ratios throughout the motion, we see that 

 the harmonics are normal vibrations. On account of the factor 

 depending upon x the s th harmonic has nodes for 



I 2Z (s-l)Z 



x = >, 2 - '-> 



s s s 



or at any instant the string has the form of a sine curve and is 

 divided by nodes into s segments vibrating oppositely, generally 

 known as ventral segments. 



In order to show how rapidly the string of beads approximates 

 to the motion of a continuous string, the following table from 

 Rayleigh's Theory of Sound is inserted. It is to be noticed that it 

 does not give exactly the ratios of the frequencies on account of the 

 variable factor s under the sine in v t> but it approximately does so, 

 and for the fundamental, s = 1, it gives exactly the ratio of frequency 

 for n beads to that of the continuous string. 



By means of an extension of the above method, Pupin has treated 

 the problem of the vibrations of a heavy string loaded with beads, 



1) Writing the factor of I/ in the form 



S7C \ 



: h 



S7t 



Q 



S7C 



since 



,. 

 lim 



= 1, 



we obtain the result. 



