STRINGS 



69 



To ascertain the normal modes of vibration of a finite string 

 we may have recourse to the general procedure explained in 

 Chapter I. In any such mode y will vary as a simple-harmonic 

 function of the time, say cos (nt + e). This makes y = ri*y, 

 and the equation (2) of 22 therefore assumes the form 



.(i) 



(2) 



The solution of this, exhibiting the time-factor, is 



/ . nx -T. . nx\ f . ^ 

 y = I A cos -- 1- B sin 1 cos (nt -f e). ... 

 \ c c / 



The fixed ends of the string being at x = 0, # = I, we must 

 have A=Q, sin (nl/c) = 0, whence 



nl/7rc = l, 2,3, ...................... (3) 



This gives the admissible values of n. In any one normal mode 

 we have, therefore, 



n . STTX /STTCt \ ,.. 



y = C 8 sm-- cos(-- + 6j, ............... (4) 



where 5 is an integer, and the amplitude C g and initial phase 

 s are arbitrary. The gravest, or fundamental mode, which 

 determines the pitch of note sounded, corresponds to s = l. 



The string then oscillates in the form of the curve of sines 

 between the two extreme positions shewn in the upper part of 

 Fig. 29. The frequency is 



