72 DYNAMICAL THEOEY OF SOUND 



that the resulting motion shall be strictly simple-harmonic, 

 and the sensation accordingly that of a pure tone. But, as 

 will be shewn more fully in 39, it is possible to suppress 

 all the tones below any assigned rank (s) by checking the 

 vibration at a node of the 5th mode, as, for instance, by 

 contact with a camel-hair pencil. The remaining nodal points 

 of this constituent are then points of rest, whilst half-way 

 between them there is vigorous vibration. The experiment, 

 which is very striking, is easily made with the monochord. 



The energy in any normal mode is easily calculated. We 

 find 



...... (9) 



...(10) 



The coefficients are equal, in virtue of 22 (3), and the total 

 energy in this mode is 



T+v 



It is further easily proved that the whole energy of the 

 string is the sum of the energies corresponding to the various 

 normal modes, viz. 



T+V-^-S.*Of-^*f(4*+Bf) ....... (12) 



This is a general property of the normal modes of a vibrating 

 system. The proof, in the present case, depends on the fact 

 that 



[ l . STTX . STT3C ..''* /-, ox 



I sin j- sm = dx = 0, ............ (13) 



JO * ' 



if s, s be any two unequal integers. See 32 (4). 



26. String excited by Plucking, or by Impact. 



The relative amplitudes of the various modes is of course 

 a matter of importance, as on it the quality of the note 

 depends ( 2). Usually a string is excited in one of three 

 ways, viz. by plucking (as in the harp, zither, &c.) f by striking 

 with a hammer (pianoforte), or by bowing (violin, violon- 

 cello, &c.). 



