Vibrations of tivitched and stroked Strings. 203 



Again /a=0, v= — 1 ; and on the right sides of (5) the ne- 



gative sign must both times be chosen, so that 

 7 __ a 2t ry*_j - u : '$? 

 L~~L + T ? L"" 5 L T 

 Equations (6) therefore become 



bL<y(y— y') 



L(S-SQ +7^-/8= ^^ 



Therefore the string now consists of the three straight lines 

 b 



y 



=b^ 



— x for O^y"— 7fr"~^r' 

 ■a ~~ xi ~~ 1 L 



-2L*) + T(L-2a> 



2a(L-a)T 

 tor T L=L~ L T 5 



?/=--(L-^)for2-£ 





(16) 



Here also only the middle line is dependent on t ; it again 

 remains parallel to (9). The straight line passing through 

 x=0 is parallel with the third; and that passing through 

 x=h is parallel with the first line in (8). 



T 



At the time t= -%■ , y becomes =7', and the string consists 



z 



of the two straight lines 



?/= — j x for 0<#<L— a, 



y= — (L— k) „ L— «<j<L. 



(17) 



Consequently it now lies symmetrical with its initial posi- 

 tion, as is always the case at the end of half the period of 

 a vibration if the initial velocity of the several points was 

 = 0. From this point the string returns in a perfectly analo- 

 gous manner to its initial position : this need not be further 

 enlarged upon. The motion thus presented corresponds with 

 the delineations given by Thomas Young*. 



* Conf. loc. cit. supra, p. 199, and 'The Theory of Sound,' by Lord 

 Kayleigh, vol. i. (London, 1877) p. 185. 



By Ch. Delagrave, Libraire-Editeur, a model (first" constructed by 

 Monge) is sold, of the surfaces which are generated by the twitched 

 string when it is moved with uniform velocity perpendicularly to its plane. 



