240 Lord Ravleiffh on tl 



o 



It 



is the conclusion arrived at. " Absolute irregularity would 

 show itself by an energy-curve which is independent of the 

 wave-length : i, e., a straight line when the energy and wave- 

 length or period are taken as rectangular coordinates . . ."*. 

 It is possible that the discrepancy may depend upon some 

 ambiguity ; but in any case I have thought that it would 

 not be amiss to reconsider the question, using a different and 

 more elementary method. 



For this purpose we will regard the string as fixed at the 

 two points x = and x = l. The possible vibrations are then 

 confined to the well-known " harmonics/'' and k is limited to 

 an infinite series of detached values forming an arithmetical 

 progression. The general value of the displacement y at 

 time t is 



v . sttx / . sirat D • sirat\ ,_. 



y=2, sin ,— ( A s cos — h B s sm — \ . . ( i ) 



in which a is the velocity of propagation and s is one of the 

 series 1, 2. 3. . . . From (7) the constant total energy (T + Y) 

 is readily calculated. Thus (< Theory of Sound; § 128) if M 

 denote the whole mass, Ts the period of component s, 



T+V=^M.2 A " 2+ 2 B/ , . . . . (8) 



Ts' 



an equation which gives the distribution of energy among the 

 various modes. 



The initial values of y and y are 



y Q = * A* sm p y = — ZsBs sm j- : 

 whence 



2 f' . 6"7T.?. ; , ^ 2 C l . . STTX , N 



A.^-j^.sm-pte; B.-— J^am-pi,. . (9) 



If we suppose that y o = throughout and that y is finite 

 only in the neighbourhood of #=£, we have A> — 0, and 



B s =A. sin f^ Y (10) 



where Y=\y dw. The energy in the various modes being 

 proportional to Bs 2 /ts 2 , or to 



1 . 2 S7rf 



A? sm " — ' 



in which 5 2 t s 2 = t/ 2 , is thus independent of 6- except for the 

 factor sin 2 (stt%II) . And even this limited dependence on s 



* " The PeriodogTain of Magnetic Declination, &c.,"' Oamb. Phil. Trans. 

 xviii. p. 108 (1899). 



