260 Lord Rayleigh on the Vibrations of 



As an example we may take a uniform string of length / and 

 density p, carrying a small load m at its middle point. If y be 

 the transverse displacement at point oo, 



, . TTX . . 27TX ,„. 



y=<£iSin-y +<£ 2 sin-y-+ ..., ... (7) 



the origin of x being at one end. In this case for the gravest 

 tone we have 



so that 

 Accordingly 



2 



«T=i»iw5 1 -4+0 6 -...) 2 , 



8 [r] =m, $ \rs] = + m. 



(8) 



4> 



since jpj : jo| — ■_/>£ =1 : s 2 — 1 . 



P,. here denotes the value of p r when there is no load. 



Now 



S 2 — 1 5—1 S + l 



in which the values of s are 3, 5, 7 9 9, &c. Accordingly 

 ^ 1 1 



(9) 



s 2 -l 4' 



and therefore 



^ =p K i_ ? + ?? +cubes }' • • (io) 



which gives the pitch accurately as far as the square of the ratio 

 m : lp. 



The free vibrations of a system subject to dissipation-forces are 

 determined in general by the equations 



±( d J.) +^1 + ^1-0 an 



dAdf) + df + d^- ' • • • • ( 1] ) 



where T and V are as before, and F, called the dissipation-func- 

 tion, is of the form 



P = |(H)f 2 +...+(12)^ 2 +...* • • (12) 



* See a paper " On some General Theorems relating to Vibrations/' 

 Mathematical Society's Proceedings, June 1873. 



