358 Lord Rayleigh on the Propagation of Waves 



A simple case included under (11) is that of a stretched 

 string, itself without mass, but carrying unit loads at equal 

 intervals (a) *. The expression for the potential energy is 



P= "- + ^(^-t-i) 2 +^'(^ + i-^) 2 + ... 5 • (13) 



T x representing the tension. Thus by comparison with (5) 



C = 2T x /a, O^Tj/a, C 2 = 0, &c. ; 



so that by (8) 



2T : 2T, 



7i 2 = — cos ka, 



a a 



n= \/(^)' 2Sin(pa) ' ' ' ' * (K) 

 fx, being introduced to represent the mass of each load with 

 greater generality. The value of V is obtained by division of 

 (14) by k. In order more easily to compare with a known 

 formula we may introduce the longitudinal density p, such 

 thnt fi = ap. Thus 



n _ //TA sin(^a) n „ 



V -F-VW'~I^~' ' • ' ( } 



reducing to the well-known value of the constant velocity of 

 propagation along a uniform string when a is made infinitesi- 

 mal. Lord Kelvin's wave-model (' Popular Lectures and 

 Addresses,' vol. i. 2nd ed. p. 1G4) is also included under the 

 class of systems for which P has the form (13). 



Another example in which again C 2 , C 3 . . . vanish is pro- 

 posed by Fitzgerald f. It consists of a linear system of rotating 

 magnets (fig. 1) with their poles close to one another and 



Fig. 1. 



disturbed to an amount small compared with the distance 

 apart of the poles. The force of restitution is here propor- 

 tional to the sum of the angular displacements {ty) of con- 

 tiguous magnets, so that P is proportional to 



. . . +(f r +f r _ 1 y+ (f r +^ r+I )2 + . . . . 



HereCjSs— ^C , and (8) gives ?i 2 = (l + cos£a), 



or 7i = « .cos (^ka), (16) 



* See ' Theory of Sound/ §§ 120, 148. 

 + Brit. Assoc. Report, 189-3, p. 689. 



