2 Lamb, Waves in a Medium with Periodic Structure. 



developed are afforded by some optical and acoustical 

 phenomena studied by Lord Rayleigh ;* but the results 

 have also an interest, on another scale, as bearing on 

 mechanical theories of reflection, refraction, &c. In parti- 

 cular, the usual assumption that the wave-length is great 

 compared with molecular intervals, is here dispensed with, 

 and we are able to trace to some extent the fate of a train 

 of relatively short waves, or of an abrupt disturbance. 

 On the other hand, some caution must of course be 

 exercised in drawing inferences as to theories of radiation, 

 absorption, and the like, from the study of a one- 

 dimensional model. 



I. Suppose we have a string of line-density p, loaded 

 at equal intervals a with particles of mass M. Taking 

 the axis of x along the string, and denoting by S the 

 displacement of any point of it, we have the equation 



dt^~^ dx^ ' ' ' • ^'^' 



where 



C^^-E/p .... (2), 



E being the coefficient of elasticity. If we assume that S 

 varies as e'^^'^\ i.e., we consider a simple-harmonic disturb- 

 ance whose wave-length on the unloaded string would be 

 27r/>^, the equation (i) becomes 



£+^^^=° • • • (3). 



If we distinguish the values of 2 corresponding to 



successive particles M by suffixes, the solution of (3) for 



the interval between the sth and (^-i-i)th particles is 



obviously 



J. J. , Bs+i - 4 cos ka . . , . 



4 = 4a cos kx + -^ — .^ sm kx . . (4), 



sin ^a ^^'' 



* P/iz/. Mag., Sept., 1888, p. 256 ; Theory of Sound (2nd ed.), vol. ii., 

 p. 311. 



