1 8 Lamb, Waves in a Medium with Periodic Structufe. 



dimensional medium which for sufficiently long waves has 

 a definite index of refraction, with coefficients of reflection 

 and transmission related to this index in the usual manner ; 

 which (again) totally reflects radiations of wave-lengths 

 lying between certain limits ; whilst for sufficiently short 

 waves there is, as a rule, free transmission, with practi- 

 cally no reflection. 



y. The above examples have been chosen for simplicity, 

 but there is no difficulty in extending the method to the 

 case where dynamical systems of any degree of com- 

 plexity, but all exactly alike, are interpolated at regular 

 intervals. 



We may suppose that the position and configuration of 

 any one of these systems is determined by means of the 

 coordinate ($) of the point of the string where it is 

 attached, and by means oi n other coordinates ^i, ^2, ... qn- 

 There is no loss of generality in supposing these latter 

 coordinates to be so chosen that the expressions for the 

 kinetic and potential energies reduce to the forms 



2 T= axq^ + «2^2^ 4- . . . + a,,q,c + 2(ai^i + a^q^ + . . . + a„^,J^ + Pl^ 



...(91), 

 and 



2 F= hq^ + V2' + . . . + b,,q^ + 2(/3i^i + /32^2 + . . . + /3.^J^ + (2^2 



...(92), 



respectively. Lagrange's method then gives n equations 

 of the type 



as^^b^^^raS,^^^l, = o . . (93), 



together with 



/'^+(2US.(ai, + /3,^,)=X . . (94), 



where X is the extraneous force corresponding to the 

 coordinate S, viz., the difference of tensions on the two 

 sides of the interruption. If we assume that all our 

 functions vary as ^*^'^^, we have, from (93), 



