6 Lamb, Waves in a Medium with Periodic Structure. 



and that of the string immediately to the left is 



{- 1. -^ A)ikEe'^'^- 

 Hence the dynamical equation is 



^U kE 



(^1 - U cos kd) + ikEly I - A)eikct. (25), 



or, 



dt'^ sin ka 



/ A\ -u / ^o COS ^^ - ^1 



(26). 



Substituting the values of ?„> ^1 from (23), this becomes 



\-A=B 



cos ka - Ilka sin ka-e ^ 



i sin ka 



^^ 



e -co^ka 



(27). 



/sin^« 



where a reduction has been effected by means of (11). 

 Solving (24) and (27) for A and ^, we find 



29 sin-f ^<a!-0 ) 



e —e 



i^ —ika 

 e -^ sin- 



ika 



sin-(/^a + 0j 

 The amplitude of the reflected wave is therefore 

 sin-l ha-0\ 



(28). 

 (29). 



(30), 



and that of the disturbance transmitted to the particles 

 Mis 



sin ka 



sin 



\{ka^6) 



(31). 



For sufficiently long waves, ka and d are small, and 

 d = Nkay where N is the refractive index ; the expressions 

 (30) and (31) then take the forms 

 N-i . 2 



^+1 



and 



iV + i 



