8 Lamb, Waves in a Medium with Periodic Structure. 



It appears that the amplitude of the reflected wave is now 

 equal to that of the incident wave. This might have 

 been anticipated from considerations of energy, since (33) 

 shews that for sufficiently large values of s the disturbance 

 to the right of the origin is insensible. The same result 

 will hold, of course, when iQ is replaced hy u — iir. 



3. The next application of our formulae is suggested 

 by the optical problem : to determine the intensities of 

 the light reflected and transmitted respectively by a plate. 

 We will suppose that to the left of the origin the string is 

 unloaded, that a series of masses M are attached at the 

 .points 



x = Oj a, 2«, 3«, , na, 



and that to the right of the last-named point the string is 

 again unloaded. 



It may be sufficient to indicate the leading steps of 

 the calculation. For x<io^ the disturbance may be 

 regarded as made up of an incident wave 



i=/^(rf-x ) . . . (37), 



and a reflected wave 



as before. For x^na, we have a transmitted wave 



The formulae applicable to the loaded portion of the 

 string will depend on the value of ka. If the value of 

 defined by (11) be real, we have 



The kinematical conditions to be satisfied at x=o and 

 x=na are, respectively, 



i+^ = C + Z) . . (41), 



and 



