4 Lamb, Waves in a Medium with Periodic Structure. 



In the alternative case we put 



Q,o^ka-\ixka^\'!\ka= ±_Q.o^u . (14), 

 where u is real, and obtain 



or 



4, = ^/^'^'+^^^ + ^/^^'-^^ . . (15), 



4 = (-)^{^/^'^'+^^^ + ^/^^^-^^^} . (16), 



according as we take the upper or the lower sign in (14). 



The two types of disturbance represented by (13) 

 and (15) or (16) are essentially different in character. 

 Either term of (13) represents a disturbance whose 

 amplitude is the same at all parts of the medium, whilst 

 in the case of (15) or (16) the amplitude of either com- 

 ponent diminishes indefinitely as we advance in the 

 medium in one direction or the other. In the former 

 case, the disturbance can be transmitted to any distance ; 

 in the latter case, this is not possible. A fuller discussion of 

 this question will be given presently (§ 2). 



In the present problem (but not in all the variations of 

 it, see § 5), will be real for values of ka falling below a 

 certain limit, and the disturbance will then be of the 

 type (13). Moreover, for very long waves, ka and ^ will 

 both be small, and we find, approximately, 



^=^{i^^).ka . . . (17). 



The disturbance represented by either term of (13) 

 may now be described as a wave travelling in the negative 

 or positive direction of x. If the wave-length (A) in the 

 loaded medium be m times the interval «, where m is a 

 large number, we have md = 27ry and therefore 



\=2TvalQ = kalQ.\ . . . (18), 



where Ao = 27r//^. Hence, for the refractive index (A^) of 

 the medium, we have 



as we should expect, since the effect of the loads is to 



