1 6 Lamb, Waves in a Medium with Periodic Structure. 



to the case oi ka = (ja\c^ which occurs always in an interval 

 of partial transmission. 



6. An instructive contrast to the state of things con- 

 sidered in § 5 is obtained if we suppose the masses M to 

 have an elastic connection with the vibrating string. In 

 the annexed illustration, each of the circles is meant to 

 represent a light rigid frame, attached to the string at 

 opposite ends of a diameter, and carrying a particle M 

 connected with it by springs. 



M#w | W i%imf m j^m 



Some addition to our previous notation is now required. 

 We will denote by 5^ the displacement of the point of 

 the string to which one of the masses is attached, and by 

 ?/ that of the corresponding particle. We have then 



M^^Ma\K-i:)-o . . (85), 



whilst, from the equilibrium of tensions, 

 IcE IcE 



Assuming that li oc e^^<'\ and eliminating g/, we find 



^s+i - 2^/ cos ka - j^ * _^^2^/ 2 sin ^«J + 4-i = o (87). 



For sufficiently small values of ka we now have trans- 

 mission, as in § 2. The critical values of ka are given by 



, \ixka . , 



cos ka - J li^^^ifP' sin ^^ = ± I . (88). 



This is satisfied by sin/^^ = o, and the remaining roots are 

 determined by the intersection of the curves (83) with the 

 curve 



This has an asymptote parallel to j^ iox x=aa\c, or 



kc=(5. 



y--^ ■ ■ ■ (89). 



