14 Lamb, Waves in a Medium with Periodic Structure. 



vibrations becomes less and less. Moreover, it is easily 

 seen that this tendency is more marked, the greater the 

 value of ju. » 



5. As a first variation of the problem stated in § i we 

 will suppose that each of the masses M^ in addition to the 

 forces which it experiences from the string on the two 

 sides of it, is urged towards its mean position by means of 

 a spring. 



The equation of motion of the s\k\ particle is then of 

 the form 



M% = ^^K^.~2K^o^ka^l,_^)-MaX . (76) 



or since 5 =<: e"^^^^ 



4+1 - 2^1 cos ka - -^lika - — 2- • 7- j sin ka\-\- 4_i = o (77). 



The constant o- here denotes the ' speed ' of the oscillation 

 when J/ vibrates under the influence of the spring alone. 



The expression whose value determines the character 

 of the motion is now 



la-^-J^m-""-^ .-^ivci'ka . . (78). 



If this lie between the limits + 1 we have in the problem 

 of § 2 a partial transmission ; in the opposite case a total 

 reflection. 



The limiting value of (78) for ka = o is 



and therefore greater than unity. It follows that for 

 frequencies below a certain limit, or for lengths of the 

 incident waves exceeding a certain limit, we shall have 

 total reflection. The fact that in a medium of the kind 

 here considered there is an upper limit to the length of 

 waves which can be transmitted is very remarkable. 



