Manchester Memoirs, Vol, xlii. (1898), No. 3. 



III. On Waves in a Medium having a Periodic 

 Discontinuity of Structure. 



By Professor HORACE Lamb, M.A., F.R.S. 



Received Decej?tber 6th. Read November soth^ iSgy. 



The main object of this paper is to illustrate the 

 selective total reflection which takes place at the boundary 

 of a medium of the above constitution. The case where 

 the periodic variation of properties is continuous has been 

 discussed by Lord Rayleigh* but the problem in this 

 form is very difficult, and it may therefore be worth while 

 to investigate in some detail the consequences of the 

 periodic structure in the more tractable circumstances 

 here considered. In the examples chosen for discussion the 

 medium is represented by a string supposed to be capable 

 of longitudinal vibrations, and the periodic interruption of 

 properties consists, in one case, of a series of equal 

 particles attached at equal intervals ; in another, these 

 particles are (moreover) urged towards fixed positions by 

 elastic springs ; whilst in a third example they are sup- 

 posed to be connected to the string (not directly but) 

 through the intermediary of springs. It is further shown 

 how the methods can be extended to the case where equal 

 and similar dynamical systems, of any degree of com- 

 plexity, are inserted at regular intervals. Exactly the 

 same analysis applies when the intervening medium con- 

 sists of a tense string capable of transversal \''^x2l\ao'!\s, and 

 to many other cases. 



Experimental illustrations of the principles here 



* Phil. Mag., Aug., 1887. 

 February ^th, ii 



