12 Lamb, Waves in a Medium with Periodic Structure. 



require to know the critical values of ka^ viz., those which 

 make 



Q,Q's>ka-\nka%\Xika= ■\:_\ . - . (67). 

 This equation requires that either 



^\wka — o . . . (68), 



or 



cot \ka = \iLka . . , (69), 

 or 



tan \]ia = - ^ytka . . (70). 



The roots of (69) and (70) are easily constructed 

 graphically by means of the intersections of the curves 



y = co\.\x, y= -\.2i\\\x . . (71), 

 (where Ji:=kd) with the straight line 



y = ifix .... (72). 

 The positive roots of {6S) are given by 



ka = O, TT, 27r, 37r, 



We will denote the positive roots of (69) by 



ka = ai, as, ag, , 



and those of (70) by 



ka = l3i, /32, A, 



It appears on reference to the figure that the positive 

 roots of (6y), when arranged in ascending order of 

 magnitude, are given by the first line of the following 

 scheme : 



ka = Oj ai, TT, /3i, 27r, aa, 37r, /^g, ... ) /^ n 



6 = 0, TT, TT, O, O, TT, TT, O ... j * \^jJ> 



and it is easily ascertained that the equation in question 

 has no multiple roots. Hence each of the roots in (73) 

 marks a transition from partial transmission to total 

 reflection, or vice versa. The upper brackets indicate the 

 ranges of ka for which there is partial transmission, and 

 the lower ones mark the intervals of total reflection. 



If the masses M be increased, the straight line 



