964 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



In the region of positive ^' and a^ and of the inequalities (7), this is impos- 

 sible, but when a^ is negative the condition may be satisfied. If it is satis- 

 fied in the higher branches, however, it must be satisfied an even number 

 of times in any branch, so that the branch exhibits as many maxima as 



3.0 



1.5 2.0 2.5 3.0 3.5 



r( PROPORTIONAL TO FREQUENCY) 



5.0 



Fig. 3 — The beginnings of the dispersion curves inferred from Figs. 1 and 2. The solid 

 shaded lines are the curves about which the branches oscillate, intersecting them at the 

 triangles and lying on the shaded side of them elsewhere, and the dashed extensions are 

 the branches themselves. For increasing t these branches all become asymptotic to the 

 base line. The dashed lines at the top are the true cut-off frequencies; the solid "cut-off" 

 line^ are the asymptotes of the shaded curves. The beginning of the lowest branch is 

 shown at the lower left; it becomes asymptotic to a line below this plot. 



i-^/'- 



2 



after passing through a shallow minimum. 



minima, for clearly the phase velocity is a decreasing function of frequency 

 near the cut-off, and the velocity can also be shown to approach its asymp- 

 totic value at high frequencies from above in the higher branches. 



In finally displaying the dispersion curves (Fig. 3) it is convenient to use 

 as reduced variables r, the number of plane transverse wave lengths in 



