962 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



all materials. At those points the phase velocity is that of a plane longi- 

 tudinal wave. 



(c) Setting sin i3 = reduces (3) to a^Qi^^ - {2n + \)a^) sin a = 0, 

 and hence in the region of positive /S^ and negative c^ the roots do not inter- 

 sect the horizontal Unes /3^ = wV (n f^ 0). As can be seen from (6) this 

 confinement imphes that the velocity is asymptotic to -x/ji/p, that of a 

 plane transverse wave, with increasing frequency. Notice that, in the re- 

 gion of positive jS^ and negative oi^, /3 takes the values Cos /3 = only where 

 XjS^ + (2)Li + X)«^ = and that the roots have zero slope there. At those 

 points the waves have a phase velocity \/2 times that of a plane transverse 

 wave. 



(d) In the region of positive jS^ and positive a^ the roots exhibit a some- 

 what more complicated behavior, but confining lines can again be found: 

 the diagonal fines ^ = {n + J)^ — a. It is the nature of such critical lines 

 as these which can be better exhibited on the linear than on the quadratic 

 plot. Alternatively those lines can be written cos a cos jS — sin a sin ]8 = 

 (and thus will be shown to have analogues in the case of the cylinder), and 

 substitution of this expression into (3) shows that if the roots intersect 

 these lines they must do so for values of a and (3 satisfying the relation 



40i + X)a|8(iUi82 - (2)u ^- X)^^) = -(XjS^ + (2/x + \)a'y cot^ a. 



But the inequahties (7) make such values impossible. 



(e) This suggests that in that region the roots may osciUate in a some- 

 what irregular manner about the diagonal fines /3 = wtt — a. Indeed it is 

 immediately evident that they pass through the points cos jS = cos a = 

 and sin /8 = sin a = 0. 



(f) Expressing those fines as sin a cos /S + cos a sin /8 = 0, and sub- 

 stituting into (3), shows that additional intersections may be afforded by 

 any roots of the quartic equation 



(X^ + (2m + X)a2)2 - 40u -I- X)a/30u^2 _ (2^ + x)o;2) = 



which obey the inequalities (7). Discarding the root a + /3 = 0, and dividing 

 by o^, yields the cubic equation 



X2/3 - {2n -h XyP + (2m + X)(2m + S\)l + (2m + \y = 0, (8) 



whose roots are the negatives of the roots of the cubic equation for the 

 Rayleigh surface wave velocity. It is weU known that the Rayleigh cubic 

 always yields one and only one significant positive root, and hence equation 

 (8) can afford at most two additional significant intersections of any root 

 of the secular equation with the fine about which it oscillates. Although it 



