968 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



o^ = at the points Ji{l3) = 0, traversing them with -^^ = -4m(m + X) 



a{a-) X^ 



or half that of their analogues. Again it is only at those points that the 

 phase velocity has the value 's/(2/i + X)/p- 



(c) Setting /i(/3) = reduces (9) to a/30xi8' - {2fi + \)a'')Ji{a) = 0, and 



hence the roots are confined between the horizontal lines /i(j8) = in the 

 region of positive 13^ and negative a^, and the velocity is asymptotic to 

 \//x/p with increasing frequency. They meet the line XjS^ + (2n + \)c^ = 

 at the points /3/o(/3) — Jiifi) = 0, or in other words at the maxima and 

 minima of Ji(0),* where again the phase velocity is \/2ju/p. 



(d) In the case of the cy Under confining lines** are 



since substitution shows that intersection of (9) with these lines would re- 

 quire 



40i 4- X)a^(M^' - (2m + \)a) . ,. . 

 (Xi82+ (2M + X)aT •'^^"^ 



which cannot be satisfied by permitted values of a^ and jSl 



(e) This suggests that in that region the roots may oscillate about the 

 lines 



J.MfM + A(^) [j'M + ,(,/f(;;\)„, -^.(«)] = 0. 



In fact, in view of the equivalence fi{x) = Jo{x) Ji{x), those lines 



X 



can be seen to have points in common with the roots of the secular equation 

 at Ji{a) = 0, Ji(j3) = 0, and at 



(f) Substituting the expression for the lines of (e) into (9) shows that 

 again additional intersections may be afforded by suitable roots of the 

 cubic equation (8). 



* The analogy to the corresponding intersections for the slab at the maxima and 

 minima of sin /3 is noted by Lamb, ref. 2, p. 122, footnote. 



** There are infinitely many such families of lines but none carries the analogy with the 

 slab to the point of being independent of the elastic constants. The families used in (d) 

 and (e) serve the present purpose as simply as any. 



