LONGITUDINAL MODES IN CYLINDERS AND SLABS 963 



is not feasible to exhibit the roots of the Rayleigh cubic explicitly for arbi- 

 trary fi and X, it is of some interest to exhibit its discriminant 



Z) = ^ \\2n + X)'(m + X)'(11X' + 4XV - 9Xm' - 10m'), 



and to note that for real positive values of X and n it changes sign only once, 

 at approximately X//x = 10/9. Hence for \/fx > 10/9 two roots of the Ray- 

 leigh cubic are complex, while for \/ii < 10/9 two roots are real and nega- 

 tive. For a material obeying the Cauchy condition X//i = 1, the roots of 

 the Rayleigh cubic are —3, —3 db 2\/3; thus I = 3^ 3 + 2\/3, both of 

 which obey the inequalities (7), are relevant to intersections of each branch 

 of the roots of the secular equation with the hne about which it oscillates. 



(g) The results of (e) and (f) suggest the value of a similar investigation 

 in the region of imaginary a. Here (denoting a ^ iA and L = ^/A where 

 A is taken positive and real) intersections occur between the branches and 

 the Hnes sinh A cos ^ — cosh A sin ^ = when (XL^ — (2^ + X))^ = 

 4(jLi + \)L{p.D -\r (2m + X)). Clearly this quartic in L has two and only 

 two positive real roots, one greater and one less than \/(2m + X)/X. In the 

 case X = Mj those roots are approximately 9 and 1/3. This information, 

 taken with that of (c), establishes that the branches are confined in the 

 region of imaginary a to bands determined by wx < j8 < (w + J)7r, having 

 one tangency to the Hnes /3 = (« + Dtt; and that at values of A greater 

 than correspond to the smaller root of /, the branches lie in the bands mr < 

 /?<(»+ i)7r. 



It is convenient to obtain assurance that in general the branches do not 

 intersect at any point by noting that the confining lines of paragraphs (c) 

 and (d) define bands within each of which in general one and only one cut- 

 off point falls. Pivoting a ruler about the origin of Fig. 1, and recalling the 

 cut-off conditions, avails. Degenerate cases arise when the elastic constants 

 satisfy a condition 2m + X = «V where n is an integer; in those cases some 

 cut-off points coincide in pairs on some confining lines. Calculation of de- 

 rivatives at those points shows that the cases are not otherwise exceptional: 

 the pair of roots forms a continuous curve which is tangent to the cut-off 

 Hne at the double cut-off point. 



From (6) it follows that the phase velocity will have a maximum or a 



minimum with frequency if -jrr-r. = -5 . That condition requires 



. 2 ^ _ (X/3^ + (2m + xWfi'K'^' + (2m + X)(2m + S\W) 

 ^^"^ " 4(m + X)(2m + X)V(^2 _ ^2) (^^2 « (2^ 4. x)«2) • 



