960 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Hence, denoting / = jS/a, the phase velocity c = w/y is given by 



where £ is an "effective stiffness", a function of the elastic constants 

 and /. 



Since jS = is a trivial root of equation (3), it can be divided by j8, and 

 the expression on the left then becomes even in both a and /5 and (3) can be 

 regarded as an equation in a^ and jS^. From (4) and (5) it is evident that, 

 if CO and y are both to be real, o? and ^"^ must be real and must obey the ine- 

 quaHties 



)S2 > a\ M/32 > (2m + X)c^, (7) 



and thus the root /3 = a can be neglected. The general character of the 

 desired roots can consequently be exhibited on a plot of ^ against o?. Evi- 

 dently on that plot lines of slope unity are lines of constant frequency 

 (equation 4), and lines radiating from the origin are Unes of constant veloc- 

 ity (equation 6). As will appear later, however, it is more convenient to use 

 a Unear rather than a quadratic plot, real a being measured to the right, 

 imaginary a to the left, of the vertical axis, and real jS upward, imaginary 

 j8 downward, from the horizontal axis. Here radial lines are still lines of 

 constant velocity, but lines of constant frequency are no longer simple. 



In Fig. 2 such a plot has been sketched for the first few modes of a mate- 

 rial obeying the Cauchy condition X = /*; the properties shown are restricted 

 to those derived in the following paragraphs, and are lettered in Fig. 1 to 

 correspond with those paragraphs. 



(a) By virtue of (7), the significant portions of the roots lie above and 

 to the left of the lines ^ = a^ ^l^ = (2/^ -f \)o?. Setting /z/S^ = (2;x -f X)^^ 

 in (3) reveals the cut-offs at sin /8 = and at cos a = 0: in other words at 



/32 = «V, c^ = — ^ «V, and also at 0" = '^±±1 (n + -\ ir\ 

 2/i 4- X M \ 2/ 



/ iv 



o? = V^~^9/ ^» where n is any integer. 



(b) Setting a = in (3), it can be seen that the roots intersect the line 

 a^ = at the points sin /3 = 0. By calculating the derivative of ^ with re- 

 spect to o?j those points (at which a changes from pure real to pure imaginary) 



dQ 

 can be shown not to be multiple points, and the branches to have 3- = 



da 



and •Tr4s — ^rs > independent of branch number and negative for 



d\fi^) X^ 



