LONGITUDINAL MODES IN CYLINDERS AND SLABS 965 



one slab thickness (which is proportional to the frequency), and ~~, the 



Co 



ratio of the velocity to that of a plane transverse wave. Evidently 

 2 _ fccaV _ 1 2m + X,^2 2^ AV (2m + X)(/'-1) 



_{coaV 1 2M + X..2 2. fcV 



nP - (2m + X) ' 



where Co = 'x/fi/p. 



For completeness, the lowest branch will be briefly sketched: that which 



originates at a = /8 = 0. A calculation of -j^^ at that point yields only one 



non-trivial root, — (2 /z + X)(2/i + 3X)/X2, and thus the phase velocity at 

 low frequencies is found to correspond, as would be expected, with that 

 given by the stiffness (a semi-Young's modulus, so to speak) of a material 

 displacement-free in the ic-direction but not in the 3;-direction, E = 

 4/xOu + X)/(2 /i + X). Since lines radiating from the origin of the (/S^, a^) 

 plot are lines of constant velocity, the dispersion curve for this branch 

 starts with zero slope. The root curves over, intersecting the line 



X/32 + (2m + \W = at /3 = |, and intersects the line jS^ = again at 



A^ (a = iA) where (2^ + X) ^4 cosh A = 4(m + X) sinh A. For large 

 negative o^ and /S^, equation (3) approaches 



\H* + 4m(m + X)/3 -f 2X(2m + \)P - 4(2m -f X)(m + \)l + (2m + X)^ = 0, 



which after discarding the trivial root 1=1 leaves the Rayleigh cubic. In 

 the case of this one branch, the phase velocity approaches its as3anptotic 

 value at high frequencies from below, and hence the dispersion curve must 

 have an odd number of maxima and minima, and in particular at least one 

 minimum, as was discovered by numerical calculation for the corresponding 

 branch in the case of the cylinder by Bancroft.** 



The complicated behavior of the displacements in the higher'branches is 

 sufficiently illustrated by a brief consideration of their nodes: the values of 

 X at which the displacement is entirely along or entirely across the slab. 

 From (1) and the boundary conditions, the rr-dependence U of the displace- 

 ment component perpendicular to the slab will be given by 



Ki U = {\^^ + (2m + \W) sin ^ sin — 



a 



(la) 

 + 2{jjl(^ - (2,1 + X)a')sin a sin ^ 



