LONGITUDINAL MODES IN CYLINDERS AND SLABS 969 



(g) In the region a = iAy the analogous lines are given by 



iMiA)m + M,) [aha) - ^,J^li\,^.^ iMiA)] = 0, 



with which intersections occur for the same values of ^/A as in the slab. 



These results pennit visualization of a counterpart to Fig. 1 for the cylin- 

 drical case. In it the critical Hnes radiating from the origin are the same. 

 The horizontal Hnes, instead of being evenly spaced by 7r/2, are spaced as 

 the zeros, maxima, and minima of Ji(j3). The vertical hnes, again no longer 

 evenly spaced, are replaced alternately by straight vertical hnes /i(a) = 



2(m + \)a 

 and by the curved "vertical" hnes Joia) = —;—, — jz — rm JiM which 



X/3^ + (2ai -t \)ot 



He between their straight companions and approach /o(«) = as /3 be- 

 comes large. FinaUy the confining Hnes, and the Hnes about which the 

 branches oscillate, become the curves defined in (d) and (e), which can be 

 seen to foUow a course not dissimilar to the diagonal course of their prede- 

 cessors, passing through the intersections of the new horizontals and 

 verticals. 



Again the branches do not intersect, except for pair-wise coincidence of 

 cut-offs on one or another of the Hnes (d) when the elastic constants obey 

 special relations. Thus the dispersion curves to which Hudson^ assigns 

 certain of Shear and Focke's data cannot be taken (and indeed Hudson 

 does not suggest that they must be taken) as corresponding to higher 

 branches of the longitudinal modes, since the former curves intersect one 

 another, and the latter cannot unless anisotropy modifies their behavior 

 quahtatively. The assignments could represent modes other than longitu- 

 dinal. The more recent results of Hueter^ show essentiaUy the behavior of 

 Fig. 3. 



In view of the closeness of the analogy thus revealed, it may be taken as 

 probable that quahtative correspondence wiU obtain quite generally be- 

 tween the longitudinal modes of the slab and those of the cyHnder. 



References 



1. Lord Rayleigh, Proc. Lond. Math. Soc. 17, 4 (1885). 



2. H. Lamb, Proc. Roy. Soc. Lond. A, 93, 114 (1917). 



3. L. Pochhammer, /. reine angew. Math. (Crelle) 81, 33 (1875). 



4. D. Bancroft, Phys. Rev. 59, 588 (1941). 



5. G. E. Hudson, Phys. Rev. 63, 46 (1943). 



6. S. K. Shear and A. B. Focke, Phys. Rev. 57, 532 (1940). 



7. R. W. Morse, //. Acous. Soc. Am. 20, 833 (1948). 



8. A. E. H. Love, "Mathematical Theory of Elasticity," Cambridge 1927, 4th Ed., p. 



287. 



9. T. F. Hueter, Jl. Acous. Soc. Am. 22, 514 (1950); Zeit. angew. Phys. 1, 274 (1949). 



