958 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



and Hueter* has used graphical methods. The alternative adopted here is 

 a semi-quantitative treatment, assisted by extensive reference to the be- 

 havior of longitudinal waves in slabs,* for which the secular equation is 

 simpler and closely analogous. The analogy in the case of flexural modes is 

 considerably less close and will not be discussed. 



The general consequences of the inquiry are that the higher longitudinal 

 branches have phase velocities which are not necessarily monotonic func- 

 tions of frequency. With increasing frequency, however, those velocities all 

 approach that of the plane transverse wave,** not that of the Rayleigh sur- 

 face wave (nor that of the plane longitudinal wave, as some investigators 

 had guessed), a fact reflected perhaps in the experimental observation that 

 driving a rod transversely usually provides purer transmission than driving it 

 longitudinally. t Variation of nodal cylinders in location and number with 

 frequency persists in the higher branches. 



2. The Slab 



The slab extends to infinity in the y, z plane and has a thickness 2a in the 

 ^-direction. The displacements of its parts in the x, y, z directions are w, v, 

 w. Its material has density p and Lame elastic constants X and /x, so that 

 its longitudinal wave velocity is ■\/{2y. + X)/p and its transverse wave veloc- 

 ity is y/n/p. That ju should be positive is a stability requirement of ener- 

 getics; X will also be taken as positive since no material with negative X is 

 known. 



The equations of small motion are, in vector form, 



(2/x -j- X) grad div (w, v^w) — ji curl curl (w, v, w) = p —z («, v^ w). 



or 



Solutions representing longitudinal waves propagated in the z direction can 

 be of the form 



where U is an odd function, and W an even function, of x alone, co is the 

 frequency in radians per second, and 7 = co/c where c is the phase velocity. 

 Solutions independent of y are chosen here because they provide the simplest 

 analogues to the case of the cylinder. Substitution shows that U = Ae "" ^ 



* I am indebted to Dr. W. Shockley for the suggestion that this behavior might dis- 

 play a close enough analogy to that of the cylinder to provide insight; the work of Morse 

 bears out the analogy. 



** The fact is noted by Bancroft. 



t Private communication from H. J. McSkimin. 



