LONGITUDINAL MODES IN CYLINDERS AND SLABS 957 



component of displacement along the rod, as well as for the "torsional" 

 modes in which there is no displacement along the rod whatever. The clas- 

 sical results are for no such nodes, the "longitudinal" (or "elongational") 

 modes, and for one such node, the "flexural" modes. The secular equation 

 for modes with any number of such nodes has been exhibited by Hudson.^ 

 For any one of these types of mode, it may be expected that the secular 

 equation will define a many-branched relation* between frequency and 

 phase velocity, and that a different number of interior cylindrical nodal 

 surfaces for the displacement components might be associated with each 

 branch. Apart from the relatively simple torsional modes, the only branches 

 whose properties have been intensively studied are the lowest branch of 

 the longitudinal'* and the lowest of the flexural^ modes, because they (and 

 the lowest torsional branch) are the only ones extending to zero frequency, 

 the others exhibiting "cut-off" frequencies at which their phase velocities 

 become infinite and below which they are rapidly attenuated as they prog- 

 ress through the medium. 



Three qualitative results of these studies are of especial interest. In the 

 first place, with increasing frequency the phase velocity in the lowest lon- 

 gitudinal and flexural branches approaches the velocity of the Rayleigh 

 surface wave, and the disturbance becomes increasingly confined to the 

 surface of the cyUnder. In the second place, the dispersion is not monotonic 

 as it is in the electromagnetic case: the phase velocity exhibits a minimum 

 in the lowest longitudinal branch* and a maximum in the lowest flexural 

 branch^ with varying frequency. Finally, in the lowest longitudinal branch 

 at least, the cyUndrical nodes of the displacement components vary not only 

 in radius but even in number with the frequency.* 



The last result suggests that it would be difficult in practice to drive a 

 cyUndrical rod in that pure mode represented by its lowest longitudinal 

 branch over any extended frequency range, since it is difficult to visualize 

 a driving mechanism having suitable nodal properties. Longitudinal drivers 

 which can be readily constructed may be expected to deUver energy to all 

 longitudinal branches, in proportions varying with frequency. How satis- 

 factory such a transmission device could be would depend importantly on 

 how much the phase velocities at any one frequency differed from branch 

 to branch. 



This paper sketches the behavior of the higher longitudinal branches. 

 That behavior could, of course, be determined exactly; Hudson^ has shown 

 how the calculation of the roots of the secular equations can be facilitated, 



* This is true in particular of the flexural type of mode, and in his otherwise excellent 

 treatment of flexure Hudson's statement to the contrary must be disregarded. Recent 

 writings in this field have tended to distinguish as "branches" what in allied problems 

 are commonly called "modes". 



