Longitudinal Modes of Elastic Waves in Isotropic Cylinders 



and Slabs 



By A. N. HOLDEN 



The general properties of the longitudinal modes in cylinders and slabs are de- 

 veloped with the aid of the close formal analogy between the dispersion equations 

 for the two cases. 



1. Introduction 



THE classical exact treatments of the modes of propagation of elastic 

 waves in isotropic media having stress-free surfaces but extending 

 indefinitely in at least one dimension are those of Rayleigh^ for semi- 

 infinite media bounded by one plane, of Lamb^ for slabs bounded by two 

 parallel planes, and of Pochhammer^ for sohd cyhnders. Rayleigh showed 

 that a wave could be propagated without attenuation parallel to the sur- 

 face, in which the displacement amphtude of the medium decreased expo- 

 nentially with distance from the surface, at a velocity independent of fre- 

 quency and somewhat lower than that of either the plane longitudinal or 

 plane transverse waves in the infinite medium. Such ''Rayleigh surface 

 waves" have received appUcation in earthquake theory. 



For slabs or cyhnders the treatments lead to a transcendental secular 

 equation, estabhshing a relation (the "geometrical dispersion") between the 

 frequency and the phase velocity, which for some time received only asymp- 

 totic appUcation in justifying simpler approximate treatments. The past 

 decade, however, has seen a revival of interest in the exact results'*- ^ stimu- 

 lated by experimental appUcation of ultrasonic techniques to rods^- ® and 

 slabs,' by the use of rods and the like as acoustic transmission media, and 

 perhaps by curiosity as to what quaUtative correspondence may exist be- 

 tween such waves and the more intensively studied electromagnetic waves 

 in wave guides. That this correspondence might not be close could be antici- 

 pated l^y observing that an attempt to build up modes by the superposition 

 of plane waves in the medium reflected from boundaries would encounter 

 an essential difference between the two cases: the elastic medium supports 

 plane waves of two types (longitudinal and transverse) with different veloc- 

 ities, and reflection from a boundary transforms a wave of either type into 

 a mixture of both. 



On grounds both formal and physical it may be expected that solutions 

 to the equations of smaU motion of the medium with a stress-free cyUndrical 

 boundary can be found with any integral number of diametral nodes of the 



956 



