118 DYNAMICAL THEOEY OF SOUND 



lateral contraction adjusts itself instantaneously through the 

 thickness. This is not quite exact, as there is a certain degree 

 of lateral inertia, but the error is insignificant so long as the 

 wave-length is large compared with the diameter. In the 

 modes of very high order it might become sensible, but these 

 are in any case of no importance from the point of view 

 of acoustics. ~~ A correction has been investigated by Lord 

 Rayleigh. 



44. Plane Waves in an Elastic Medium. 



The theory of plane waves in an unlimited isotropic elastic 

 medium is so closely analogous to that of longitudinal waves 

 in a rod that it may be briefly noticed here. It is assumed 

 that the state of things is at any instant uniform over any 

 plane perpendicular to the direction of propagation (#). 



Such waves may be of two types, which are distinguished 

 as " dilatational " or " longitudinal," and " distortional " or 

 " transversal," respectively. In the former class the displace- 

 ment is wholly in the direction of propagation. Denoting it 

 by f, we have, in the notation of 42, 



6! = dg/diK, 6 2 = 0, 3 = 0, 



and therefore p l = (\ + 2/i) l = (K + f /*) d/dx .......... (1) 



Considering the portion of matter corresponding to unit area 

 of a stratum of thickness 8x, we have 



whence = 



< 2 > 



if tt = (* + iri)lp ...................... (3) 



Some numerical values of the wave-velocity a are given on 

 the next page, and it will be observed that they are in all 

 cases greater than the corresponding values of c, as was to be 

 expected, since the potential energy due to a given extension 

 8f/3# is now greater owing to the absence of lateral yielding. 



In the second type of plane waves the displacement is 

 everywhere at right angles to the direction of propagation. 

 It may be resolved into two components parallel to y and z, 

 respectively, which may be treated separately. Considering 



