66 Prof. C. Gr. Knott on Ttrfleccion and Refraction of 



may be compared, so far as direction of motion is concerned, 

 to waves of light in the luminiferous aether. In the con- 

 densational wave the vibrating particles move to and fro 

 in lines parallel to the direction of motion of the wave. In 

 the distortional wave the particles move to and fro in lines 

 perpendicular to the wave's direction of motion. 



In all cases these two types of wave are propagated with 

 different velocities, which depend upon the density and the 

 elastic constants of the material. For an isotropic elastic 

 solid there are two independent elastic moduli, known 

 respectively as the bulk -modulus, or resistance to com- 

 pression, and the rigidity, or resistance to distortion. The 

 velocity of the distortional wave depends on the ratio of the 

 rigidity to the density. The velocity of the condensational 

 wave, however, is not so simply related to the other modulus, 

 but depends for its value upon the rigidity as well. 



Take, for example, a uniform cylindrical rod of iron. By 

 giving the one end of this rod a slight twist we may set up a 

 series of torsional vibrations, whose velocity of propagation 

 along the rod is to be measured by the square root of the 

 ratio of the rigidity to the density. The velocity of pro- 

 pagation of longitudinal vibrations, which may be supposed 

 to be given by an impact on the end, is to be measured by 

 the square root of the ratio of the so-called Young's Modulus 

 to the density. Young's Modulus is a definite function of 

 the principal moduli already mentioned, being given by the 

 formula 



2nk/(3k + 7i), 



where k is the resistance to compression and n is the 

 rigidity. 



Again, if we consider the case of plane waves in an infinite 

 solid, we find that here also the velocity of propagation of 

 the distortional wave is given by the ratio Vnj s/ p ; while 

 that of the condensational waves is measured in terms of a 

 mixed modulus which is not necessarily the same as Young's 

 Modulus. Its value is # + fn, which is equal to Young's 

 Modulus only if 3£ = 2n. 



According to Navier's and Poisson's theory of elasticity 

 we should have ?>k = 5n. This is usually expressed by saying 

 that, when a bar is stretched under a longitudinal pull, its linear 

 contraction at right angles to the pull is one quarter of the 

 elongation in the direction of the pull. So far as experi- 

 ments with hard metals go, this ratio may vary from '2 to *4. 

 Nevertheless '25 may be taken to be a pretty fair mean 

 value. 



