Mastic Waves, with Sehmological Applications. 91 



and in which p is the density, n the rigidity, and (m — £n) the 

 bulk-modulus or resistance to compression. 

 The stress-components have the values 



P = (m + n)V 2 4>-2/>!^ S=n|^ 



Q^m + iOV^-an!!, T=»|? j (II.) 



/ Q B 2 <£ 3V <3V\ 



where P, Q, R, S, T, U have the same meanings as in Thomson 

 and Tait's ' Natural Philosophy.' 



The components of stress on the plane whose normal has 

 direction-cosines A, /£, v, arc 



F=PX+U/*fTi/^ 



G=U\ + Q/*+Sv (III.) 



H=T\+SH-Ev j 



The waves of the <f> type are condensational-rarefactional 

 waves travelling with a speed equal to \/(m + n)/p. The 

 waves of the y\r type are purely distortional waves travelling 

 with speed \/n/p, the vibrations being in the plane XY. 



The £ displacement belongs also to a purely distortional 

 wave, the vibrations being at right angles to the plane XY. 



Let the plane interface between two media, mnp and m'n'p', 

 be perpendicular to the X-axis, then any incident wave of the 

 £ type will, in general, break up into two waves, a reflected 

 wave and a refracted wave, of the same type. 



But any incident wave of the <j> type will, in general, break 

 up into four parts, two distortional (reflected and refracted) 

 as well as two condensational-rarefactional (reflected and 

 refracted) . 



Similarly any incident wave of the yjr type will also, in 

 general, break up into four parts, two condensational-rare- 

 factional as well as two distortional. 



The necessity for this duplication of reflected and refracted 

 waves may be easily shown by a simple consideration of the 

 boundary conditions which must be satisfied at the interface. 

 Even in the simple case of a solid bounded by an im- 

 passable barrier, we must assume the two reflected waves as 

 derived from the one incident wave, or we encounter an 

 absurdity. 



