REFLECTION OF DIVERGING WAVES 383 



would have if it were rotated about y through tt, its torque along y is the 

 same as that of the first, and that normal to it is equal and opposite. Hence, 

 if the gyrostats are properly oriented, the resultant torque will be parallel 

 to the displacement and the medium will be isotropic. The y component of 

 the opposing torque will be S(p cos- a. Thus if the B axes are uniformly dis- 

 tributed in space the total torque will be one third what it would be if they 

 were all parallel to the axis of the applied displacement. Hence if there are 

 .V gyrostats per unit volume the vector restoring torque T per unit volume 

 will be 



7 = — - ~ <p. (4aj 



The next step is to derive the wave equations for a medium having this 

 stiffness to rotation. If the vector velocity q is very small, 



VX9=2^, (5a) 



where ^ is a vector angular displacement of an element of the medium at 

 the point under consideration. 2<p plays a role analagous with that of the 

 dilatation in compressional waves. Then, from (4a) and (5a), 



where the generalized stiffness of the undisturbed medium, 



^V (C - A)' 2 ,. ^ 



r?o = j2 "^4 '''■ ^^^^ 



To get the companion equation, we interpret the torque exerted by an 

 element in terms of the forces it exerts on the surfaces of neighboring ele- 

 ments. Let the x axis Fig. 2 be in the direction of the torque TAx^ which is 

 exerted by the medium within the small cube. This very small torque can 

 be resolved into the sum of two couples, one consisting of an upward force 

 FyAx- on the right face and an equal downward force on the left one, and 

 the other of a leftward force FzAx:' on the upper surface and a rightward one 

 on the lower one. But, if there is not to be a shearing stress, Fy and F^ must 



T . 



be equal, and each equal to — . Thus a torque per unit volume T is equivalent 



T 



to a set of tangential surface forces per unit area of — each. 



Now consider the force exerted on an element by its neighbors, through 

 the adjoining surfaces. To take the simplest case, let T in Fig. 2 be every- 

 where in the x direction and independent of z but varying with y. Then 



