REFLECTION OF DIVERGING WAVES 371 



where po is the constant density, rjo is a generaUzed stiffness of the undis- 

 turbed medium, given by (7a), q is the vector velocity, and T is the torque 

 per unit volume. In a plane wave q is normal to the direction of propagation. 



T . .... 



— is a tractive force per unit area m the direction of q, which acts on a surface 



normal to the direction of propagation. 



If, however, the amplitude is finite the equations become much more 

 complicated. For present purposes we need consider only waves for which 

 there is no component of velocity or torque in the direction of propagation, 

 and we need consider only plane polarized waves for which the direction of 

 the velocity is the same at all times and places. Also, as will appear below, 

 we are concerned with the equations which describe a wave of infinitesimal 

 amplitude which is superposed on a finite disturbance. This description need 

 cover only infinitesimal ranges of time and position. It can therefore be 

 expressed in terms of wave equations in which the constants of the medium 

 have local instantaneous values which depend on the finite disturbance. 



Subject to these restrictions it is shown in the appendix that (2) is to be 

 replaced by (23a) 



where Iq is a unit vector in the fixed direction of the velocity, and p is an 

 instantaneous local density, defined in terms of the finite disturbance by 

 (20a). And, in place of (3), (22a) 



where l^ is a unit vector in the direction of the axis of rotation, p is again 

 an instantaneous local density, c is an instantaneous local velocity derived 

 in the usual way from p and an instantaneous local stiffness ??, while / is a 

 function defined by the relation, (13a), 



T = -IM^P, 0. 



This function takes account of the fact that when the spin axis of the rotor 



is given a constant finite displacement, the restoring torque is not constant 



as in (1), but changes with time as the spin axis rotates toward the axis of 



displacement, and so reduces the component of the sjjin which is normal 



f>f 

 to the displacement axis and so is effective in producing stiffness. — 4 — 



dt 



represents the rate of this change in torque for a fixed angular displacement. 

 — 4— is to be interpreted as the rate of change of torque with angular 



