REFLECTION OF DIVERGING WAVES 



387 



write 



T = -IM<P, t) 



(13a) 



where l^ is a unit vector in the direction of the axis of rotation. 



The derivation of the wave equation is much simpler if we consider only 

 the case of present interest where the direction of the rotation is everywhere 

 the same so that l^ is constant. Then (9a) can be written as 



V X 9 = hlFi 



dip\ dip 



dt dr 



(14a) 



and (13a) as 



T = -if(^, /). 



We wish now to replace — by — I - 

 dt dt \2 



These partial derivatives refer to a 



constant position so we are interested in the total time derivatives of T as 

 given by (12a). To get the desired relation we need to express T explicitly 

 in terms of (p and /, that is, we must evaluate (p. Since the variables are 

 small, we neglect their products of higher order than the third. Then 



cos ^ = 1 



where 



Putting 



1 



C - A 



j pdt\, 



C03 



(15a) 



T = —4770 cos I —- cos d dt, 

 J at 



in accordance with (12a) and substituting for cos 6 gives 



T 



■47J0 



ip — a(p 



\ <pdt -\- a <p- I ip dt) dt 



Then 



JT 

 ^ 



= —47/0 



1 - a 



f <pdt y^ - a^' \ ^dt 



When ip is constant the tirst term is zero, so the second term can be inter- 

 preted as the partial derivative of T with respect to /. Physically this de- 

 scribes the change in torque for a fixed displacement which results from the 



