Doctors 



and 



Q' = Q (w,u,t) . 



It is convenient to calculate the P and Q functions using 

 an axis system x y z that rotates with the craft rather than the xyz 

 system, in which the x axis lies in the direction of motion. This is 

 illustrated in Fig. 12. The yaw angle e (t) is taken positive for clock- 

 wise rotation of the craft, when looking down on it. If w* and u* 

 are the induced wavenumbers relative to these craft axes, then 



w*(t) = w cos] e (t) } - u sin je (t) J = k cos J +e (t)[ 

 and 

 u* (t) = w sin j « (t) ( + u cos j e (t) } = k sin | + « (t) ( . (47) 



Then it may be shown that 



P(w 

 Q(w 



S 

 analogous to Eq. (26). 



,U,t)_ / */* -frvCOS/** * *\ , # j # /„ > 



,u,t) " / P ^ ' Y ' sin (W X Y ' Y ' ( ; 



For the pressure distribution given by Eq. (5), it immediately follows 

 from Eq. (33) that 



„, v _ 7T*sin(aw*) 7r»sin(bu # ) 



lW,U ' J " P o a.sinh(7rw # /2a) -sinn^u */Z /3) 



and 



Q(w,u,t) = (49) 



We now consider a craft travelling at a constant velocity at 

 a fixed angle of yaw from time -T to , and then allowed to yaw 

 up to time t . The t integral in Eq. (46) for just the first phase 

 of the motion is 

 



I = / sinJ7^t - r)| • [(OP- - PQ') cos|wc(t -t )}+... 

 . . . + (PP 1 + QQ') sin)wc(t - r )( I dr 



54 



