Forces on an A.C.V. Executing an Unsteady Motion 



<P 



. . cosh \ k(z + d) f 

 (w, u;z, t) = ' v ,,. ,/' 



. . . + 



p . cosh(kd) 

 sin (7 t) 



t 



/ P(w, u; t ) • cos|-y,(t -rHdr 

 L 

 P(w,u;0) 



We express P by means of the Fourier transform, and then the 

 inverse Fourier transform is taken : 



t 00 00 



<M £ »y. z. t) = 



bzff'H. 



S' -co 



dw I du p U'.y*, r) 



cosh I k(z + d) f ( »/~ , ' ., , x , v ) 



T-TTTx — •coslVgk. tanh(kd) ' (t - t)\ 



cosh (kd) 



exp{i(w(£ - *'-|+u(y -y'))} + — ^2-//dS' / 



00 S' -°° 



/^i,, J 3 / £' , r i n\ cosh{k(z + d) } 

 dup U ' y ' 0) cosh(kd) 



— 00 



sin(\gk'tanh(kd)'t) 

 -y/gk*tanh(kd) 



dw 



{i(w(« - n + u(y-y'))| (21) 



5 5 



3 = p 



and y' are dummy variables in the stationary reference frame 



where p 3 = p J (^'.y', t ) , defined over the area S' , while £' 



Eq. (21) is the potential for an arbitrary time-varying 

 pressure distribution starting at t = . Thus the solution for the 

 general motion of an ACV is obtained. In the following sections, we 

 shall consider special motions of a pressure distribution which is 

 non-time varying with respect to axes rigidly attached to the vehicle. 



III. RECTILINEAR MOTION IN HORIZONTALLY UNRESTRICTED 

 WATER 



III. 1. The Potential 



We now consider motion of a craft starting from rest at 

 t = . The expression for the potential, Eq. (21), may be simpli- 

 fied by partial integration of the five-fold integral with respect to r : 



43 



