Doctors 



II. BASIC THEORY 



II. 1. Problem Statement 



We represent the ACV by a pressure distribution p(x, y, t) 

 acting on the free surface, and travelling with the speed of the craft. 

 Two right-handed coordinate systems (reference frames) will be used, 

 and are shown in Fig. 2. A third coordinate system that rotates with 

 the craft during a yawing motion will be introduced later. The axis 

 system £yz is fixed in space, and the system xyz moves with the 

 craft, z being vertically upwards while x and £ are in the direc- 

 tion of the rectilinear motion. The relation between the coordinates 

 is then given by 



= « - s(t) 



= £ 



/ 



c( t) dr 



(6) 



where s is the distance that the craft has moved. The pressure in 

 the stationary reference frame is denoted by p 5 ( £ , y, t). The 

 velocity potential in the stationary frame, <f> (such that the velocity 

 is its positive derivative), satisfies the Laplace equation, so that 



V 2 (J) = 







(7) 



The kinematic boundary condition on the free surface requires 

 that a particle of fluid on the surface remains on it (for example, see 

 Stoker (1957)), so that 



D 



Dt 



r({ 



■**>].«* 



= 



where f is the elevation of the free surface. Now we have the sub- 

 stantial derivative : 



D 



= ~- + <t>, 



+ <t> 



oy z 



Dt at '£ as y 



so that the exact kinematic condition becomes 



* z - * { r t 



y y 



z=r 



f = o 



* t 



(8) 



The linearized kinematic condition is obtained by dropping the second 



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