Linearized Potential Flow Theory for ACVs in a Seaway 



We shall presently be expanding all the physical variables 

 describing the motion of the ACV and that of the water in powers of the 

 perturbation parameters. Taking the velocity potential of the water 

 as an example, the correct expansion would be 



/ ►/! \ ^ V k„l m n i crpt (x, v. z) 



$(x, y, z;6;/3;a; € ;t ) = Re la 8 a e e F * , . v ' y ' ' 



* klmn 



k, 1, m, n, p 



Corresponding to excitation by waves with frequencies 



a , 2cr, 3 cr , 



However, as the algebra will become extremely complex, the 

 whole solution can first be carried through for one frequency component 

 with say, p = 1 . The final result can then be extended to any number 

 of Fourier components in the wave system. It may be observed in this 

 connection that in simulating an irregular seaway in a towing tank a 

 finite number (of the order of ten) Fourier components is usually 

 selected. In this case, when we desire a verification of the theory 

 from experimental results, the solution should cover the same number 

 of Fourier components. 



III. 1 Coplanar Motion 



The analysis will be restricted to a study of the ACV moving 

 in a longitudinal plane. This is by no means a requirement of the 

 linearized problem, but this simpler study will reveal clearly the 

 general features of arbitrary motion in all six degrees of freedom. 



III. 2 Body-fixed Axes 



The third co-ordinate system mentioned in section II- 1 is the 

 (x 1 , y' , z' ) system fixed in the ACV. The origin o' coincides with 

 the origin o of the moving system when there are no oscillations. 

 Also, the z' -axis (like the z-axis) contains the C. G. (on the negati- 

 ve side) and the x' , z' -plane (like the x, z -plane) is the fore-and-aft 

 plane of lateral symmetry of the ACV. 



It is clear that the x' , y' -plane is the load waterplane (LWP) 

 of the side hulls when the ACV is on its air cushion. 



All the three systems of axes are illustrated in Figure 2. 



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