singular Perturbation Problerns in Ship Hydrodynamics 



5) 4^j: This is another useful normalized velocity potential. 

 It is related to mj the way ^\ is related to n, . It 

 satisfies: 



^'xx '^ ^'yy ''" ^'zz ' ^ ^^ ^^^^ region; 



■^ = mj on S(x,y,z) = 0; } (2-76) 



|V4^i I ~* at infinity. 



In particular, it can be seen that these conditions are 

 satisfied for i = 3, 5 if: 



il;3(x,y,z) = <^, (x,y,z)j (2-76') 



i|jg(x,y,z) = - <j>^(x,y,z) - (z<^| - x<j>| ). (2-76") 



(The last term does satisfy the Laplace equation.) 



Now we can write down the velocity potential for the combined 

 translation and oscillation in terms of the above -defined quantities. 

 It is a well-known fact of classical hydrodynamics that the fluid 

 motion can be expressed as a superposition of six separate motions, 

 each of which would be caused by the motion of the body in one of the 

 rigid-body degrees of freedom. However, it is essential for the use 

 of this fact that the description be made in terms of a coordinate 

 system fixed with respect to the body. Note that there is no lineari- 

 zation Implicit in this superposition. In the sense that there Is no 

 requirement that motions be small In any way. In the body-fixed 

 reference frame, the velocity potential Is: 



[ - U cos ^5 - l<43 sin y <^|(x,y,z) 



+ [ - U sin ^5 + Iwljcos ^g]<)»^x,y,z) + Ic4g4>g(x,y ,z) . 



The nature of the superposition Is obvious when we compare the 

 first two coefficients here with (2-70). However, It must also be 

 recalled that the velocity potential obtained In this way gives the 

 absolute velocity of the fluid, that is , the gradient of this potential 

 Is the velocity In a reference franne fixed to the fluid at Infinity, 

 Thus, we must add to this potentlcil an extra term to provide for the 



This can be concluded also by recalling the definition of ^^: Its 

 gradient vanishes at Infinity. 



725 



