251 



w 



3x 



z=Ef (x,y) 



z=ef (x,y) 



3<j>* 

 W 



JzT 



+ 0(e^ 



z=0 



+ O(e^) 



z=0 



the order of E . Moreover, ijj/p can be considered 



(28) as an acceleration potential as is obvious from the 

 relationship with (I)1! . 



Finally, we proceed to convert Eqs . (25), (30), 



(29) (33), (34), and (35) for (J)t to equations for ip by 

 using the relationship (38). Using (25) and (38), 

 ijj must satisfy the following equation: 



Hence, by substituting (26) -(29) into (14), we can 

 obtain 



"3i 



z=0 



in S * . 

 s 



(30) 





3y2 





(39) 



On the hull surface, S , we can obtain from (26) 

 the following equation: 



Further, for the boundary conditions on the free 

 surface, the following equation can be obtained by 

 substituting (20)~(24) into (15): 



"^ " ^ " ^^^ 



(31) 



y=0 



And, in correspondence with the Eq. (15), the follow- 

 ing equation is also obtained: 



3n 



on S 



3i|j d\b d\b 



1 TT^+n 7r'-+n tt-*- 



X dx y oy z dz 



on S 



3z 



+ 0(£'^) in S* 



(40) 



z=0 



On the other hand, if (x,y) is a point on Sg, the 

 following equation can be obtained from (30) and 

 (38): 



3?* 



9y 3t 



(32) 



y=0 



Hence, eliminating 5i from (31) and (32), we can 

 obtain the boundary conditions on the free surface: 



i2x* 



* 2 32^* 1 32^* 



3* 



3x2 



U 3x3t U2 9t2 u2 gy I y=o 



= . (33) 



Moreover, at infinity, boundary conditions are given 

 as follows by (17), (21) and (24): 



Derivatives of (j); -> when i^xZ + y2 + z2 ->. „ , (34) 



1 8tp I , . /„ 3 3 



Pf 3^ z=0 z-0 ^ ^^ ^^^5^ 



dx 3t 



3*t 



-{z=0) 



(41) 



Thus, the hull surface condition for (J)i can be con- 

 verted to that for i)J as follows: 



8n 



= 



(42) 



on S 



Similarly, the free surface condition (33) for (j)i* 

 can be converted to that for i> as follows: 



Si ->■ when y-x.^ + z2 



(35) 



Now, let us denote the pressure of the flow field 

 in the towed condition and that in the self-propulsion 

 condition by Ps(x,y,z) and Psp (x,y,z; t) respectively. 

 By substituting (20) and (21) into Bernoulli's 

 expression, we can obtain 



^2 

 3x2 



3!l + 2_ 



3x3t U^ 



3^ ^ g3t 



3t^ 



= 



U^Sy y=0 



(43) 



Moreover, for the boundary condition at infinity, 

 the following equation is given from (34) and (35) 



Ps 



Pf 



-gy 



eU— +0(^2), 



(36) 



who 



n /^ 



+ y-^ + z-^ 



(44) 



- sp _ 



- gy 



3<j)j 3,f,* 



V dx 3x 



3(f)* 

 - £3^ + 0(e2) , (37) 



where p represents fluid density. Hence, the 

 pressure change, )> (x,y ,z;t) , due to the interaction 

 of the hull and propeller is given by the following 

 equation: 



Integral Equation 



We proceed to seek the solution of \li which is the 

 harmonic function in the region bounded above by 

 the plane y=0 and elsewhere by the hull surface and 

 satisfies boundary conditions (42) , (43) , and (44) . 

 At first, we separate the solution into the two 

 parts and write it as follows: 



l}j(x,y,z;t) = V(x,y,z;t) + W(x,y,z,- 1) , 



(45) 



\b 1 



^ = ^ 'Psp - Ps' 



3<j)f 3*1 



3x 3t 



(38) 



This equation shows that the magnitude of Jp is of 



where both V and W are the harmonic functions in 

 the region as indicated above. Moreover, let W 

 represent the pressure induced by a rotating pro- 

 peller moving straight ahead with a constant speed 

 in still water and a free surface. Now, we have 



