250 



On the free surface, the condition of constant 

 pressure and the kinematical condition can be given 

 as follows: 



3x 3: 



\ 3 X d X 



3()) 



3** 



U + ^ + r 



3x 3x 



(j) 3(|)< 



+ ^^-^ + a— 



^dz OZ 



3<t>* 



'3i|> 3it)* \ 2 

 ^ \d^ "^ 3^y "^ \dT' '*' 3z 



3(j)* 



3*;, 



sp 



3x 



3(|) 



sp 



3(1)* 



3y 3y 



8C 



sp 



3^, 



sp 



3z 



3t 



y=C 



sp 



(11) 



(12) 



Derivatives of <()* ^ when /x^ + z ->- «> , 



?* ->• when /x^ + z^ ->■<«> , 



(17) 



where C* (x,z;t) represents change of wave height 

 due to the propeller action and the following 

 relationship must be satisfied: 



Csp - Cs. 



(18) 



Acceleration Potential and Approximate Calculation 

 Method 



Acceleration Potential 



where ^sp represents the wave height in the self- 

 propulsion condition. At infinity, the following 

 boundary conditions might be given: 



+ y'^ + z^ 



C ->■ when /x^ + z^ ^ °= . 

 sp 



(13) 



The purpose of this section is to indicate that the 

 equation and the boundary conditions for (()* derived 

 in the previous section can be expressed in the 

 terms of acceleration potential on the assumption 

 of thin hull. 



At first, using the assumption of thin hull, we 

 express the shape of the hull as follows: 



Finally, using the equations derived under the 

 towed condition and the self-propulsion condition 

 described above, let us derive the equation and 

 boundary conditions for (j)* which express the change 

 of the flow field around the hull due to the pro- 

 peller action. At first, (()* must satisfy the Laplace 

 equation (9) . Next, let us obtain the boundary 

 conditions for (j)*. On the hull surface, the fol- 

 lowing relationship is given from (5), and (10): 



3i}i* 

 3x 



3y 



'on S 



. 



(14) 



On the free surface, the following equation is 

 given from (6) and (11) in correspondence with the 

 condition of constant pressure: 



3ij)* 

 3x~ 

 3(|)* 

 V 3x 3x 



3x 



3* 



3()) 

 s 



3y 

 3(j)* 



3y~ / "^VaT" + Yi 



" aT- " 5^ y=, 



sp 



3^x 2 /^*^^^ 2 



3x ' A3y 



3<Ps\ 2 



3(t) 



+ U- 



gy 



y=Csp 



(15) 



z = £'f (x,y) 



in S 



(19) 



where £ represents a small quantity of the first 

 order and S * represents a projected plane of the 

 hull surface, Ss , in the xy-plane . And, it seems 

 reasonable to develop all our quantities in powers 

 of e, as follows: 



= Ecpi + e (|)2 + 



s 



ti* = 



£*1 



e <)>2 + 



sp 



ESl + £ C2 + 



eCi + e C2 + 



Thus, 5* can also be developed as follows: 



eC.1 + £ S2 + 



(20) 

 (21) 

 (22) 

 (23) 



(24) 



Next, we proceed to obtain the equation and 

 boundary conditions for (j)i* which correspond to the 

 first order of £ by substituting the development 

 (20)~(24) into the equation and boundary conditions 

 for ())* in the previous section. The following 

 equation in Q* can be obtained from (9) and (21): 



And, using (7) and (12) , the following equation is 

 given in correspondence with the kinematical 

 condition: 



32<f>t 



3^^ 





3z2 



(25) 



U + 



34 34* 

 s 



3x 3x 



3()) 



sp 



3x 



3(j) 



) 3())* 



+ It-^ + ^— 

 ^3z 3z 



3x 



3; 3? 

 sp ^ sp 



3z 



3t 



3y 3y ■ 

 i'=^sp 



U + 



dc, di 34 3c 

 s _ s s _ s 



3x 3y 3z 3z 



y=5= 



(16) 



At infinity, the following boundary conditions 

 might be given: 



Let us consider the boundary conditions for (|)i*. 

 First, using Eq. (19) , we can estimate the magni- 



tude of n , n , n in the Eq. (14) as follows: 

 x y z 



1 = 0(e) , n 

 X y 



0(e), n = 0(1) , 



(26) 



where denotes the order symbol. In addition, we 

 obtain from (19) and (21) 



3<i)* 

 3x 



z=Ef (x,y) 



3** 



+ 0(e-^) , 



z=0 



(27) 



