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APPENDIX 



Let us examine the definite integral in Eqs . (74) 

 and (78) for the balance of force and the definite 

 integral in Eqs. (77) and (81) for the balance of 

 energy. At first, we denote these integrals by F 

 and F as follows: 



where H represents total head as follows: 



P 1 



P^g 2g 



2 2 2 



U+u + V + w 



(98) 



dS 



PO - p + 2y, 



3u 



e 3x 



PfU(U + u) 



Now, using Oseen's approximation, the following 

 relationship can be written: 



u = 7r^ +u',v = 7r^+v',w 

 dx dy 



8z 



+ w'. 



(99) 



+ i Pf^ 



dz Z 



where (|) represents velocity potential, and u' , v' , 

 (94) and w' represent velocity components of rotational 

 motion which are zero at other than OJ. Then, pres- 

 sure, p, and wave height, ?, can be expressed as 

 follows : 



Fg = ^ I dS <{ {u2 + v2 + w2) (U+u) + u(p-po) 



„ 3u /3v 3u\ fdw 9u 

 2uT— +VT— + T— +w(^ + : 

 3x \3x 3y/ \3x 



UPfg 



(95) 



dz^;^ 



If the terms to which y is related are assumed to 



e 

 be small, F and F can be rewritten as follows: 

 f e 



F = p g I da)(Ho - H) + -| I dS(v^+ w^-u^) 



3* 

 -P^gy - PfU g^ , 



s = c + e' , 

 p 



(100) 



(101) 



where ?„ is due to a potential motion and t, ' is due 

 to a rotational motion. 



Substituting (99), (100), and (101) into (96) and 

 (97) , we can get 



■f 2 



dS 



dy J \oz/ \ox 



P^g 



dz C , 



(96) 



p^g 

 dz C ^ + ^ 



P 2 



dz Cp?' - p^ 



da3 



dz 5/ 



2-^ u' + u 

 dx 



,2 



p^gu 



dz s' , 



(97) 



+ (V(|))^ + Uu' 



(102) 



