5 5 5 



Substituting in (10), it follows that 



X, 



//(■ 



dn 



dS 



(12) 



Thus we have found the exciting forces in a form de- 

 pending only on the incident-wave potential ipo and the 

 radiated-wave potentials <pj, and which is independent 

 of the diffraction potential v^. Finally, we note from 

 Green's theorem that if S„ is any control surface in the 

 fluid outside S, then since (po and <pj both satisfy the free- 

 surface condition, 



S + So. 



and therefore, Xj may be evaluated from a surface in- 

 tegral at infinity : 



X,= -i.^-'Jj[,.^^-.,^)dS (14) 



Thus Xj depends only on the asymptotic behavior of <pj 

 at large distances from the body. (Haskind reached this 

 conclusion by the introduction of Kochin functions.) 

 We note that in (14) the direction of the normal n is 

 inward, or from the control surface into the fluid. 



If we take as the control surface <S„ a vertical circular 

 cylinder about the z-axis of large radius R, then with 

 (R, 6, z) as polar coordinates, it follows that 



dS = Rdz de 



d/dn = -b/dR 



and thus 



^' = ^'^"' loli (^° U' - ^^ Sf ) ^ '' "' ^''^ 



Exciting Forces on a Submerged Ellipsoid 



For the submerged ellipsoid with semi-axes (oi, 02, 03), 

 defined by the equation 



5l I ^ J. ?L 



1 



(16) 



the asymptotic representations of the radiation potentials 

 ipj were derived in reference [2], for the case of sinusoidal 

 oscillations with constant forward velocity. These re- 

 sults are an approximation based on the assumption of a 

 moderately large depth of submergence. In order to 

 study the exciting forces on a fixed ellipsoid we use the 

 potentials derived in [2], setting the forward velocity 

 equal to zero. Thus, from equation (21) therein, we 

 obtain 



<P) = -i (^^ ' Pi(' + e) exp[/^(2 - h- iR) 



-t- «74] (17) 

 where 



Pi(u) = — 2iriu/faia2a3Di cos u 



Piiu) = —2inuKaia2aJi2 sin m 



Piiu) = 2irijL>Kaia2a3 D3 



9 



Pt{u) = — 27ria)iC^aia2a3(a2^ — a3^)D4sin u — - 

 Pi{u) = 2riwK^ uichaziai^ — a,^)^^ cos u — ^ 



Pt{u) = —2ToiK^aiaia3(ai' — 02^)1)6 cos u sin u 



hiq) 



and 



q = K[{ai^ — a,') cos- u + (02' — 03^^) sin^ u]'^' 



Here Oi, a-t, and 03 are the semi-axes of the ellipsoid, with 

 2ai the length, 2a2 the beam, and 2a3 the depth, h = 

 depth of submergence of the centroid, jniq) is the spheri- 

 cal Bessel function, and the coefficients D, are related to 

 the virtual-mass coefficients of the ellipsoid in an in- 

 finite fluid, and are defined by 



D, = (2 - a,)-> (j = 1,2,3) (18) 



■>'" = [<^~^') + - - -]"' 



(7= 1, 2, 3) (19) 

 and 



Jo (a/ 



d\ 



(20) 



For regular incident waves of amplitude A , progressing 

 in a direction which makes an angle S with the x-axis, 

 the velocity potential is 



V50 = — expIiCz - iKR cos(0 - /3)] (21) 



'J) 



Substituting equations (17) and (21) in (15), we obtain 



X, = -pAKe'"' (^ girj'^ £' j [1 - cos (6 -/3)] 



.exp{2ii:2 - Kh - iKR[\ + cos(e - /3)1 



-h irt/4}P/7r -f e)dzde 



= -ipA (^ gR\^' e-A*+.<-«+«74 



I [1 - cos(9 - /3)]'exp{-iA'ff 

 Jo 



[1 -i-cos(e- ^)]]PjiT + e)de 



