spheroid, due to axisymmetry, is then studied in detail, 

 for the general case of an ellipsoid. As a second illustra- 

 tion we present the exciting-force amplitudes of various 

 two-dimensional floating elliptic cylinders which are 

 easily obtained from the corresponding damping charac- 

 teristics. In the special case of a circular cylinder, the 

 results obtained check with direct calculations made by 

 Dean and Ursell [4], and for more general bodies, the 

 method is consistent with the extensive calculations pre- 

 sented by Grim [5]. 



Haskind's Relation* for the Exciting Forces 



We consider two independent problems involving a 

 floating or submerged rigid body; i.e., the diffraction 

 problem of regular incident waves moving past the fixed 

 body, and the radiation problem of forced sinusoidal 

 oscillations of the body in otherwise calm water. In 

 both cases, assuming small disturbances of an ideal fluid, 

 there exists a velocity potential *(z, y, z, satisfying 

 Laplace's equation and the free-surface condition 



--^,- =0 on z = 



(1) 



Here (x, y, z) is a Cartesian-coordinate system, with z 

 = the plane of the undisturbed free surface and the 

 3-axis positive upwards. For incident waves of fre- 

 quency o) or forced oscillations with the same frequency, 

 we can write 



*(x, y, z, t) = <p(x, y, z) e" 



(2) 



where the real part is to be taken in complex quantities 

 involving e'"'. From (1), the potential <p satisfies the 

 condition 



^ - K<p = on 

 oz 







(3) 



(% + ^7) = 



onS 



(4) 



where K = u^/g. 



For the diffraction problem, with the incident wave 

 system given by a known potential, ipo, the total potential 



>f> = <Po -\- <Pl 



must satisfy the boundary condition of zero normal 

 velocity on the body, or 



dn 



where n is the unit normal vector into the fluid and S de- 

 notes the submerged surface of the body. For the radia- 

 tion problem, there are six degrees of freedom, including 

 surge, heave, sway, roll, pitch, and yaw, the velocities 

 of which we denote by 



v,e'"' (7 = 1, 2, 3, 4, 5, 6) 



respectively. For oscillations in the ;th mode we can 

 write 



ip = ipjVj 



and in general the velocity potential wiU be of the form 



v» = X v^<p)(x, y, z) 



(5) 



where, due to the presence of the free surface, the po- 

 tentials ^^ win be complex. On the body, the potential 

 ip, mast have the same normal velocity as the corre- 

 sponding mode of the body, or 



dn 



= /j(x, y, z) on S (7 = 1, 2, . . . 6) 



(6) 



where 



/i = cos(n, x) 



fi = cos (n, y) 



fi = cos (n, z) 



ft = y cos (n, z) — z cos (n, y) 



fi = z cos (n, x) — X cos (w, z) 



/e = z cos (n, y) — y cos (n, x) 



Finally, the radiation potentials ^>( j = 1, 2, . . .6) and 

 the diffraction potential fi must each sat'sfy the radia- 

 tion condition of outgoing waves at infinity, and must 

 vanish at infinitely large depth in the fluid. In view of 

 these conditions and the free-surface condition (3), which 

 must be satisfied by each potential independently, it 

 follows from Green's theorem that 



//('■ 



dn dn 



A rfS = (i = 1, 2, . . . 6) (7) 



Now we consider the six exciting forces and moments, 

 which we denote by Xj, following the same designation 

 of index as for the velocities. Thus, Xi is the surge force, 

 ^2 the sway force, Xt the roll moment, and so on. Then 



X>= - jfpfjdS 



(8) 



where the hydrodynamic pressure p is given by the 

 linearized Bernoulli equation 



d* 



dt 



-twpipe 



(9) 



For the exciting forces on the fixed body in waves, 



<P = <Po -^ <PJ 



and thus 



Xt = tope'"' jj (<po + •pi)fjdS 

 s 



= i.pe^"'j^[f.+ ^^^dS (10) 



.5 



However from (7) and the boimdary condition (4) for ^^ 

 onS, 



