In this manner we can find a relation for each of the excit- 

 ing forces in terms of the corresponding damping co- 

 efficient. Without loss of generality we can set |8 = 0, 

 and we thus obtain the expressions 



|X,<o'(0)| = A (^' BuJ' 

 |Z6<»'(0)| = A (^^ 555)'" 



(31) 

 (32) 

 (33) 



Thus for bodies with a vertical axis of symmetry, the 

 damping coefficients are sufficient to determine the 

 amplitudes of the exciting forces (although not their 

 phases), and vice versa. 



Two-Dimensional Case 



We consider now the case of plane two-dimensional 

 motion, such as beam waves incident on an infinitely 

 long cylinder. Then equation (14) is replaced by a line 

 integral at infinity, and if we take as the control contour 

 a large rectangle consisting of the free surface, the bottom 

 at 3 = — 00 , and two vertical lines — <» ^ z ^ at 

 a; = ± 0°, ve then obtain, in place of equation (15), 



X> 



For the two-dimensional incident-wave system, pro- 

 gressing in the positive ^-direction. 



gA 



exp {Kz — iKx) 



(35) 



while the asymptotic radiation potentials, analogous to 

 equation (17), are 



>Pj = Pj± exp(A"2 - iKx ) for X -► ± CO (36) 



Here the functions Py± depend only on the wave number 

 K and the particular body under consideration, and the 

 superscript ± corresponds to the case a; -*- ± <» . In 

 general the two functions Py+ and Pj~ will be unequal, 

 but in the practical cases of importance, involving 

 bodies symmetrical about the y-axis, the magnitudes of 

 these two functions will be the same, or more precisely, 



Pi+ = -Pi- (surge) 



P3+ = P3- (heave) 



Pii+ 



(pitch) 



(It is -more conventional to apply these concepts in the 

 two-dimensional x — y plane, and j = \ may be thought 

 of as either surge or sway, while j = 5 corresponds to 

 either pitch or roll.) 



We now proceed as before to find the exciting forces 

 and damping coefficients as functions of Pj. Substituting 

 (35) and (36) in (34) we obtain 



:) [P,+{-iK + iK) 



- PriiK + iK)] (37) 



= pgAs'^'Pr 



which is the two-dimensional analog of equation (23). 

 Thus the exciting force is proportional to the amplitude 

 of the radiation potential at infinity, in the direction from 

 which the waves are incident. 



The damping coefficients are given in terms of energy 

 radiation, by the expressions 



B,, = iPo.[|P>|^ + \PrY] 



or, for a symmetrical body, 



B,, = p^\PrV (38) 



These are the two-dimensional analogs of equation (27). 

 Comparing (38) with (37) it follows that for an arbitrary 

 two-dimensional body with transverse symmetry, 



PQ'A^ 



|A/' 



X/»)=^(f B,,)'" 



(39) 



which is the desired relation for the amplitude of the 

 exciting force in each mode, in terms of the corresponding 

 damping coefficient. This equation is to be compared 

 with equations (31-33) for a three-dimensional body of 

 rotation with vertical axis. We emphasize again that 

 within the framework of linearized water-wave theory, 

 equation (39) is an exact expression which holds for any 

 two-dimensional body with transverse sjonmetry, in 

 each of the three degrees of freedom. 



■ In the two-dimensional theory one frequently uses the 

 "wave-height ratio" rather than the damping coefficient, 

 especially for heave, where the wave-height ratio A is de- 

 fined as the wave amplitude at infinity, per unit ampli- 

 tude of heave displacement. Thus in the present nota- 

 tion, where the velocity potential is the potential per 

 unit heave velocity, and the wave height is given by the 

 expression 



f = - - v(x, 0) 



it follows that the wave-height ratio for heaving oscilla- 

 tions will be 



^l^3| 



(40) 



where we delete the superscript (±) since for a sjon- 

 metric body (P3+I = |P3i. Substituting (40) in (37), 

 it follows that 



|A'3<»'| = %AA 



(41) 



