Understanding and Prediction of Ship Motions 



Since cp. and cp^ satisfy the same radiation condition and the same free surface 

 condition, Green's theorem yields the fact that 



'd 3n 



dS 



I- 



Bn 



dS 





dS 



The second equality follows from (18b). 

 from the expression for X-: 



With this formula, we can eliminate rp^ 



^i 



Re 



\ icope^""^ J 



Bcpj 



i dn . 



dS 



(19) 



Green's Theorem can be used again to show that the integral need not be 

 evaluated over the actual hull surface, S, but may be evaluated over any control 

 surface enclosing the ship. In particular, we may choose a surface arbitrarily 

 far away, say a cylindrical surface extending from the free surface far down 

 into the water, closed on the bottom by a horizontal surface. (The latter, as its 

 depth becomes infinite, will contribute nothing to the value of the integral.) This 

 is a particularly valuable result, because we avoid considering all of the local 

 disturbance effects in cp-. In fact, we need only asymptotic expressions for cp^, 

 valid far away from the ship, and such expressions will represent simply the 

 radiated waves in the forced oscillation problem. These asymptotic forms of 

 the potential will be the same functions which are needed to predict the damping 

 coefficients, as will be seen in the next section. 



Calculation of Damping Coefficients 



In an oscillating ship problem, the existence of damping implies that the 

 ship is performing work on the water, that is, energy is being put into the water. 

 Since we consider always a nonviscous fluid, this energy cannot be converted 

 into heat but must be radiated in outgoing surface waves. Thus we expect to 

 find a relationship between the damping coefficients and the outgoing waves far 

 away from the oscillating ship, and this relationship will be based on the law 

 that there can be no non-zero average rate of accumulation of energy in any 

 region of the fluid. In the derivation which follows, due to Newman (1959), it will 

 be shown that there is a simple formula giving the diagonal elements of the 

 damping coefficient matrix in terms of the velocity potential at infinity. Also, it 

 will be possible to obtain a formula which relates the sum of symmetric pairs 

 of the same matrix to the potential at infinity, but it has not yet been found pos- 

 sible to determine these off-diagonal elements completely separately except in 

 the special case of zero forward speed. 



We establish formulas for three energy flow rates. First, we assume that 

 the ship is being forced to oscillate sinusoidally in some mode or combination 



45 



