Ogilvie 



We can interpret the various terms here readily. The first term of the first 

 integrand is just the hydrostatic pressure at equilibrium, and the second term 

 yields the hydrostatic disturbance force. The following three terms are vin- 

 steady force contributions, and the last term in the first integral yields the re- 

 sistance and the unsteady force which results from interaction between the 

 steady flow and the displaced position of the hull. The line integral represents 

 an interaction force arising from the superposition of steady and unsteady mo- 

 tions; the factor Bcp^Bxj can be recognized as being proportional to the wave 

 height in the steady motion problem, and 



— i - V — ■ 

 Bt 3X] 



is proportional to the unsteady wave height (omitting a term containing both cp^ 

 and cpj. 



It may be noted that the line integral is zero for j = 3. This follows from 

 the assumption that the ship is wall- sided near the waterline, which means that 

 n lies in the plane of i^ and i^ for points on L^. Furthermore, if we consider 

 only motions in the longitudinal plane, then the line integral is also zero for 

 j = 2 . Finally, if the steady motion problem is linearized in any of the usual 

 ways (thin, flat, or slender ship approximations), then n-i-i is small, of the 

 same order as cp^, and the whole integral is of second order in terms of the 

 perturbation parameter for the steady motion problem. 



The moments acting on the hull can be calculated in a similar manner, with 

 only slight complications appearing. Let us again choose to work directly with 

 the moments about the unsteady (body) axes: 



^j+3 



= I (n-i j X x' ) p dS 



Then the moments about the steady axes will be given by: 



M = M' + ^ xM' + ^x X' , 



where M = (X^.x^.x^) and x = (X^,x^,x^), and similarly for the primed quantities. 



We proceed to calculate x]^3 in the same manner as we did Xj and so the 

 details will not be repeated. The factor (n • i ! x x') must initially be kept within 

 the inner integral in the correction terra, since x' depends on X3, but if we set 

 X3 = in this factor, we cause only higher order errors, as is easily verified. 

 The result is: 



.._,.., .. . . . ■. . „ ^^1 



Xi+3 = P 



J dS(n-ijXx) |-gX3- g(f3+ x/i- Xj^j) ~ ^ + '^ 



72 



