Ogilvie 



the hull in the usual way, with direction cosines and lever arms for weighting 

 functions, as appropriate. We can distinguish three separate parts of this 

 force: (1) the force due to the incident waves, which is a Froude-Krylov force, 

 (2) the hydrostatic restoring force due to the disturbed position of the hull, (3) 

 the hydrodynamic force due to the ship motions and to the diffraction wave. The 

 sum of all these is set equal to the inertial reaction of the ship, in accordance 

 with Newton's Law, to yield the equation of motion. 



These different kinds of forces involve various functions of e. We obtain 

 the lowest order theory by considering only those forces which are homogene- 

 ous in the lowest order of e . The precise form of these expressions is not par- 

 ticularly interesting except to someone who wants to make quantitative predic- 

 tions. The details can be found in Newman (1964). The qualitative conclusions 

 which can be drawn are, however, worthy of some comment. 



In yaw and sway, there is no hydrostatic restoring force or moment. The 

 other three kinds of forces, that is, inertial, excitation (Froude-Krylov), and 

 motion -induced forces, are all of order e% and so all must be included in the 

 equations of motion for these modes. It may be noted that there are no free 

 surface effects in the motion- induced forces (except inasmuch as the rigid wall 

 may be considered as a free surface condition). Also, it has already been 

 pointed out that there are no interaction effects between sections in these modes. 

 So here the theory becomes exceptionally simple: The excitation is calculated 

 from the Froude-Krylov formula, the body inertia comes from the ordinary the- 

 ory of the dynamics of a rigid body, and the hydrodynamic reaction is obtained 

 from the solution of a fairly simple two-dimensional problem. In fact, since 

 there can be no radiation of waves in the presence of a rigid wall at X3 = 0, the 

 motion -induced hydrodynamic force is simply the added mass force for the 

 simplified two-dimensional problem. Therefore the whole resistance to the yaw 

 and sway excitation force has the nature of an inertial reaction. 



In pitch and heave, the Froude-Krylov excitation force and the hydrostatic 

 restoring force provide the leading terms in the equations of motion; these 

 forces are both of order e, and all other forces are of higher order. The low- 

 est order theory is accordingly even simpler than for yaw and sway. There is 

 no hydrodynamic force (except the excitation) in the first approximation. Also, 

 the inertia does not enter into the calculation. The response is entirely con- 

 trolled by the "spring" term. 



However, in heave and pitch, it is fairly straightforward to derive higher 

 order forces, and Eq. (26) leads directly to formulas for the next approxima- 

 tion. It can be shown that the interaction term in (26) yields a force of order 

 e^ log e in the heave and pitch equations of motion. This force includes added 

 mass, damping, and diffraction effects. The inertia of the ship itself is of order 

 e^ in these modes, and the calculation can be extended to take this into account. 

 This is a much more interesting situation, for the system now has the proper- 

 ties of a damped spring-mass system. However, it must be recognized that the 

 occurrence of a resonance is a higher order effect superimposed on the simple 

 effects described in the previous paragraph. If the amplitudes of response are 

 very large near resonance, or if there are large phase shifts, then we can hardly 

 pretend that these are higher order effects, and the theory is of questionable 



64 



