Ogilvie 



The eqviations of motion in the longitudinal plane are all found to contain only 

 second and higher order terms. 



They then restrict their attention to the case of a ship in sinusoidal head 

 waves. These incident waves have an amplitude with order of magnitude /3. The 

 resulting second order problem is then straightforwardly separated into a time- 

 independent problem and a time-dependent problem. The solution of the former 

 leads to the Michell-Havelock solution of the resistance problem, unaffected by 

 the incident waves or the motions. The solution of the latter, the time -dependent 

 problem of lowest order, is carried out without the necessity of treating any 

 more boundary value problems. The resulting ordinary differential equations of 

 motion represent simply a two-degree-of-freedom coupled spring-mass system, 

 without damping. Even the coupling is removed if we assume that the centroid 

 of the waterplane is in the same cross section as the center of gravity of the 

 ship. In this case, the solution predicts an undamped resonance in heave and in 

 pitch. In the heave mode, the spring constant is the hydrostatic restoring force 

 per unit deflection, and, in the pitch mode, the spring constant is the hydrostatic 

 restoring moment per unit pitch angle (plus a non-hydrodynamic contribution 

 which results from the condition that the center of gravity is generally below 

 the origin, i.e., below the pitch axis). The resonance frequencies are then ob- 

 tained as the square roots of spring constants divided by mass and moment of 

 inertia, respectively. The disturbing force is just a "Froude-Krylov" force. 

 That is, the heave or pitch excitation is obtained simply by integrating the pres- 

 sure in the incident wave over the hull, with direction cosines and lever arms 

 as weighting functions, as appropriate. The presence and motion of the hull do 

 not affect the values to be used for the pressure. 



Obviously, some of these results must be rejected on physical arguments, 

 especially the prediction of undamped resonances and thus of infinite amplitudes 

 of motion. Unfortunately, heave and pitch resonance frequencies quite often oc- 

 cur within the important range of wave excitation frequencies, and, if this hap- 

 pens, it is evident that the narrow spectral band around resonance covers the 

 frequencies of most interest. Even if this theory is valid for other frequencies, 

 it is not of much help in predicting real phenomena. 



In spite of these difficulties, the results come directly out of the hyptheses. 

 There can be no arguing with the logic used by Peters and Stoker in deriving 

 conclusions from their formulation of the problem, and so the difficulty must be 

 sought in the formulation. This situation will be resolved presently, but for the 

 moment let us note that the anomalous behavior at resonance can be explained 

 non-mathematically. There are three types of quantities which are assumed to 

 be of order /?, and we can start a catalog of orders of magnitude by listing these: 



ship beam, waterplane area, volume, mass /3 

 amplitude of incident waves /? 



amplitude of oscillations of the ship yS 



Speaking in terms of orders of magnitude, we can say that: (a) exciting force = 

 (amplitude of incident waves) x (waterplane area); (b) restoring force = (ampli- 

 tude of ship motions) x (waterplane area); (c) ship inertial reactions = (ampli- 

 tude of ship motions) x (ship mass); (d) amplitude of motion-generated waves = 



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