Current Progress in the Slender Body Theory- 

 magnitude of the potential can only be established rigorously by solving the 

 problem, but it can be estimated heuristically by considering the corresponding 

 two-dimensional problem in the transverse or "cross-flow" plane, where the 

 ship's submerged area is pulsating at a rate proportional to the longitudinal 

 rate of change of sectional area, and it is easily verified that a pulsating cir- 

 cular cylinder of radius R will have a potential, on its surface, of magnitude 

 proportional to R logR times the normal velocity. The hydrodynamic forces 

 follow from Bernoulli's equation and the fact that the longitudinal projected area 

 of the ship is OCe^). (The potential of the incident wave is of course 0(1) since 

 it doesn't depend on e.) There is no hydrostatic restoring force in surge and 

 the inertial force is proportional to the displaced volume, or 0( e^) . The leading 

 order forces are the Froude-Krylov exciting force and the inertial restoring 

 force. Thus the leading order equation of motion for surge oscillations does not 

 depend on the hydrodynamic disturbance generated by the body. We note that a 

 similar conclusion holds for heave, roll, and pitch. Thus, at least in these four 

 modes, the familiar damping and added mass forces are secondary and the 

 Froude-Krylov hypothesis has a rational basis. 



Certain fundamental conclusions follow from Table 1: 



1. In every mode the leading-order equations of motion are homogeneous 

 in e. Thus the response in each mode, to incident waves, will be 0(1) in terms 

 of e, and of the same order as the wave height. 



2. For surge the dominant forces are inertial and Froude-Krylov, with the 

 effects of the ship's own disturbance small by the factor e^ log e. 



3. For sway and yaw the ship's hydrodynamic disturbance must be ac- 

 counted for even in the first-order equations of motion. 



4. For roll the Froude-Krylov exciting moment and hydrostatic restoring 

 moment are dominant, with other effects small by the factor e . 



5. For pitch and heave the dominant forces are Froude-Krylov and hydro- 

 static, with effects from the ship's hydrodynamic disturbance small by a factor 

 e log e. It follows that the first-order equations of motion for pitch and heave 

 will not contain resonance effects, but there will be a bounded resonance in the 

 second-order equations. 



At first glance the above conclusions may seem trivial, at variance with 

 physical observation, and a step backward in our scientific development. One 

 noted critic has even stated that "at least thin-ship theory predicts resonance." 

 The best counter -argument is to show some results from the application of the 

 first-order theory for pitch and heave (Figs. 1 and 2). These show the pitch and 

 heave response of an aircraft carrier at zero speed. The solid curves are the 

 results of solving the coupled first-order equations of motion, equating the 

 Froude-Krylov exciting force and moment* to the hydrostatic restoring force 



^Actually we employ the slender body limits of the Froude-Krylov force and 

 moment, in which the surface integrals over the hull are replaced by simpler 

 line integrals. 



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