132 



THEORY OF SEAKEEPING 



Fig. 



k B»250 -^ 



24 Sketch of symmetrical model (from Haskind and 

 Riman, 1946) 



face of the element under consideration. In addition, 

 cos a = cos ;8 = 1. 



The second assumption consists of taking the per- 

 turbation velocity, d(f>, dn{=d(t>/dij),-^ to occur at the 

 ship's plane of symmetry, instead of at the half-breadth 

 y. It is common in hydrodynamics to represent a body 

 by a distribution of sources on its surface. The velocity 

 potential (/> for a single pulsating source is known, and 

 </) for the entire ship can be obtained Vty integration of 

 the potentials of all sources. In the described approxi- 

 mations the sources are considered distributed o\^er the 

 plane of symmetry (taken to be rectangular) instead of 

 the true ship surface. The two assumptions just de- 

 scribed make the integrations tractable. In the Kotchin- 

 Haskind method these approximations permit evaluation 

 of the function H. Once H is evaluated, all character- 

 istics of the fluid motion connected with body motion are 

 simply expressed in terms of it. 



The definition "thin ship" often is used interchange- 

 ably with "Michell ship." This is done since approxi- 

 mations become closer with reduction of the beam B and 

 the process becomes exact as the beam approaches zero. 

 The term thin ship is, however, unfortunate, since such 

 important characteristics as resistance, damping and 

 cross-coupling in pitch and heave vanish as S — >- 0. 

 INIost of these A'ary nearly in proportion to B". In the 

 literature on ship-wave resistance and ship motions, a 

 ship always is regarded as having a finite and not too 

 small beam. Merely, a certain error has to be accepted 

 as a result of Michell's assumptions. 



30 



ze 



2G 



ZC 



22 



12 



' I I 



2^ On the basis of the first assumption. 



16 



Fig. 2 5 Variation of added mass, m", and damping coeffi- 

 cient, b, with frequency ai (from Haskind and Riman, 1946) 



An exception to the foregoing is found in the work of 

 Peters and Stoker (1957) and of Xe^^^nan (1958). In 

 these cases an attempt is made at a rigorous solution, 

 starting with a thin ship and approximating finite ship 

 width by a series expansion in a small quantity B/L. 

 As in all ship-motion theories, an infinitesimal amplitude 

 of motions has been assumed. So far only a first-order 

 solution has been achie\'ed, and in Peters and Stoker's 

 case it gave the paradoxical result that there is no damp- 

 ing in the pitching and hea\ing of a ship, as well as no 

 cross-coupling between these motions. Newinan, (1958) 

 attributed this result to the fact that Peters and Stoker 

 expressed the beam and the motion amplitudes as in- 

 finitesmial quantities of the same order. While assum- 

 ing these to be infinitesimal, NeA\7nan allowed them to 

 l)e of different orders and arri\'ed at finite \'alues of damp- 

 ing and ship resistance. He compared the results with 

 the unpublished experimental data of Golovato. The 



