Joosen 



Simultaneously Joosen [5] derived the solution of the same problem without 

 waves, also using Green's function for two conditions. More precisely for the 

 case where the frequency parameter is of order unity and for the case where 

 the frequency parameter is of order e' ^. In the first case the final formula for 

 the velocity potential consists of two terms, one term corresponding to the prob- 

 lem of a pulsating double body in an infinite fluid and another term representing 

 the longitudinal interference effects. 



In the second case the result leads to the conclusion that the flow in each 

 cross section is independent of the flow at other sections. 



It is therefore a rigorous justification for the use of the two-dimensional 

 strip theory such as is applied by Grim [6] and Tasai [7], who calculated the 

 added mass and damping coefficient for a family of cross section curves. The 

 agreement between theoretical values and experimental data is very good, of 

 course, especially for the higher frequencies. 



The problem of a slender body moving at a steady speed on the water sur- 

 face has also drawn attention. Vossers was the first who attacked the problem 

 starting from the three-dimensional formulation with Green's theorem. Using 

 the method of inner and outer expansions. Tuck [8] solved the problem for a 

 body of revolution. Starting from the formulation with Green's function Joosen 

 [5] obtained the solution for a body of arbitrary shape under the condition of 

 straight vertical lines at bow and stern. It appears that the influence of the end 

 point terms is dominant and that the series expansion is not uniformly conver- 

 gent for arbitrary shape of the bow and stern line. From the numerical value of 

 the wave resistance it can be concluded that the results are not in closer agree- 

 ment with the experiments than the Michell theory. The reason for this seems 

 to be the behaviour near the end points and the fact that the Froude number is 

 in most practical examples of the same order of magnitude as the slenderness 

 parameter. The more satisfactory formulae for this case will be obtained as a 

 by-product of the present work. The result contains only integrals along the 

 bow and stern line. 



In the following sections the full problem of an oscillating slender body at 

 forward speed will be considered. In the usual strip theory forward speed ef- 

 fects and three-dimensional effects are not present. In the past several authors 

 have considered the forward speed effect in damping and cross-coupling co- 

 efficients; see Grim [6], Korvin-Kroukovsky [9]. Although this work seems to 

 be in good agreement with experimental data (Vassilopoulos [10]), a consistent 

 theory, based on a rigorous asymptotic expansion of the three-dimensional for- 

 mulae is still lacking. 



Recent experimental work of Gerritsma [ll] has shown the relatively small 

 effect of forward speed on the total value of damping and added mass coefficient 

 for heave and pitch, but an important influence on the distribution of the damp- 

 ing over the ship length. In order to verify these results an asymptotic theory 

 is set up in this paper with the assumptions that the Froude number is of order 

 e^/2 an(j ij^Q frequency parameter of order e' '. It is expected that the result 

 consists of that of the two-dimensional strip theory extended with some terms 

 representing the three-dimensional and forward speed effects. 



168 



