pioneered to such an extent that classical hydrodynamics has again been established 

 as a powerful tool for solving so complicated a problem as marine propulsion. We 

 are referring particularly to the theory of interaction between hull and propeller and 

 the theory of wake-adapted propeller. Not all in the audience may have been aware 

 of the fact that the researches in the last hundred years in hydro- and aerodynamics 

 were essentially concerned with the motion of bodies where the disturbance on external 

 forces due to the presence of the bodies was usually negligible. If we consider, however, 

 the fact that in the case of a self-propelled vessel such disturbance is of first order, 

 we need an analysis to study the interaction of flows and bodies. To Dr. Lerbs must 

 go our deep appreciation, for it was he who introduced us to these subjects and whose 

 wise counsel and encouragement were vital for whatever improvement and understand- 

 ing we have been able to achieve on these problems. 



We should like to mention now some of our recent results and their interrelation 

 with Dr. Lerbs' work, by which they were inspired. 



a. The application of the sphere theorem by the discussors to the problem of 

 hull-propeller interaction is of course an approximation. Nevertheless, it was the 

 simplest conceivable model of interaction available and it was applied with the objective 

 of searching for a relationship which can utilize model testings. The reward was 

 indeed surprising, as a relationship could be found which in order to obtain the inter- 

 action force requires only measurements of the flow field at the domain of the pro- 

 peller location of a towed vessel and the free-running propeller itself, which complies 

 with the conventional testing practice. This result encouraged our endeavor to study 

 more complicated bodies such as oblate and prolate spheroids, and ellipsoids, as well 

 as general bodies of revolution, with the objective of finding similar laws which may 

 govern the problem of interaction. Indeed, in our Final Report II [4] a theorem has 

 been developed for the prolate spheroid which was not directly stated, but was indi- 

 rectly a result of the analysis *, as was found later by inspection. This theorem opens 

 the way to finding a general method which will lend itself to the evaluation of the 

 disturbance potentials and velocities for those cases where bodies of more complicated 

 shapes are encountered. It indicates that the customary testing procedure is essentially 

 sound and needs only a slight modification. Most important of all, it becomes apparent 

 that more extended and refined measurements at the propeller disc of a self-propelled 

 model, which to date have defied any such attempt, are not necessary. For the prolate 

 spheroid we can demonstrate this in a few lines. The potential function of the totality 

 of singularities which includes discrete ones as well as distributions in an infinite ideal 

 fluid domain can always be expressed in the form 



co n 



VH \-^1 [B m nQn m (£) (£ > £o 



0o = 7.7. [ cos ( m ^) + A ™n sin (m<p)]P n m (v) < for < (33) 



LjLU \C mn Pn m (0 U<h 



n=0 171=0 



(lo is £ °f the singularity) 



The disturbance potential fa due to the presence of a prolate spheroid £ = | x < £ D is 



a solution of the Laplace equation and can be written as 



0x = T^ J^ [cos (m<p) + D mn (m<p)]EmnPn m (ri)Qn m (i;) for £ > fc (34) 



* of finding an exact solution for the interaction potential due to the action of an 

 infinite-bladed propeller in the presence of a hull of prolate spheroidal form. 



175 



