6. Yeh, G. C. K., and Martinek, J.: '■Disturbance Potential due to the Presence of a Pro- 



late Spheroid (Exterior Theorem)", to be published (1956). 



7. Hunziker, R.: Final Report III on effective wake and thrust deduction for deeply sub- 



merged bodies. "The influence of body shape variation on the thrust deduction coeffi- 

 cient". Contract NOnr 1445(00) Bureau of Ships Fundamental Hydromechanics 

 Research Program. Aug. 30, 1956. Reed Research Inc. 



/. J. Eisenhuth 



The paper by Dr. Lerbs was indeed very interesting and touched on some impor- 

 tant aspects of the theory of marine propulsion. The comments which are about to 

 be made are only intended to bring to the attention of the readers some other com- 

 paratively recent work that has been done in this general field, specifically in the field 

 of torpedo propulsion. It is felt that certain phases of this work might be of interest 

 to the ship propeller designer. The torpedo propeller work that will be mentioned here 

 will reflect primarily the writer's direct familiarity with what was done at the Ordnance 

 Research Laboratory of the Pennsylvania State University. The phases that will be 

 touched upon will not be all-inclusive since some of the work has been of a classified 

 nature. 



Mention should be first made of work done on the theory of optimum pro- 

 pellers. In Reference [1], F. Lane describes a variational method applied to the prob- 

 lem of optimum wake-operating propellers. His method simply involves the variational 

 problem of minimizing the integral of power input while holding constant the useful 

 or thrust work integral. The solution of the problem establishes the condition of 

 optimum distribution of bond circulation along the blade for minimum power loss at 

 any required value of thrust or useful power. 



On the problem of representing a blade of finite chord with a bound vortex 

 or lifting line, Reference [2] presents an interesting way of handling an otherwise very 

 complicated calculation. In this reference B. W. McCormick applies an electromagnetic 

 analogy to the Ginzel-Ludwieg theory for the correction of cambers of wide-bladed 

 propellers. McCormick has also reported on the investigation of such subjects as the 

 effects of a finite hub on the optimum propeller and the minimum pressure in a trailing 

 vortex system ( Reference [3 and 4] ) . 



In Reference [3], Goldstein's analysis of a propeller satisfying the Betz condi- 

 tion is extended to the case where hub radius is appreciable relative to the propeller 

 radius. The boundary condition which is satisfied is that of an infinitely long cylinder 

 representing the hub. It is shown that for a trailing vortex sheet of given pitch, the 

 insertion of a hub increases the value of the bound circulation of the line vortex 

 generating the sheet. This change is more pronounced the smaller the number of blades. 



In Reference [4], a report is made on a fairly extensive study made on the 

 subject of tip vortex cavitation. In that study, the problem of predicting the minimum 

 pressure in the trailing vortex system of a planar lifting surface was investigated both 

 theoretically and experimentally. A semiempirical method of making these minimum 

 pressure predictions was finally evolved, and later applied successfully to the design of 

 torpedo propellers. 



The items mentioned thus far are summarized in Reference [5]. This reference 

 was more or less intended as a handbook for the design of torpedo propellers. 



REFERENCES 



1. "Optimum Single Propellers in Radially Varying, Incompressible Inflow," F. Lane. 



Journal of Applied Mechanics, Vol. 19, 1952. 



2. "The Application of an Electro-magnetic Analogy to the Determination of Induced 



177 



