HYDRODYNAMIC FORCES 



in 



1.1 



0.9 



O.B 



0.7 



0.2 



0.3 0.4 



5 



0.6 0.7 



0.8 0.9 



1.0 



Fig. 3 Added mass coeflficients for vertical motion (from Landweber and de Macagno, 1957) 



Landweber and de iVIaragno (1957) have .shown that 

 F. M. Lewis' tran-sforniation is a pai'ticiUar case of a more 

 general transformation form 



+ a 



i.'S 



+ fl-./r"- 



(18) 



in which the indices m, n, etc., are odd numbers in the 

 case of symmetrical .sections. I'rohaska (1947) com- 

 puted the properties of sections with indices {m, n) of 

 (1,5), (1,7), and (3,7). 



The transformation from a circle, described in the fore- 

 going, is not suitable for section forms with sharp edges. 

 The water- flow pattern around polygonal forms and the 

 resulting added masses can be obtained by means of the 

 Schwartz-Christoffel transformation. A good description 

 of this with examples of application will be found in 

 Wendel (1950). Rectangles, rectangles with bilge keels, 

 and rhombuses w'ere analyzed b}' this method. The 

 data on these will be found in Figs. 4 and 5. 



Experiments, based on the analogy between the elec- 

 tric potential and the \'elocity potential of a fluid flow 

 (Koch, 1933), gi\'e results identical in principle to those 

 obtained from conformal transformations. They can be 

 useful where conformal transformation becomes labori- 

 ous, as for instance in the case of the added mass of a 

 floating body in shallow water. They could also be 

 u.seful in cases to which the available computational 

 methods do not apply. A ship section compo.sed of 

 curves and equipped with bilge keels can be cited as an 

 example, as can also typical sections in the stern por- 

 tions of commercial ships. These sections are usually 

 bounded by curvetl lines and poissess either a large \ertical 



deadwood, or propeller-shaft bo.ssings, or both. The 

 hydrodyiianiic properties of such sections are not known, 

 and their evaluation by means of electrical analogy can 

 be recommended. 



An extended discussion of added mass (neglecting free- 

 surface effects) will be found on pages 417-441 of Volume 

 II of "Hydrodynamics of Ship Design" by Harold E. 

 Saunders published by The Society of Naval Architects 

 and Marine Engineers, 1957. A listing of 54 references 

 is included in this work. 



3.12 Effect of the free water surface. The evalua- 

 tion of the added mas.ses in the foregoing was based on 

 the assumption that the water flow about the submerged 

 part of a floating profile is identical with the flow about 

 a submerged profile consisting of the original one and its 

 reflection in the water surface. This means that the 

 wavemaking by an oscillating floating body was neg- 

 lected. Subseciuent mathematical work (Havelock, 

 Ursell, Haskind) has shown that the foregoing assump- 

 tion is valid for a high oscillation fre(|uency. The values 

 obtained on this basis are therefore directly applicable to 

 ship vibrations, which was indeed the application in- 

 tended by F. IVI. Lewis and Prohaska. 



A different situation is found at the low frecjuency of a 

 ship's pitching and hcaxing in waves. Two wave sys- 

 tems are generated Ity these motions: The standing-wave 

 system in proximity to the ship, and the progressive 

 waves running away from the ship. The mean energy 

 content of the first of these systems remains constant 

 and defines the added mass. The progressive waves 

 carry the energy away from the ship ancl represent the 



