3.3. Damping 



Beginning with Froude [3], considerable efforts have been made to determine 

 the damping of roll by experimental methods. It is known that Froude possessed sound 

 ideas on the wave damping in this case, but his plotting of extinction curves without 

 deriving therefrom orthodox dimensionless damping coefficients has hampered progress. 



To the present writer's knowledge, pertinent experiments on heave and pitch 

 were first carried out by him at the Berlin Tank on a very full and medium full cargo 

 ship model. They yielded added mass coefficients of about unity for heave and some- 

 what less for pitch. Dimensionless damping coefficients were K ~ 0.4 — > 0.5 for both 

 motions; there being no measurable difference in these values for F = and F = 0.20. 

 These results were later checked by Igonet [35]. 



Experiments conducted in Gottingen by Schuler and his school to check theo- 

 retical findings yielded a frequency dependence of damping as well as of added masses. 

 Havelock [18] interpreted an important result due to Holstein by indicating the under- 

 lying source distribution, and treated the three-dimensional case of heave and pitch by 

 alternating sources. Almost all work mentioned below refers to the damping experi- 

 enced by bodies moving in calm water and performing forced oscillations. Such is 

 in fact, the mechanical scheme used at present in this kind of research, (except for 

 model investigations in a seaway). Similar studies were independently performed by 

 Kochin and followers primarily on the damping of submerged bodies including shallow 

 water effects. Haskind gave an extended general treatment of the subject and developed 

 formulas for the heave and pitch damping of thin ships in a regular very long seaway. 

 We mention Brard's important work on pulsating singularities advancing in calm water. 

 In the two-dimensional case Ursell and Grim satisfied more accurately the boundary 

 conditions on the body which otherwise remained somewhat undeterminate. 



Further work of Haskind [25] discusses approximate methods to determine 

 hydrodynamic forces. Finally, Havelock [36] has made an investigation on the limits 

 of applicability of the strip method for determining the damping of submerged bodies 

 as function of the frequency parameter. So far no systematic investigations are known 

 on the magnitude of damping coefficients in a regular seaway. 



a. Damping in the two-dimensional case 



A strip method is now widely in use to determine pitch and heave damping of 

 ships; it can be applied to roll also. Results have been given in the form of rate of 

 energy dissipated, wave amplitude at infinity, and damping coefficient N, from which 

 the dimensionless damping coefficient k can be derived. 



In Holstein's case of a uniform alternating source distribution, the wave ampli- 

 tude referred to the heaving amplitude amounts to 



Therefrom 



N = 



co% 



,*H 



2o>2 



- — H 



(4) 



(5) 



,B 



y/a+K.)Hg 



exp 



2a>2 



- — H 



9 



C0£ 



V-- 



1 H1 + 



K. 



exp 



COB' 



2H 



B 



(6) 



Holstein's result has been generalized by St. Denis [37] for arbitrary section 

 forms. 



We note that for small w , the damping and its dimensionless coefficient is 

 proportional to w. The same applies to results found by Ursell for the circular cylinder 



69 



