DISCUSSION 

 PREDICTION OF DAMPING-IN-HEAVE 



Recent papers on ship motions have utilized the principle that the damping of the 

 vertical motion of a ship's section can be related to the waves generated by a suitable dis- 

 tribution of pulsating sources along its surface. The attractive feature of this method is its 

 essential simplicity. 



One of the characteristics of this solution is the existence of an infinite succession 

 of the values of the frequency at which the damping is zero. This phenomenon is due to 

 interference effects between the wave systems generated on both sides of the body. 



Ursell^ has pointed out that at higher frequencies, when the draft is several times the 

 wavelength, there is a shielding effect and there cannot be any interference. In fact, he has 

 demonstrated that the exact solution does not reveal any such zeros and showed that the 

 pulsating source technique is valid only at low frequencies. Ursell was able to find the ex- 

 act solution for several oscillating cylindrical forms. 



Grim'* has found a nearly exact solution for a wider class of two-dimensional forms. 

 The section is reflected about the waterplane and the potential for this double body in an in- 

 finite medium is found. To this is added a potential such that the sum of the two satisfies 

 the free-surface condition. To these are then added a sum of other potentials so that the 

 total still satisfies the free-surface condition and satisfies the boundary conditions at certain 

 points on the body. Only a few such terms are found necessary in the calculation. From 

 these results the added mass and the damping can be calculated. Grim's results for the added 

 mass compare very closely with the exact solutions of Ursell. No multiple zeros are found 

 in the damping. 



Grim has computed the wave amplitude (and consequently the damping) of a class of 

 ship-like forms of beam-draft ratios of 3, 2, 4/3, and 0.4 with varying fullness. An effort 

 will be made to compare these results with the experimental values. A procedure is used 

 which is analogous to that used by Prohaska^ for finding the added mass. 



It is assumed that ship-like forms having the same beam-draft ratio and fullness would 

 have the same damping characteristics. Thus for any section, Figures 9, 10, 11, and 13 of 

 Reference 4 can be used, with suitable interpolation, to find the damping of each section of 

 the model. Integrating these results along the length of the body gives the total damping. No 

 correction was made for the three-dimensional nature of the flow. Grim suggests that such a 

 correction is not necessary for ship forms when the frequency is large enough, say oP'L/gy^. 

 For the model being tested, this corresponds to w > 4 rad/sec. 



In Figure 9 the experimental damping coefficient B for 0, 1, and 2 knots is given. Also 

 shown is the damping as computed by two methods; using the Grim curves and the distributed 

 source technique. Both the Grim method and source method over estimate the damping but the 



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