HYDRODYNAMIC FORCES 



119 



2.0 



1.50 



1.00 



060 _ 



0-V 



A 



-V 



+ -t + + i CO = ll.3Il/sek 

 ooooo oj = I3.IO/sek 



• •.•• CO = 15.00/sek 

 xxKyx CO = 18.00/sek 



• •••• cjj =24.70/sel< 



X 



"**. 



30 40 



GU^M 10"^ 



50 



60 



70 



Fig. 10 Variation of logarithmic decrement, i5, with frequency, 

 w, in damping of heaving oscillations of 60-deg wedge at sub- 

 mergence d (from Dimpker, 1934) 



contour. Haveloc^k suggcstctl that a. ship section can he 

 replaced by a rectangular one of draft / corres])omling 

 to the mean draft of the section; i.e., /' = A/H, with 

 sources di.strilnited along the bottom. The resultant 

 expression for the ratio A is 



A = 2e-*-.i^ sin (hy) 



(22) 



where'^ /co = ojc'/iJ '^'if^ !l '■'^ ^^^^ h'^'' beam of a sliip sec- 

 tion under consideration. Fig. 9 taken from Korvin- 

 Kroukovsky (iy55a) gi\Ts a comparison of the ratio 

 A computed by three methods: As given for a semi- 

 cylinder by Ursell (1949) (this agrees with Grim's values 

 for a semi-cylinder); as computed by a source distri- 

 bution over the contoiu*; and as computed l)y a source 

 distribution over the bottom of a rectangle of the same 

 sectional area. The results of all methods are in reason- 

 ably good agreement at low frequencies, but differ con- 

 siderably at high fre(|iiencies. iMirtunately, the fre- 

 quency of oscillation at synchronism of normal ships is 

 generally in the region in which the disparity is not ex- 

 cessive, and both Havelock's and Grim's damping coeffi- 

 cients have been used with reasonable success. It should 

 be rememl)ere(l that damjiing is most important in the 

 evaluation of motifin amplitudes near synchronism, and 

 that, at frequencies widely different from synchronism, 

 large errors in estimated damping have relati\Tly little 

 effect on the amplitude. The effect of the damping on 

 the phase lag of motions is, howe^-er, mo.st pronounced at 

 fre(|uencies different from the synclironous one. 



3.21 Experimental verification of the sectional damp- 

 ing coefficients. The complexity of advanced forms of 

 the solution of the foregoing problem, .such as Ur.sell's and 

 Grim's, necessitates adopting various approximations, 

 the effect of which is difficult to appraise. Therefore, 

 experimental verification is desirable. In Holstein's and 

 Havelock's method of calculation, no investigation is 

 made of boundary conditions at a body, particularly as 



" It is necepsary to distinguish lictween the frcciueiicy ay of the 

 oncoming waves and the trec[uency oic of the wave encounter which 

 is also the frequencj- of the ship-radiated waves. 



Fig. 1 1 Dependence of net logarithmic decrement, d — 6(i, 

 on submergence, i/, with frequency, w, as parameter for a 

 cylinder lOcmdiam (from Dimpker, 1934). 5 is the decre- 

 ment measured in water, 5,, is decrement measured in air in 

 preliminary calibration 



these are modified l)y the wave formation. Accept- 

 ance of this method depends entirely on succes.sful experi- 

 mental verification. H(jlstein (1936) made expeii- 

 ments to \'erify his theory. These were limited, how- 

 ever, to a rectangular pri.sm varying in degi'ees of initial 

 immersion, and were made in a small test tank 0.70 ni 

 (2.:-! ft) wide by 3 m (10 ft) long. 



The \'alues of .4 were established by comparison of 

 directly observed wave amplitudes with the amplitudes 

 (half strokes) of the heaving prisms. The re.sults of a 

 large number of experiments appear to be consistent, thus 

 uispiring confidence. The small size of the tank, how- 

 ever, makes the data questionable. It should be re- 

 membered that to evaluate A the wave amplitudes 

 .should be measured far enough from a body for the i)ro- 

 gressive wa\'c system to be completely free from the 

 standing waves. Furthermore, one must be certain that 

 the progressive waves are not contaminated by reflec- 

 tion from the test-tank ends. These aspects of the test 

 are not discussed sufiiciently l)y Holstein, and, in view 

 of the shortness of the test tank, they may be suspected 

 as having affected the results. The use of a rectangular 

 prism is also questionable, since a certain disturljance 

 can emanate from its sharp edges. This effect is not 

 pro\'ided for in the theory. 



In his experiments Holstein also attempted to deter- 

 mine virtual masses, but the r(>sultant data were too 

 erratic to be useful. 



Dimpker (1934) pulilished data on exjieriments with a 

 (JO-deg wedge and a cylinder with various degrees of 

 immersion. The wedge was tested with an initial immer- 

 sion from to 12 cm, and the lO-cm-diam cylinder with 

 an initial immersion \arying from to 8 cm. The 

 floating body was connected by springs to a motor-dri\'en 

 eccentric, so that either free or forced oscillations could 

 be investigated. Only the free oscillations were dis- 



