and by Grim for the class of Lewis sections. There is a good agreement between 

 Ursell's and Grim's findings in the case of the circular cylinder; however, a wide dis- 

 crepancy has been stated between Grim's results and those found by the "source 

 theory" for larger frequency parameters o B *. In the latter case, one should have more 

 confidence in Grim's and Ursell's results since they rest upon a more advanced approach; 

 recent measurements by Golovato support the correctness of Grim's findings. Further 

 investigations will be conducted to settle this problem finally. 



From the point of view of ship design, the range of frequency parameters 

 close to synchronism only is of primary importance, say corresponding to 0.6^ A^ 1.2, 

 or expressed in ship parameters roughly 0.5 ^v B *^2. 



b. The Three-Dimensional Problem 



The velocity potential is known for the harmonically pulsating source and 

 doublet advancing with a constant speed, from which in principle a solution can be 

 built up for a system picturing the actual ship [38]. Neglecting the speed of advance 

 Havelock [18] has given his well-known analysis of heave and pitch. The damping 

 coefficients for an elliptic waterline are : 



N s = -- B*L*co 3 exp [ - 2u*H/gW* (7) 



N* = B^LW exp [ - 2a*H/g]N+* (8) 



16 g 



for small w N~ <-~ w 3 , as against N r- > w in the two-dimensional case. N z * and Ny* are 

 dimensionless coefficients. 



For a wholly submerged body, oscillations are pictured by dipole distributions. 

 In the range of small w 



N z tt CO 5 , N Z M tt CO 5 , N* « CO 7 , (9) 



where NJ S) is the damping coefficient of the body calculated by a strip method. While 

 it had been earlier established [16] that for heave the strip method is appropriate for 



a>* 2 =: ^ 6, it follows that in pitching the corresponding lower limit of applica- 



8 

 bility is higher, say w * 2 ^8. It should be remembered that for application damping 

 values become decisive in the neighborhood of synchronism only. 



Haskind's results for low frequency parameters are given in a different form 

 [16]. However, an agreement with Havelock's expressions can be obtained. 



From [25] we reproduce two diagrams. The full lines in Fig. 1 represent 



N 33 

 dimensionless damping coefficients of heave, C 33 r C z r =I; , for F = 0, as 



irp\/gL B 2 

 2l function of the frequency parameter ©*, apparently based on Reference [16], for a 

 full (a = 0.8) and a fine waterline (a = 0.64). The marked points in this diagram 

 refer to damping at the Froude numbers indicated. This diagram and another showing 

 experimentally-obtained heaving amplitude ratios plotted versus w *, support Haskind's 

 assertion, that within a wide range the damping coefficient is roughly independent of 

 the Froude number. Phase lag curves corroborate the statement. 



In principle there is, however, no reason to expect that Haskind's finding is 

 more than an approximate rule which may be valid within certain ranges of the fre- 

 quency parameter and Froude number. Eggers, of my institute, has investigated this 

 question using Brard's results for simple combinations of singularities and found that 

 outside of the range considered by Haskind important deviations from the rule can 

 occur. Of special interest in this connection appears to be Japanese work [29]. 



70 



