Thus the genera^ character of the curves m (d) is different from that of m, while the 



damping curves iV (d > and N display a certain similarity. 



_ 2tt 



Approximate relations for N jm (d) are given when — 2/1 sin Xl is small. Formulas 



_ " A 



are developed from which the forces Y (d \ Z (d \ M/ d ^ per unit length can be calculated 

 when the transverse dimensions are small compared with the length of the ship, and 

 therefrom by a strip method all resultant forces and moments experienced by the vessel 

 (except, obviously X< d) ). 



In the case X — or -n- in which we are strongly interested, the present simplified 

 method leads to a purely vertical force per unit length, which coincides in type with 

 that for the well-known relative-motion effect. However, instead of fn g the diffraction 



mass m 



(d) 



appears. 



If we assume for the time being that especially for larger — , 

 _ g 



mJ d) < m z , one sees that the relative-motion correction over-emphasizes the actual 

 effect. 



In [25], the inconsistency of the relative-motion correction is emphasized. Very 

 probably, however, this criticism applies to an erroneous computation of the velocity 

 and acceleration field in a wave. 



Exciting forces and moments calculated by Haskind depend linearly upon the 

 orbital velocities of the wave particles. The important hydrodynamic reactions due to 

 yaw and sway of an advancing ship cannot be determined by the method proposed. 



Finally, we consider an unpublished investigation by Grim already referred to 

 [24], "Die Schwingungen von schwimmenden Zweidimensionalen Koerpern." For sim- 

 plicity we treat here all hydrodynamic forces involved in heave, sway and roll. While 

 emphasis is laid on the determination of hydrodynamic characteristics when Wo ~> 0, 



(1) ~ o 



the method is extended to finite small frequency parameters . 



o 



a 



Using Lewis' sections the velocity potential for forced roll in calm water is 

 established when Wo ~^ 0, and the hydrodynamic forces are determined. The potential 

 for the lateral motion (sway) is obtained for O j ~* and added masses calculated; 

 results agree with those given by Haskind. 



The equation for roll without damping, 



J OX I « 2 



(m OG + m v h q y 

 m + m y 



4> + D GM <f> = 0, 



(ID 



agrees with the findings by Woznessensky based on Haskind's work [25]. J ox is the 

 moment of inertia of the ship referred to the waterline point O (axis of symmetry of 

 the load water line) ; OG is the distance of the center of gravity; and h q is the lever of 

 the lateral added mass force. The position of the axis of rotation is derived herefrom. 



A strip method is suggested to compute pertinent values for the actual ship. 

 For small w the damping due to the side motion and the roll can be approximately 

 determined by adding an appropriate potential. The results are compared with those 

 obtained for elliptical and circular sections by a more rigorous method [24], and a 

 reasonable agreement is stated. In connection with an earlier investigation by Ursell, 

 it is interesting to state, that for the profile considered the sway contribution to damping 

 is much larger than that due to roll. Thus the loss of wave damping effect which has 

 been found by Ursell for certain sections may not be so tragic as anticipated. 



The influence of the effective added moment of inertia upon the roll frequency 

 is communicated as function of GM. Beside, the influence of large bilge keels on the 

 period is shown as the result of a rather intricate investigation. In addition, expres- 



73 



