130 



THEORY OF SEAKEEPING 



can be coiif^idered as applicable to rolling but not to 

 A'ibration of ships. 



The brief summary in the foregoing paragraphs is 

 entirely inade([uate to illustrate the importance of Ur- 

 sell's (1949(7) paper. The mathematical formulation of 

 the water flow presented in this paper is the fundamental 

 information on the basis of which added mass and damp- 

 ing forces and moments can be deri\-ed for many non- 

 circular cylinders. 



5.34 Effects of viscosity and frequency. T. B. Abell 

 (1916) made experiments on attenuation of rotary oscilla- 

 tions of a sciuare prism. A 6 x 6-in. prism was suspended 

 vertically in a water tank with ends fitting closely to the 

 tank bottom and to the boards covering the water sur- 

 face. These boards prevented surface-wa\'e formation, 

 so that the water flow corresponded to that at an in- 

 finitely long prism in an infinite fluid. The prism was 

 suspended on three wires, forming a rotary pendulum, 

 and the oscillation frequency was governed by the prism's 

 weight and moment of inertia and also by additional 

 radially disposed masses. Various initial deflections 

 and various fre(|uencies were tried. The rate of attenu- 

 ation of oscillation was expressed by equation (31), and 

 the resulting coefficients a and b are shown in Table 4. 



Table 4 



Period 



T 

 No keels 



3.35 



2.94 



2.58 



2.14 

 With keels 



3.54 



3.15 



2.82 



2.41 



(From T. B. Aliell, 11(16) 

 Small 

 -amplitudes — . ■— 



Large amplitudei- 



a 



n 031 

 04 

 042 

 054 



007 

 0091 

 0157 

 0122 



0007 

 0007 

 0007 

 0007 



0Uiti4 

 0.02380 

 0305 

 0430 



0.017 0.0954 



0.026 0.1278 



0.053 0.1584 



0.083 0.1996 



h'r- 



2098 

 2062 

 2066 

 2000 



1 . 196 

 1.266 

 1-260 

 1.106 



The attenuation is expressed in degrees per full cycle. 

 The upper half of the table refers to the bare prism, and 

 the lower one to the prism with V2-in-wide bilge keels at 

 the four corners. 



Since the time of Froude (ISlil, 1872, 1874) it has been 

 understood that the linear term of equation (31) is 

 caused by the energy dissipation in waves and the ciuad- 

 ratic term by the effect of A-iscosity.-* Under the con- 

 ditions of Abell's experiments, therefore, a zero value for 

 a would be expected. No explanation has been found 

 for the existence of the small values of a shown in Table 

 4. 



Ideal fluid conditions are approached in the case of the 

 bare prism at small amplitudes and low period, in which 

 case both a and b are small. The coefficient b becomes 

 appreciable at large amplitudes, and is greatly increased 

 by bilge keels. 



" This opinion appears to be contradicted by Watanahe an<l 

 Inoue(1958). 



T. B. Abell called attention to the fact that, with vis- 

 cous resistance proportional to the square of the velocity, 

 the product b4>-T- is constant. For a series of experi- 

 ments made with constant initial amplitude <j>o, the 

 damping cofhcient b will \-ary with frequency, but the 

 product bT- is expected to be constant. This expecta- 

 tion is confirmed by the experimental values listed in the 

 last column of Table 4. This observation of T. B. Abell 

 appears to have escaped the attention of later experi- 

 menters, and the coethcients a and b have usually been 

 reported without regard to the oscillation frequency. 

 For the sake of consistency the \-alues bT- .should be re- 

 ported instead of b. 



6 Direct Three-Dimensional Solution Including Wavemaking 



Apparentl\' the uK^st complete analyses of the hydro- 

 dynamic forces and moments acting on a svirface ship 

 oscillating in waves have been made by Haskind and 

 Hanaoka. Haskind's (1945a, b; 1946, 1954) work is 

 based on a method initially developed by Kotchin (1937, 

 1940). Attention was concentrated on forces due to 

 body motions, while exciting forces due to waves were 

 not analyzed. Ship-wa\'e formation, resistance, damp- 

 ing and motions in waves were treated by Hanaoka 

 (1951, 1952, 1953), but most of this work has not yet 

 been translated from the Japanese, and therefore is 

 largely inaccessible. At the NSAIB Symposium (1957) 

 Hanaoka presented a broad outline of his work in Eng- 

 lish, but the exposition is too sketchy for complete under- 

 standing without reference to his previous, more de- 

 tailed work. This \\ork includes also calculation of the 

 bending moments acting on a ship's hull. 



A \-aluable exposition of the basic principles of ship- 

 motion analysis was given by Fritz .John (1949). Other 

 investigators have not treated the l)road problem of 

 ship motions, but have concentrated instead on elucida- 

 tion of certain aspects of the broad problem. Wigley 

 (1953) and Havelock (1954) determined the forces and 

 moments acting on a submerged ellipsoid moving under 

 waves without oscillations. Havelock (1955) also dis- 

 cussed the coupling of heave and pitch due to speed of 

 advance, in support of the findings of Haskind (1946). 

 Havelock (1956) and ^^ossers (1956) compared damping 

 computed by three-dimensional theory with that ob- 

 tained by the strip theory. Havelock (1958) discussed 

 the effect of speed on damping. 



6.1 Statement of the Problem. In the approaches 

 taken by the foregoing writers, a frictionless fluid is 

 assumed, so that the velocity potential </> exists and satis- 

 fies Laplace's equation. This velocity potential is to be 

 evaluated subject to the "boundary conditions." The 

 boundaries are formed by (a) the mo\'ing surface of the 

 ship, (b) the wave-covered free surface of water, (c) the 

 ocean bottom, and (c/) the infinitely distant parts of the 

 ocean in fore, aft, and lateral directions. The last two 

 are simply expressed. No vertical velocity of water 

 {b<j>/c)z = 0) exists at the ocean bottom. At infinite 

 distance all space derivatives (i.e., fluid perturbation 



