HYDRODYNAMIC FORCES 



131 



velocities) vanish, i.e. d(f>/d.c = d4>/d!j = d<t>/dz = 0. 

 The term "perturbation velocities" is ii.sed here to desig- 

 nate fluid velocities caused by the presence of a ship. 



The boundary condition at the free water surface con- 

 sists of statements that the air i)re.ssure is uniform and 

 that surface particles of water remain on the surface. 

 This is expres.sed in .suitable mathematical form in terms 

 of the velocity potential <f). 



The bounilary condition at the shi])'s surface consists 

 of the stipulation that water flow is tangential to the 

 surface. Mathematically, this is expressed by the state- 

 ment that the fluid-velocity component normal to an 

 element of ship surface is etjual to the \-elocity of the sur- 

 face itself in the direction of its normal. To give a simple 

 illustration, consider a wedge moving with velocity 1' 

 through initially calm water. Let a be the half-angle 

 of the wedge. The component of the mf)tion of the 

 wedge surface normal to itself is then expressed on the 

 basis of body geometry as V sin a. The fluid velocity 

 induced by the foregoing motion is expressed in terms of 

 the velocity potential by the vectorial sum of the fore- 

 and-aft and lateral components as (d<^,'d.c) sin « -|- 

 {d4>/dy) cos a. The boundary condition is then 



d4> . ,£)</) 



— sni a + — cos a 

 OX Ol/ 



V sin 







(,-t^) 



As a somewhat more complicated example, consider a 

 surface element of a normal ship mo\ing forwartl with- 

 out oscillations. This can be described by the angle 

 a of this element to the fore-and-aft :r-axis, and by the 

 angle IS of the rotation of this, initially imagined as 

 vertical, element about the .r-axis. The boundary con- 

 dition then becomes 



— sm a -\ cos a cos l3 



ox OIJ 



+ — cos a sm 13 

 oz 



V sin a = (45) 



The next step is to introduce oscillations of a ship 

 usually described by six components — three translational 

 ones along the .c, y, and ^-axes, known as surging, side 

 swaying, and heaving, and three rotational ones, roll- 

 ing, pitching, and yawing. Haskind (194(3) wrote, in 

 vector notation, 



U„(i\I,t) = Un + Q X rn (46) 



where t'„ is the normal velocity of the surface element as 

 a function of the position of the element il/ and time /. 

 U is the resultant velocity vector of all translational l)oily 

 motion, i.e., 



U = U^i+ U,j + l\k- 



and il, is the resultant angular velocity \-ector of the body 

 rotations about three axes, i.e., 



= U,i + l\j + l\k 



i, j, k, and n arc the unit vectors along the co-ordinate 

 axes and normal to the ship surface S, respectively. 

 r = OM is the radius vector from the center of gravity of 



the ship to the surface clement J\I on the botly surface 

 ,S'. 



Expression (4(j) thus is a geometrical description of 

 the normal velocity of an element of body surface just 

 as r sin a was in the first example of a simple wedge. 

 In the expression for the normal fluid velocity in terms 

 of the velocity potential, it is con\-enient to consider the 

 total potential $o as composed of two parts: 



<i)„(.r./y,,r,0 = <P(.f.n.:J) + <i'*(x,y,zj). (47) 



where $* is the known potential of the simple harmonic 

 wave train, and 4) is an as yet unknown potential due to 

 the presence of the body and its motions. The bound- 

 ary condition is then expressed as 



d* a/t = i'JM.I) - d**/dM 



(48) 



The forced oscillations of a floating body which are 

 established after damping the free oscillations can be 

 taken as 



c = r*c'"'; n = n*c'"': r„ 



(/( 



and the time-dependent ijotential $ can correspondingly 

 be taken as 



*(.r,.v,^,n = <j>(x,y,zj)e- 



(4'J) 



where co is the circular fretjuency. 



6.2 Evaluation of the Velocity Potential 0. K\alu- 

 ation of the fujiction 4> is a difiicult mathematical prob- 

 lem for the solution of which the reader is referred to the 

 works listed under the authors mentioned pre\'iously. 

 Evaluation of the function apparently has been possible 

 in only two applications; i.e., to submerged ellipsoids and 

 to "thin" or "Alichell" surface ships. 



Wigley (1953) and Havelock (1954, 1956) have con- 

 centrated on analysis of forces acting on submerged ellip- 

 soids moving under waves. The results of their work can 

 be applied directly to sulimarines and torpedoes, and are 

 t[ualitati\'el3' indicati\-e of what can be expected in the 

 case of surface ships. 



Haskind (1946) first de\'eloped \'arious basic expres- 

 sions for wave formations and for hydrodynamic forces 

 and moments acting on an oscillating surface ship in gen- 

 eral form. These are applicable to any body form and to 

 any mode of motion. The expressions take simple form 

 in terms of a certain function H originally defined by 

 Kotchin (1940). The //-function for a given freeiuency 

 depends on a body's form and is given as a doulile inte- 

 gral taken over the surface of a ship. This function gen- 

 erally is prohibitively complicated, but is simplified and 

 becomes tractable for either ellipsoids or Michell .ships. 



The term "Michell ship" designates a mathematical 

 model of a ship for which theoretical computations be- 

 come tractable after making simplilying assimiptions 

 formulated by Michell (1898) in his theory of ship-wave 

 resistance. In the first of these assumptions a ship is 

 considered as ha\ing small beam/length and beam/draft 

 ratios so that it is possible to assume that angles a and 

 l3 in equation (45) are small. Thus, sin a = tan a = 

 dy/dx, where y is the half-breadth at the water sur- 



