114 



THEORY OF SEAKEEPING 



The values of coefficient ki, computed by I'rsell for a 

 floating cylinder, are listed in Table 2. 



Table 2 



Note: For ver>' high-frequencN' parameters the eoefficient kt 

 asymptotically approaches unity. 



In the absence of data for other ship sections, Korvin- 

 Kroukovsky (1955f) and Korvin-Kroukovsky and Ja- 

 cobs (1957) have assumed that the coefficient k,, initially 

 computed by Ursell for a circular cylinder, will apply to 

 other profiles. Additional theoretical research is evi- 

 dently needed in order to provide values of /i'4 for F. M. 

 Lewis' and other ship sections. 



Attention .should be called to the particular surface 

 effect occurring in the case of sections with inclined sides. 

 Such inclinations exist in the bow sections of ^'-form ships 

 and are particularly pronounced in stern sections of most 

 ships. When such a section penetrates the water sur- 

 face, the adjacent water level ri.ses and the wetted beam 

 becomes greater than the one indicated by the inter- 

 section with the undisturbed water level. This causes 

 an increase in added mass and probably in damping 

 forces. A solution of this problem was given by Wag- 

 ner" for the asymptotic case of an impact in which grav- 

 ity forces are neglected; i.e., the instantaneous water 

 rise is taken into account but not the subsequent wave- 

 making. In Wagner's solution, the effect of a deep 

 draft in conjunction with a V-section has not been con- 

 sidered. .1. D. Pierson (1950, 1951) u.sed a computa- 

 tional method (initially suggested by Wagner) in which 

 the draft of a V-section is included, subject to the grav- 

 ity-free assumption. No information is available on 

 the water flow, and the added masses and damping con- 

 nected with it, for the water-surface penetration by a 

 wedge at the frequencies of a .ship's heaving and pitching 

 in waves. Also, the available impact theories treat only 

 the hydrodynamic force acting on a body when it pene- 

 trates the water. The theory' of ship motions and bend- 

 ing moments requires also knowledge of the forces during 

 body emergence from the water. The added masses in- 

 volved in these motions will evidently be functions of 

 oscillation fr('i|uency. IJescarch in this field is needed. 



3.13 Three-dimensional effects. In the evaluation of 

 hydrodynamic forces by the strip theory, the initial as- 

 sumption is made that the water flow at each strip is 

 two-dimensional and is not influenced by the adjacent 

 strips. Subsequently, it is desired to verify this as- 

 sumption and, if practical, to establish a correction fac- 



tor for the deviation of the physical conditions from the 

 initially assumed ones. This problem can be con- 

 sidered in connection with three applications: 



o) A body \ibrating with various mmibers of nodal 

 points. 



b) A body of a certain LB ratio pitching and heaving, 

 in smooth water. 



c) A body subjected to waves. 



The first problem [a) was treated by F. M. Lewis 

 (1929) and .1. Lockwood Taylor (ly^O^) with varying re- 

 sults. Macagno and Landweber (1958) have investi- 

 gated these solutions and demonstrated that the results 

 are strongly affected by the completeness with which the 

 shear and flexural deflections of a body are described. 



The second problem [b) has apparently received no 

 attention. The third problem (c) was solved for a sphe- 

 roid mider waves by Havelock (1954) and Cummins 

 (1954a, /)) using advanced mathematical methods, and 

 by Kor\-in-Kroukovsky (19556) using the strip theory. 

 The agreement between calculational methods was satis- 

 factory (as shown in Figs. 15 and 16), and apparently 

 no correction for three-dimensional effect is needed for 

 the length beam ratios normally used in ships and for 

 wa\-es of length approximately ecjual to a ship's length. 

 This conclusion applies, however, to the coefficient of 

 accession to inertia k, for the entire body in the analysis 

 of body motions. Three-dimensional effects, in all proba- 

 bilitj', do affect the distribution of hydodynamic forces 

 along a ship; i.e., the /c.-values for individual strips. 

 These effects are, therefore, significant in the analysis 

 of ship bending moments and research toward their 

 evaluaticin is recommended. 



3.14 Inertial forces caused by waves. It appears 

 that satisfactory estimates of forces exerted by waves on 

 a submerged body can be made considering inertial forces 

 alone and neglecting viscous forces. This means that 

 hydrodynamic-force components in phase with the waves 

 can be expressed in terms of added mass. The added 

 mass is expressed, in turn, in terms of body displacement. 

 In a strip theory the displacement is taken per unit 

 length. In considering the wave action on a body it is 

 necessary to remember that a velocity gradient and a 

 corresponding pressure gradient exist in waves. 



The force acting on a small submerged body in long 

 wa\'es can be calculated most conveniently on the basis 

 of this gradient and the body's volume. The force 

 exerted by a hydrostatic pressure gradient is equal to 

 the product of the gradient and the body's volume. 

 G. I. Taylor (1928) showed that this relationship is modi- 

 fied when the pressure gradient in a fluid results from the 

 acceleration of fluid particles. The relationship becomes 



Force = {I + k) 



X (pressure gradient) X (body volume) (20) 



where k is the coefficient of accession to inertia. The 

 factor (1 + k) is the result of modification of an acceler- 

 ating fluid flow by the presence of a body.'" 



' To be discussed in greater detail in Section 7 on slamming. 



'° Forces exerted on a body b. 

 investigated by Tollmien (1938). 



fluid accelerations were also 



