Motora and Koyama 

 Fj = pghAj e' cos (kx - wt ) (4) 



The total force is Fj + F 



Fj + Fj = pgh cos (kx - ojt ) 



kT ,, -kF 



A^e ^-Ve 



(5) 



Note that if e"'^'^ % e'*"^, the wave-excitationless circular wave number is 



which is just Motora and Koyama' s result without all their "correction coeffi- 

 cients." 



When the body is not made of cylinders, it can be analyzed by subdividing 

 it into "elemental cylinders" and Eq. (5) is still correct. 



The results from the simple, straightforward theory I have given above 

 should be accurate when the scattered wave is small. With fixed, surface- 

 piercing two-dimensional bodies like those of Motora and Koyama, the scattered 

 wave will not be small so the simple theory would be inaccurate for them. How- 

 ever, when a two-dimensional body is totally submerged with only small append- 

 ages piercing the surface, the simplified theory may be accurate. Ogilvie (3) 

 has considered a submerged two-dimensional horizontal circular cylinder in 

 waves with the top of the cylinder more than one cylinder diameter below the 

 free surface, and he has shown that the forces when considering the scattered 

 wave agree within 10 percent error with the forces found when neglecting the 

 scattered wave. His results for no scattered wave reduce to Eq. (5) when ka is 

 small, a being the cylinder diameter. 



The above facts as well as the very good agreement between the simple the- 

 ory and experiments with the FLIP buoy shown by Kerr (4) have prompted me to 

 calculate by the simple theory the heaving force on the sphere with vertical cylin- 

 der used by Motora and Koyama. The results are shown in Fig. D6. The simple 

 theory does not fit the experimental results of Motora and Koyama as well as 

 their theory, but their theory is "adjustable" by means of their four arbitrary 

 constants y^, y^^ ^zj 3-^^^ K^* 



The disagreement of the simple theory with the experimental result is prob- 

 ably not due to the scattered wave alone. The diameter of the thinnest vertical 

 strut considered here is hali the diameter of the sphere so there is a large as- 

 sy metry about a horizontal plane through the center of the sphere. Therefore, 

 the body can develop vertical lift associated with circulation, in the presence of 

 horizontal currents, and the horizontal fluid velocity in the wave is in phase with 

 the pressure forces thereby maximizing the possible discrepancy due to lift. 

 There are many structures in use today where the bulk of the buoyancy is wholly 

 submerged and the surface-piercing elements are relatively small. For struc- 

 tures of this type the simple theory should give predictions suitable for use in 

 the engineering office. 



410 



