HYDRODYNAMIC FORCES 



115 



In estimating the force acting on a l)ody l)y the strip 

 theory, the foregoing relationship is applied to each sec- 

 tion using the appropriate value of the coefficient /r. 



A detailed (ierivation of the wave-caused force by 

 means of surface-pressure integration was gi\'en by Kor- 

 vin-Kroukovsky and Jacobs (1957) and is reprinted in 

 Appendix C. In the derivation carried out for a semi- 

 circular ship section, neglecting surface elfects, the prod- 

 uct of the sectional volume and the mean presstu-e gradi- 

 ent was found to be multiplied l)y 2. Since A,, = 1 for 

 a semi-circular section, the factor of 2 w-as interpreted 

 as 1 -f A',, on the basis of G. I. Taylor's result. Cirim 

 (1957c) has confirmed this intuitive conclusion by appli- 

 cation of F. M. Lewis' transformation. 'I'o correct 

 for the surface effects neglected in the formal analysis, 

 Korvin-Kroukovsky and Jacobs (1957) interpreted /(■„ 

 as kiki in evaluating the force exerted l)y the vertical 

 wave pressure gradient on a surface ship. The calcu- 

 lated wave forces on a ship's model were confirmed l>y a 

 towing tank test (Appendix 2 to Korvin-Kronko\'sky, 

 1955c). 



Attention should be called to the fact that the direction 

 of the pressure gradient is suc'h that the \ertical force is 

 acting downward on a submerged body under the wa\'e 

 crest, and upward under a trough. In a body floating on 

 the water surface these pressure gradient (or inertial) 

 forces are subtracted from the displacement force caused 

 by water-surface rise in wa\'es. The net force is thereby 

 considerably reduced. 



It is often convenient to think of water acceleration in 

 waves as algebraically added to the acceleration of grav- 

 ity. The water at wave crests ajjpears then to be lighter 

 and at wave troughs heavier than normal. 



This modification of the effective weight of water in 

 waves is often referred to as the "Smith effect," since 

 attention was called to it by Smith (1883) in connection 

 with ship bending-moment evaluation. Estimation of 

 the wa\-e forces acting on a ship b.y the buoyancy forces 

 modified by the Smith effect is referred to as the "Froude- 

 Kriloff hypothesis." The effect of the ship in disturl)- 

 ing waves is neglected in this case; i.e., the added term 

 k in ecjuation (20) is not taken into account. Since its 

 inclusion is a simple procedure, there is no justification 

 for neglecting it in the future. 



3.15 Experimental data on inertial forces. Very few 

 experimental data are a\ailal)le on added masses, and 

 these, while confirming the general ideas outlined in the 

 previous paragraphs, do not provide exact information. 

 Experiments have been made for the following cases: 



a) Deeply submerged prisms and cylinders. 



b) Prisms oscillating on the free water surface. 



c) Ship forms oscillating on the water surface. 



d) Restrained ship forms and other bodies acted upon 

 by waves. 



1 Deeply submerged prisms and cylinders. Tests 

 in the first category (a) are of interest for confirmation 

 of the classical theory The reasonableness of neglecting 

 viscosity is the particular assumption to be verified. 

 The frequency and amplitude of oscillations would be 



irrelevant if water were a truly non viscous fluid. The 

 existence of a small viscosity, however, may cause eddy- 

 making in certain experimental conditions, particularly 

 in the case of a l)ody with sharp edges. In such a case 

 scale relationships may become significant. 



Moullin and Browne (1928) experimented with two- 

 node vibrations of flat steel bars submerged in water. 

 The bars were from \/i to 1 in. thick, 2 in. wide, and 78 

 in. long, so that three-dimensional effects probably were 

 insignificant. The vibrations were excited b.v an electro- 

 magnet, and the added mass was obtained by comparison 

 of the resonant frc(|uencies in air and in water. It was 

 concluded that the added mass is ef(ual to the water mass 

 of the cylinder circumscribing the rectangular profile of 

 the bar. This is in agreement with the theoretically indi- 

 cated added ma.ss of a thin plate considering the ex- 

 pected increase with thickness of the rectangular section. 



2 Prisms oscillating on the water surface. Moullin 

 and Browne (1928) also experimented with bars V2 in. 

 thick and .3 in. wide, set on edge and partially sub- 

 merged. They concluded that the added mass is inde- 

 pendent of the vibration freriuency. This conclusion is 

 in agreement with theoretical expectations for high 

 frequencies. Browne, Moullin and Perkins (1930) tested 

 the vertical vibrations of rectangular and triangular 

 prisms partl.v immersed in water. The prisms were 

 attached to a flat steel spring and were vibrated by an 

 electromagnet. A 6 X X 54-in. prism vibrated at a 

 frequency of about 15 cps. This frec[uency corresponds 

 approximately (to scale) to the usual two-node frequency 

 of ship vibrations. The theoretical added mass (for a 

 submerged donlile i)rofilc) was computed by a Schwartz- 

 Christoft'el transformation. The experimentally deter- 

 mined added mass was found to be about 90 per cent of 

 the theoretical one. Experiments were made with vari- 

 ous lengths of prisms and the authors stated that, above 

 a length/beam ratio of 4, the added mass was inde- 

 jx'ndent of the length. Todd (1933), in applying these 

 and Lewis' (1929) results to ship-vibration analysis, 

 attributed the reduction in added mass to the effect 

 of the length/beam ratio. 



It should be emphasized that the frequencies in the 

 experiments just outlined correspond to ship-vibration 

 frequencies. These are about 10 times as much as the 

 usual freciuency of a ship's pitching in head seas. Theory 

 (Section 3.12) indicates that at lower frecjuencies a pro- 

 nounced dependence of added mass on frequency can be 

 expected. 



Prohaska (1947) rejiorf cd on oscillation te.sts of prisms 

 of several profiles partly submerged in water. The ex- 

 perimental data indicatetl added-mass values about 90 

 per cent of the theoretical ones computed for the sub- 

 merged double profiles. ITnfortunately no information 

 was furnished as to the freciuency of oscillations. The 

 description indicates that the test apparatus was of the 

 same type used by Dimpker (1934) and Holstein (193()). 

 With such an apparatus, the high frequency of Moullin's 

 experiments can hardly be expected. Without infor- 

 mation on the frequency, Prohaska's tests can be ac- 



