110 Samuel Karp, Jack Kotik, and Jerome Lurye 
[14] Stoker, J.J., and Peters, A.S., “The Motion of a Ship, as a Floating Rigid Body, in a 
Seaway,” Institute of Mathematical Sciences, New York University, Report IMM-NYU 
203, 1954 
[15] Magnus, W., and Oberhettinger, F., “Formulas and Theorems for the Functions of Math- 
ematical Physics,” Chelsea, 1949 
[16] Yourkevitch, V., “The Form of Least Resistance,” A.T.M.A., 31:687 (1932); see also 
the Shipbuilder and Marine Engine Builder, p. 235, Apr. 1933 
[17] Karp, S., Kotik, J., and Lurye, J., “On Ship Forms Having Minimum Wave Resistance,” 
TRG, Inc., Report TRG-119-SR-1, 1959 
DISCUSSION 
C. Wigley (London) 
Unfortunately advance copies of these papers were only available yesterday. This pre- 
vents any criticism of the mathematical work, but there are some points of importance which 
should be mentioned regarding the practical use of these calculations of forms of minimum 
resistance. 
Firstly the wave resistance as calculated by the Michell or equivalent formulas is that 
wave resistance which would exist in a perfect fluid. 
Secondly it is assumed that for a ship the wave resistance is to be simply added to an- 
other resistance due to viscosity which does not change greatly with a change of form. 
Regarding the first point raised, in fact, except at very high speeds where the Froude 
number is above 0.4, the effects of viscosity on the wave formation are serious, causing a 
decrease in the efficiency of the afterbody as a wavemaker. 
Regarding the second point, the nonwave resistance may be considered as consisting 
of the sum of two terms, the first term depending only on the wetted surface and speed and 
being some 90 to 95 percent of the total, and the second term, sometimes called the form re- 
sistance, depending as well on the shape of the form. Very little is known as to the varia- 
tion of this form resistance with speed, although its value at very low speed is shown by 
the difference between the calculated frictional resistance and that actually measured, since 
the wave resistance can be neglected at such speeds. It is suggested by M. Guilloton (see 
Trans. I.N.A., London, vol. 1952, pp. 352 and 353) that the form resistance is increased at 
the higher speeds owing to the effect of the wave motion on the flow round the form, a symp- 
tom of this influence being the change of attitude of the form during motion. 
The effects of these considerations on the actual minima of resistance are well shown 
by the simple question of the optimum position for the center of buoyancy. From the math- 
ematical theory, as given by these papers today, the optimum position of the center of buoy- 
ancy would be amidships at all speeds. In practice, it is found that, at the lower speeds up 
to a Froude number of about 0.25 where the wave resistance is small the optimum position of 
the center of buoyancy lies forward of amidships, since the form resistance is thus dimin- 
ished. As the speed increases the optimum position moves aft and lies aft of amidships for 
