Pien 



obtain a large family of T7 surfaces. At present, B(0, Pb^^^j and PgCO are 

 chosen to be constant. Later, if necessary, the general case will be examined. 



We choose Eq. (2) as the expression for the singularity density, which is 

 defined as the singularity strength per unit velocity of a moving ship. 



M(^, O = I I ai; ^\' for -1 < ^ < 1, -T < ^ < (2) 



i J 



These surface singularities can be either source or doublet. For the purpose of 

 generating a bulb, a line source and line doublet located at the end of the i? -surface 

 are also included in our scheme. Equations (3) and (4) define the line source and 

 line doublet strength, respectively. 



line source S(Q = 2j s.^ v"/ 



j ^ 



line doublet D(0 = Z] d.^^ (4) 



j 



To obtain a flat keel line or a flat bottom, an additional surface source and 

 doublet are placed on the horizontal bottom of the 7]-surface. 



Theoretical Analysis of Wavemaking Resistance 



We first assume that our hull form, theoretically represented by Eqs. (1) 

 through (4), has a very low level of wavemaking resistance. Under this assump- 

 tion, the theory can be used to analyze wavemaking resistance characteristics 

 of the forebody of a hull form quite accurately. If, at the end, the theoretical 

 wave-resistance level of the hull form under consideration is not low, we reject 

 such singularity distributions. 



We are interested in two different kinds of theoretical analysis. First we 

 must obtain the theoretical wavemaking resistance curve as well as the free- 

 wave amplitudes of a given singularity distribution. Second we must find the 

 optimum singularity distribution under a set of design conditions. A computing 

 program has been developed to perform both kinds of theoretical analysis. 



The general scheme and procedure for performing the double integrations 

 for free-wave amplitudes and triple integrations for wavemaking resistance 

 numerically have been fully discussed in Ref. 4. Computing the free-wave am- 

 plitudes and wavemaking resistance curve of a given singularity distribution is 

 a relatively straightforward procedure. To find the optimum singularity distri- 

 bution under a given set of design conditions is more complicated. Our aim in 

 such theoretical analysis is to develop a hull form with both low wavemaking 

 resistance and a satisfactory hull geometry. It should be emphasized here that 

 the wavemaking resistance tiieory is used to obtain a hull form with low wave- 

 making resistance rather than to predict the wavemaking resistance. The wave- 

 making resistance of a final design is obtained by model experiments. It should 

 also be mentioned that when we write down a set of design conditions, we have 



1116 



