Wave Resistance 

 THEORETICAL CONSIDERATIONS 



The mechanism of bulbs at a ship bow has been studied quite extensively 

 (Wigley, 1936; Inui, 1962; Yim, 1963). The wave resistance is known to be due 

 to the regular waves which carry out energy far away. The regular waves con- 

 sist of bow waves, stern waves, and possibly shoulder waves, ail of which can 

 be considered to originate from the discontinuities of a function representing 

 the hull shape of a ship. These waves are a composition of sine waves and co- 

 sine waves propagating in all directions from the originating point and are called 

 elementary waves (Havelock, 1934). In particular the bow waves of a sine ship; 

 i.e., one represented by the source distribution 



i(xi) - ag cos(77Xi) - a^ J2 



n=0 



2n 



(77Xi) 



in 



0<Xi<l;y=0 and < Zj < 1 



consists only of positive sine waves originating from the bow. The regular waves 

 due to a point doublet below the free surface are known to be negative sine waves. 

 Since the point doublet can be represented by an approximately spherical bulb, 

 such bulbs, properly selected, can cause cancellations of the bow waves. Yim 

 (1964) determined, for various Froude numbers, the optimum radii r^ of point- 

 doublet bulbs located at various depths at the stem of sine ships. For low Froude 

 numbers, it was shown that the same results applied to any general ship (Yim, 

 1965). This theory was applied to a practical bulb design with reasonable suc- 

 cess (Yim et al., 1966). However, in practice, such bulbs must be made roughly 

 cylindrical in order to avoid separation due to necking down of the bulb. This 

 can be expected to reduce the bulb effectiveness. 



In connection with problem, Yim (1967) has proposed a simple method for 

 constructing an approximate source distribution and corresponding doublet dis- 

 tributions for an optimum ship with a cylindrical bulb. He assumes that for the 

 ship with a cylindrical bulb, represented by the source distribution for the ship 

 and the doublet distribution for the bulb, the volume of ship is the linear super- 

 position of the volume of the ship without the bulb and that of isolated bulbs 

 which would be produced by the doublet in an infinite medium. He considers a 

 cross section of the ship with a cylindrical bulb whose center is located at 

 y = 0, z = -H in Fig. 2. Using the notation in Fig. 2 the area inside the circle 

 of the given radius r and outside the cross section of the ship model is 



2 / ■^ 2 



A = 77r - + T + yr cosi^ 



where 9 is in radians. In Fig. 2 the Michell ship assumption (Michell, 1898) has 

 been made insofar as the waterlines are taken to be given by dy/dx = 2rTm(x^) . 

 Actual stream surface calculations (Yim, 1966c) show that the Michell assump- 

 tion does not give the correct ship sectional shape. However, since the influence 

 of draft is small at the low Froude numbers considered here, the simple 

 attachment of a semicircular section to the bottom as shown in Fig. 2 has 



. . 1562 



