Developments in Theory of Bulbous Ships 



stream under a free surface as in Fig. 8. As mentioned already, a point source 

 produces positive cosine waves while a point doublet produces negative sine 

 waves. By using Lagalley's theorem, he obtained forces at the doublet point and 

 the source point separately as shown in Fig. 8 corresponding to the optimum 

 distance "a" between the two points (shown in Fig. 9), which was calculated so 

 as to minimize the total force in the x direction. If we consider only the wave 

 phases along the centerline through the two points, the distance "a" should al- 

 ways be one quarter of the wavelength a^ for cancellation of phases. 



k 



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However, it is shown in Fig. 9 that the optimum "a" is always less than a^/4. 

 Figure 8 shows the remarkable reduction of the total wave resistance in this 

 case. In addition, the negative force at the doublet is rather an interesting phe- 

 nomena. The shape of bulbs made of these singularities can be produced by 

 plotting the body streamlines as Inui does for his double model, or we may use 

 an approximate sphere for a point doublet and the head of a Rankine ovoid for a 

 point source. 



Wave resistance of half body with optimized bulb 

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