Yim 



the opposite direction of that at the bow. Thus, we understand how the wave ef- 

 fect in the vicinity of the submerged bulb at the stern acts to decrease the wave 

 resistance. 



We cannot use the ideal bulb in practice, since it is infinitely long. How- 

 ever, this idealized situation not only introduces an easy analysis but also intro- 

 duces us to a deeper insight of the mechanism. Since the lower part of the bulb 

 deeply submerged under the free surface does not very much influence the reg- 

 ular waves, in general all the theory here can be applied approximately to prac- 

 tical bulbous ships (see Ref. 2). 



APPLICATION 



Michell's ship theory has too much discrepancy with experiments for a 

 given surface ship; hence it was suggested to build a model from a singularity 

 distribution by tracing the body streamlines. However, since it is so compli- 

 cated and time consuming to plot streamlines for the ship singularity under the 

 free surface with the free surface condition satisfied, Inui used a double model 

 for which the free surface is considered to be flat. Inui (1) claims that the dou- 

 ble model is better than Michell's ship for common Froude numbers. Pien (8) 

 also used double models for his optimum singularity distributions to build his 

 bulbous ships which showed better performance than the corresponding ships in 

 the Taylor series. 



Here, as an application of the theory on the image system of a zero wave 

 ship in an earlier section we may propose to use a small wave image system 

 that is composed of a double model ship with a point doublet for a bulb and its 

 negative mirror image as shown in Fig. 3. The approximate optimum radius of 

 a bulb represented by a point doublet which minimizes the bow wave resistance 

 for each Froude number (0.15 < F < 0.3) and each entrance angle is given by 

 Yim (7). For the case of the ideal zero wave singularity system as in Fig. 1, 

 the free surface condition is satisfied without having any wave image. Since the 

 general behavior of the flow field near the free surface due to the long vertical 

 doublet distribution and its negative mirror image is similar to that due to the 

 concentrated doublet and its negative mirror image, the free surface condition 

 may be almost satisfied by this small wave singularity system, and the approx- 

 imate local disturbance will be represented by the free surface streamlines. 



CO 



+ + + 

 + + + 



+ + + 

 + + + 



Fig. 3 - Singularity and image 

 system of a bulbous systenn 



696 



