Yim 

 4k. 



^1 = -^ I A(O,0) sin (UgX sec 6) + B(O,0) cos (k^x sec 9) - dd , (16') 



-77/ 2 



and 



4k r"/2 r'- 



^2 = ~^ d^ U(koH,x,0) cos [ko(x-Xi) sec 6] dx j 



•^77/2 -^X 



= ^ A(L,0) sin(koX-Lsec6') + B(L,0) cos (kgX-L sec^)- d^ (17') 



•^-77/2 



have been called (1) regular bow waves and regular stern waves respectively at 

 X > L. (In this report, we use simply bow waves and stern waves in this sense.) 

 Another reason for this can be attributed to the concept of elementary waves (4). 

 Namely, Eqs. (16') and (17') can be considered to be equivalent to i9-wise super- 

 positions of elementary waves starting from the bow and the stern respectively. 

 Thus, the wave resistance of an analytic ship is considered to consist of three 

 components: the bow wave resistance, the stern wave resistance, and the inter- 

 action of the two. This concept is very helpful to treat a bulbous bow (6) for the 

 range of low Froude numbers. 



If we consider the source distribution 



m(Xj,z j) = ag 



in < Xj < a and < Zj < h, which is only the first term of Eq. (15) and rep- 

 resents a wedge ship bow approximately, then Eqs. (16) and (16') or (17) and 

 (17') coincide with each other, Eq. (17) representing the shoulder wave rather 

 than the stern wave. Namely, B(x,i9) in Eqs. (16) and (17) is identically zero, 

 and the actual regular waves in < x < a is represented by Eq. (16'). We no- 

 tice that a^ is a dominating term (1,7) in the wave height (Eqs. (16) and (17)) for 

 low Froude numbers (<0.3). 



It is well known that the bow bulb flattens the regular bow waves. In fact, if 

 the ship source distribution is represented in an even power series of x as in 

 cos TTx, it was shown (2) that a doublet distribution exists along the straight in- 

 finite stemlines of the bow and the stern, which cancels all the regular ship 

 waves. 



In general, the influence of the stern shape on the wave is very complicated 

 due to the effects of the boundary layer and the wakes in the actual fluid. These 

 effects seem to tend to decrease the influence of the discontinuity of the slope at 

 the stern. Therefore, in the low Froude number range, the bulb even at the bow 

 only is supposed to decrease the wave resistance considerably. 



It must be noted, however, that in < x < L the regular wave which contrib- 

 utes to the wave resistance is in general expressed not by Eq. (16') but by Eq. 

 (16). Therefore, just canceling Eq. (16') does not necessarily mean canceling 

 the actual regular waves on the ship hull. In addition, although Eq. (16') involves 



686 



