doubtful. Observations of the white plume of the wave crest around the 

 ship, may be an indication of the nonexistence of the exact solution. 



It is of some interest to observe the numerical result by Guilloton's 



7 8 



methodj, which shows a plausible improvement in agreement with measured 



wave profile and wave resistance. Unlike the consistent successive 



approximation, the various conditions involved are not satisfied at the 



same order of magnitude. This is because the field equation is satisfied 



only in the first order while boundary conditions are partly satisfied up 



Q 



to the second order. In spite of such inconsistencies, Guilloton's 

 method may be regarded more useful than the consistent second order theory 

 from an engineering point of view. 



An Expression for the Solution of 

 Exact Nonlinear Boundary Value 

 Problem 



A special feature of the free surface flow is the nonlinearity of 

 boundary conditions. The direct nonlinear analysis is applicable only to 

 the simplest case such as monodirectional waves in a channel. Because of 

 its complexity, the only possibility of an analytical method describing 

 fluid motion around the ship hull is the perturbation analysis. As 

 mentioned in the preceding section, only the first order approximation is 

 useful for practical cases because a higher order approximation cannot 

 improve the result in most cases. The first order solution is usually 

 obtained by the linearization of the free surface condition at the be- 

 ginning. However, there is a possibility of giving a formal expression 

 for the solution which satisfies the exact nonlinear boundary condition at 

 the free surface. This expression is of no use by itself for the pre- 

 diction of wave resistance, but it facilitates general discussions of 

 perturbation analysis. 



First, we assume an inviscid incompressible fluid with irrotational 

 flow. A Cartesian coordinate system with z-axis vertically upward and the 

 axes of x and y is applied to the fluid on the still water surface. It is 

 convenient to employ dimensionless quantities. The length scales, e.g., 

 x, y, and z, are normalized by a characteristic length, SL, which may be 



