equations can thus be formulated, and these can be solved in several ways, in parti- 

 cular by using an iterative solution procedure or inverting a matrix of influence 

 coefficients. The performance of a three-dimensional calculation method (measured in 

 terms of accuracy control, computing times, and complexity of implementation) must 

 obviously depend, to a large extent, upon the form and mathematical properties of 

 the integral equation and upon the solution procedure selected as the basis of the 

 calculation method. It thus may be useful to consider various alternative integral 

 equations and solution procedures. 



The object of this study is to present a new integro-dif ferential equation and 

 a related recurrence relation for determining the velocity potential. The results 



given in this study generalize those obtained previously for the particular problems 



9 10 



of wave radiation and diffraction at zero forward speed, ship wave resistance, 



and potential flow about a body in an unbounded fluid. 



The integro-dif ferential equation, defined by Equations (5.1)- (5. 4), is an equa- 

 tion for the velocity potential <^, rather than for the density of a related distri- 

 bution of sources or dipoles. This equation involves both a waterline integral 

 (i.e., a line integral around the intersection curve between the mean hull surface 

 and the mean sea plane), as in the problem of ship wave resistance, and a water- 

 plane integral (i.e., an integral of the Green function over the portion of the mean 

 sea plane inside the mean hull surface), as in the problem of radiation and diffrac- 

 tion at zero forward speed. The highly singular dipole terms (()(x)8G(^,x)/8n and 

 (!)(x)8G(^,x)/8x in the hull and waterline integrals, respectively, in the classical 

 integro-dif ferential equation defined by Equations (4.10c), (4,8), and (4.9), take 

 the forms [(()(x)-(j)(t) ]9G(t,x)/9n and [(})(x)-(f)(t) ] 9G(|,x)/8x, respectively, in the 

 modified integro-dif ferential equation obtained in this study. These modified dipole 

 terms are nonsingular, i.e., remain finite, as the integration point x approaches 

 any field point C where the hull is smooth (i.e., has a tangent plane). 



A recurrence relation is proposed for solving the integro-dif ferential Equation 

 (5.1) iteratively. This recurrence relation is defined by Equation (5.7), where the 

 initial (zeroth) approximation is taken as the nonhomogeneous term i>(Q in the 

 integro-dif ferential Equation (5.1). In the particular case of potential flow about 

 an ellipsoid (with arbitrary beam-to-length and draft-to-length ratios) in trans- 

 latory motion, along any direction, in an unbounded fluid, the first approximation. 



