Ship Waves and Wave Resistance ^ 



Derivation of the Second-Order Potential 



We introduce the dimensionless coordinates X = 2x/L, Y = 2y/L, and Z = 2z/L, 

 where L is the ship's length. The velocity potential is nondimensionalized as 

 = 20/Lc. As speed parameter we use y^ = gL/2c2. 



Let Y= ±eF(X,Z) be the dimensionless representation of the hull geometry, 

 where B is the ship's breadth, e - b/l will serve as a perturbation parameter 

 and is considered as a small quantity. Let X = X3 and X = x^ be the equations of 

 two vertical control planes S^ and S^ ahead of and behind the ship (Fig. 1). Let 

 Sj, stand for the tank bottom plane Z = - H . Let Y = ± T be the equations of the 

 vertical tank walls s^ and Sg, where T = b/L and b = tank width. Let s^ stand 

 for the free surface Z = UX,Y) for x^ < x < X3 and T < Y < T, and let s/' stand 

 for the undisturbed free surface z = with the water-plane area of the ship ex- 

 cluded. Let S^ stand for the wetted surface of the ship, and let S^ stand for 

 the part of the surface up to Z= 0. Let D stand for the domain of the complete 

 flow, bounded by s^, Sj, s^ , s^, s^ , s^ and s^, and let D° describe the corre- 

 sponding domain if S^ and S^ are replaced by 5^° and Sj' . Let 0^ ^^ stand for 

 the Michell type first approximation to the exact potential 0. Let P stand for a 

 point in D or D° with coordinates X, Y, and Z, and let P' represent a point on a 

 boundary surface with coordinates ^, 77, and ^. Let G(P,P' ) stand for the po- 

 tential of a source of output ^-n as defined in the appendix. 



The functions 0, 0^ '^ , and G of the variables X, Y, and z are subject to 

 the following set of conditions: 



A. Laplace equation: A0 = in D, A0( ^) = in D° , and AG = 4r7S(P - p' ) in D° , 

 where A stands for SVSX^+ h'^/hY'^^ sVSZ^ and S means the Dirac delta function, 

 which is zero if P is unequal P' . (Condition A implies that G becomes singular 

 at - l/|P-P'| .) 



B. On s9 we have (linearized free surface condition): 



"Xo^ 



( 1) 





-Xn G, + Gxx = 0. 



X-X, 



Y-0 Y-T 



Z-§(X,Y)on S, 



Y-fF(X,Z) on S^ 



Fig. 1 - Nomenclature 



651 



