Wave Analysis Techniques to Achieve Bow-Wave Reduction 



77 [F^Cu) + G^Cu)] du 



and the wave resistance is given by 



Rw = 2 [F^Cu) + G^Cu)] (2 - cos2 0)du , (7) 



J - CO 



as was first shown by Havelock (7). 



In previous work (4-6) the author has used, following Eggers (8), the con- 

 venient "deep tank representation," which is obtained immediately from Eq. (6) 

 by replacing the integral with an infinite series. Assuming a transverse sym- 

 metric disturbance in a deep tank with straight vertical walls y = ±b/2 one ob- 

 tains the following general expression for a linearized free wave system far 

 abaft: 



00 



U^'V) = 2__j ^^ ^"'^ ^°^ (s^x) + /3^ sin (s^x)] cos (u^y) , (8) 



v = 



u^ - 277V /b , e^ - 1/2 for v= and 1 elsewhere, 



or alternatively 



CD 



Ux,y) = 2__^ e^ [A^ sin s^(x-^^)] cos (u^y) , (9) 



v = 



A. - yf^J^ . tan (s^^^) = -ay^^ ; 



see, for instance, Ref. 6a. The coefficients A^ and f ^ in Eq. (9) are, in fact, 

 actual amplitudes and phases of elementary tank waves, each of which can be 

 considered as a separate physical entity. For any given disturbance producing 

 the free wave in a tank of width b the coefficients a^ and /3^ are simply related 

 to the continuous functions F(u) and G(u) which constitute the free wave spec- 

 trum by the proportionality 



a^ = (477/b) G(u) , /S^ = (477/b) F(u) . (10) 



Assuming any sufficiently large hypothetical tank width b, Eqs. (9) and (10) 

 can also be considered as a numerical approximation to the exact integral in Eq. 

 (6) for laterally unrestricted fluid. The corresponding approximation for wave 

 resistance is then 



CO 



R, = (b/8) 2^ e^{aj + fij) (2 - l/s^^) , 



(11) 



735 



