Wave Analysis Techniques to Achieve Bow-Wave Reduction 



ba(u) = 47rG(u) = ^ [C„ sin (uy) + S„ cos (uy)] , (^^a) 



s(2s2- I) y ^ ' y 



b/3(u) = 477F(u) = ^ [C cos (uy) - S^ sin (uy)] , (15b) 



s(2s^ - 1) ^ ' 



and 



R =-f -^ \ (C2+S2)du. (16) 



It is obvious from an inspection of the foregoing equations that in the most im- 

 portant and interesting region of the spectrum at u = o, that is at s = 1, a numeri- 

 cal zero must be multiplied with a numerical infinity on the right-hand side of 

 Eqs. (13) and (14) in order to obtain a nonzero value of the transverse wave am- 

 plitude Aq , while no such difficulty occurs with the Eqs. (15) or (16). 



Finally, it should be mentioned that Newman (9a) independently derived an 

 equivalent of Eq. (14) and presented it simultaneously with the author (4) to the 

 International Seminar on Theoretical Wave Resistance at Ann Arbor, Michigan, 

 in August 1963, while Pien and Moore (10) and Shor (11) proposed somewhat dif- 

 ferent methods of longitudinal-cut analysis on the same occasion. 



TRUNCATION CORRECTIONS 



The numerical difficulty just mentioned finds another expression in the fact 

 that under certain circumstances the Fourier transforms c, S and Cy, Sy , etc., 

 do not converge at all. In fact the theoretical free wave spectrum of any ship 

 form is such that in general the function ^(x,y) is not absolutely integrable at 

 the afterend -x-»co, andc^ + S^ becomes infinite at the point s = l . 



Similarly, for a surface disturbance (as distinguished from a submerged 

 disturbance of arbitrarily small but nonzero depth of submergence) also the 

 function ^y(x,y) is not absolutely integrable and produces an infinity of Cy^ + Sy^ 

 at the other end of the spectrum, namely, as s-oo. However, this problem can 

 be circumvented by restricting the computation to a finite range of wave number 

 s without prejudice to the practical aim of wave analysis, since this end of the 

 spectrum contributes but little to wave resistance. In one typical case, for in- 

 stance, it was estimated that truncating the spectrum at s = 3 would cause an 

 error of less than 0.5 percent in wave resistance. 



Fortunately, the aforesaid convergence and the associated truncation prob- 

 lem (arising from the practical restriction of the x integration to a finite range) 

 in the analysis of height records can be approximately solved by a simple trick. 

 Since the amplitudes are in any c ase ob tained by multiplying the Fourier trans- 

 forms C and s with the factor 4y/s^- l/(2s2- i), which becomes zero at s = l, 

 one might as well start right away with the following weighted Fourier trans- 

 forms: 



737 



