Wave Analysis Techniques to Achieve Bow-Wave Reduction 



Z- = V2(C3-XJ(S± l)/77 . (23) 



The further evaluation of the weighted Fourier transforms c* and s* , thus cor- 

 rected for any truncation errors at the after end -x -»» , proceeds as before but 

 with modified weighting factors. Thus 



ba(u) = 47tG(u) = — [C* cos (uy) - S* sin (uy)] , (24a) 



2s - 1 



b/3(u) = 47rF(u) = — ^ [C* sin (uy) + S* cos (uy)] , (24b) 



2s - 1 



and 



P. = i Jl(C')^* (S*)^l ^T^^fbr^. (25) 



It is apparent that the numerical ambiguity at s = l has been eliminated. It may 

 be noted in passing that the function ^(x,y) decays so rapidly for x->cd that no 

 truncation corrections are in practice needed at this end. 



For the sake of record it may be recalled that Newman (9) also derived a 

 certain truncation correction for wave resistance in a different way but starting 

 with an assumed asymptotic behavior of wave height similar to Eq. (18). 



NUMERICAL CHECKS ON THEORETICAL WAVE SYSTEMS 



Before trying to apply the longitudinal-cut methods in practice it appeared 

 advisable to check them first on a few theoretical wave patterns of predetermined 

 spectra. As a simple example, typical of a bulb wave system, the linearized 

 wave pattern of a submerged sphere of unit radius r = l, located at x = y = 0, 

 z = -f was investigated. This is represented to a good order of approximation 

 by the functions 



r- x/y 

 ^x,y) =2 s^ exp(-fs2) sin (sx + uy) dt , (26a) 



J _ 00 



n- x/y 

 ^y(x,y) = 2 us^ exp(-fs2) cos (sx+uy) dt , (26b) 



J _ 00 



s = yt^ + 1 , u = t /t^TT , y > , 



which are adapted from Havelock (13) after neglecting certain local terms in- 

 volving double integrals (which had to be dropped in view of limited computer 

 facilities at the disposal of the author). 



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