Sharma 



which is again "exact" for a tank of width b. The quantity A (in an actual tank) 

 or Ab (if tank width b is only a numerical artifice) considered as a continuous 

 function of the wavenumber u or s seems to be a quite reasonable definition of 

 the "amplitude spectrum" and will, therefore, be used for its graphical repre- 

 sentation in this paper, as has already been the practice in previous publications 

 of the author. 



PRINCIPLE OF LONGITUDINAL -CUT ANALYSIS 



For the sake of completeness the theory of longitudinal-cut analysis, al- 

 ready derived and discussed in Ref. 4, will be briefly recapitulated here. Con- 

 sider the following Fourier transforms 



-a, 



C(s,y) + iS(s,y) = ^(x,y) exp(isx)dx, 1 < s < w , (12) 



J- 00 



of any longitudinal cut y = const, through a transverse symmetric wave system 

 z = ^x,y) generated by a ship or model in laterally unrestricted deep water. 

 Now it was shown in Ref. 4a, by exploiting the asymptotic behavior of ^(x,y) for 

 large y, that the sine and cosine amplitude functions characteristic of the wave- 

 generating agency, as defined in the foregoing, can be directly obtained from the 

 above transforms by the simple relationship 



4n/s2 - 1 ^ ^ (13a) 



ba(u) = 47tG(u) = [C sin (uy) - S cos (uy)] ^ ' 



2s - 1 



b/Q(u) = 4-r7F(u) = — — [C sin (uy) - S cos (uy)] , '^^^^ 



2s2 - 1 



while the wave resistance can also be computed directly, without reference to 

 the amplitude or phase spectrum, by the formula 



^\:[ f ~ ^ (C^ + S^)du. 



(14) 



It may be noted in passing that if the wave pattern is transversally unsymmet- 

 ric, one longitudinal cut on either side of the disturbance is necessary and suf- 

 ficient for the determination of the entire spectrum in the range -co < u < oo and 

 hence for the computation of the original wave resistance integral in Eq. (7). 



It was further pointed out in Ref. 4 that similar relations can be derived for 

 the analogous Fourier transforms of wave slopes ^^ or i^ measured along lon- 

 gitudinal cuts. Especially, a decided numerical advantage was claimed in favor 

 of the transforms Cy and Sy of the transverse wave slope ^y over those of wave 

 height i because of a more reasonable weighting function in the following rela- 

 tions corresponding to Eqs. (13) and (14): 



736 



