Pien and Lee 



will apply, for suitable shapes f (x, y) , essentially because here 



the damping coefficient depends on an integral, with respect to the 



waveangle , of the square of the surface integral. 



T L/2 



f f -Kz + iKx cos df(x, z) , , 



J J e — ^ dxdz 



- L/2 dz 



and since the wavenumber K is independent of the parameter , the 

 above integral will vanish for cylindrical vessels having the appro- 

 priate two-dimensional shape. Based on this argument, we can con- 

 clude that zero damping can occur not only in two dimensions but also 

 in three, provided the forward speed is zero. 



Finally, let us consider the case of forward speed. Then, as 

 shown, for example, in my paper in the June 1959 issue of the Jour- 

 nal of Ship Research the threee -dimensional damping coefficient is 

 again proportional to the square of a surface integral similar to that 

 shown above, but now the wavenumber K is no longer a single cons- 

 tant, but, depending on the value of the Brard parameter wU / g , 

 takes on either two or four discrete values, each of which depends 

 on . Thus, since the square of the surface integral is integrated 

 over a continuous spectrum of K ; the probability of zero damping 

 is greatly reduced. The three-dimensional theory which I presented 

 in 1959 was eclipsed by the subsequent success of the simpler strip 

 theories, which could be more easily refined to account for the effects 

 of finite beam. But Dr. Lee has discovered a situation where the 

 three-dimensional and forward- speed effects may be more important, 

 and I hope to have the opportunity to pursue this matter further, by 

 resurrecting my 1959 theory and applying it to the catamaran configu- 

 ration. 



DISCUSSION 



Robert F. Beck 



University of Michigan 

 Ann Arbor j Michigan, U.S.A. 



Personally, I am wondering if you have done any non-head- 

 sea calculations. What you have shown in the paper is just for pitch 



542 



