Recent Research on Ship Waves 



g«t>,2 + ^^ux= on z = (10) 



where subscripts denote partial derivatives. The notation and co- 

 ordinate system are as defined in Section II. By suitably non- 

 dimensionalizing the coordinates, we may replace (10) by the con- 

 dition 



I XX 



= on z = . (11) 



The general solution of this free- surface condition and of Laplace's 

 equation, not including local effects near the disturbance, is (cf. 

 Eq. (6)) 



=£ 



27r 



de f(0)e'''*'-'^ (12) 







2 2 



where k = (g/V ) sec 0, k = (k cos 0, k sin 0) , and x = (x,y). (In 

 Section II the wave angles were restricted to the sector - ir/z < < 

 tt/Z, Here we allow all values of in the integrand of (12), in 

 order to avoid taking the real part of the complex exponential; 

 Eq. (12) will be real if f(0) = f *(tt- 0), and to avoid difficulties with 

 the radiation condition we shall assume that (12) holds only if x is 

 large and negative.) 



The second-order free-surface condition, analogous to (11), 



4'2z+ 4>2xx= 2V4>, • V(|>„ - cj>,,(4>,„ + 4>,,„). (13) 



By inserting the first-order solution (12) for ^, in (13), and replac- 

 ing products by repeated integrals, it follows that a particular solu- 

 tion of (13) will be 



.2w ^Zir 



p-" p-" k z+ik -x 



<})2=\ d0 \ d02 f(0 ) f(02) W(0, ,02)e '^ '^ (14) 



where 



k,2= k(0,) +k(02). 



The weight function W is an algebraic function, determined by the 

 various derivatives in (13), and it can be shown that this function 

 contains only removable singularities. Thus by repeated application 

 of the method of stationary phase, cjig = 0(R' ) for large distances 

 R from the disturbance, and this second- order potential will be 



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