Salve s en 



Re 



at z = X + ib 



(11) 



where v - gAJ^ is the wavenumber. Including terms of second-order, one ob- 

 tains after some manipulation that the second-order free-surface condition is 



where 



Re 



;(2) 



+ iv 



V^(2) 



irf(''> 



- f(x) = ^ Iw^'M ^ - Im w(i) Re 



at z = X + ib 



z z z 



(12) 



It also follows from these substitutions that the corresponding approximations 

 to the wave height are 



;(i)(x) = A^(l)(x,b), 



,(2) 



(X) 



U ^ 



(x,b) + -^ ^(1) (x,b) 0^'^ (x,b) 



(13a) 



(13b) 



In this work the complex potential will be used only to determine the wave 

 elevation and to obtain the wave resistance from momentum considerations in 

 the far field. We will not need, therefore, the general expansion given by Eq. (8) 

 but only the expansion given by Eq. (10), which is valid near the free surface 

 and in the far field. The expansions and boundary conditions to be used for the 

 more general problem are discussed by Giesing (5). It is seen from Eq. (10) 

 that we have two second-order terms, ewg^ and e^wp^. Tuck (4) has shown that 

 in obtaining the wave resistance for a circular cylinder it is more important to 

 include the second-order free-surface term e^wp^, than the body correction in 

 term ewg^. Assuming that the same applies to the problem treated here, we 

 will disregard the term ewBj and simply represent the body by its singularity 

 distribution in an infinite fluid. This implies that the cylinder-wall condition is 

 satisfied only to the first order of approximation. 



It can certainly be argued that the second-order theory presented here is 

 not consistent; however, it is believed that the method used will give a solution 

 far more realistic than the first-order theory and therefore that it is an im- 

 provement over the linear theory. 



MATHEMATICAL SOLUTION 



Complex Potential and Wave Elevation 



The two-dimensional submerged body will be represented mathematically 

 by a singularity distribution of the following strength and location: eleven 

 sources equally spaced along the x axis between x = and x = 1.0, and with 

 strengths m^ given by 



600 



