Lee 



and the bar signs mean the complex conjugates, I.e. 



?i = <Pic - J<Pis fo^ i = 1,2. 

 On the cylinder surface 



where 



or, in terms of the stream functions, 



*6(xo.yo) = J7(^ly- ^2y). (28) 



In the far field <p6y '"*' as y -* - oo and tp^ should represent out- 

 going waves as |x| — ^ oo. The symmetric -flow condition implies 

 that 



^fi(x,y) = <p (-x,y). 



IV. SOLUTIONS FOR <p^ AND <p^ 



It will be assumed that solutions for the first-order potentials 

 <P\ and ^2 ^^^ known. The method of multipole expansions for 

 finding <p, and <p^ is described in Appendix B. 



The main difference between the first- and second-order 

 problems Is in the free- surface conditions. A first-order problemi 

 has a homogeneous differential equation for the free- surface condi- 

 tion (see e.g. (A-1) of Appendix A) whereas a second-order problem 

 has an inhomogeneous one (see e.g. (19)). When there exists a non- 

 constant pressure distribution on a free surface of negligible surface 

 tension the first-order free-surface- condition for an inconnpresslble 

 Irrotatlonal flow is represented by an inhomogeneous differential 



914 



