APPENDIX E 



DERIVATION OF PRESSURE TO SECOND ORDER 

 FOR TWO PROGRESSIVE WAVES AT DIFFERENT FREQUENCIES 



Consider the problem of the nonlinear interactions o£ waves at two 

 distinct frequencies traveling in the same direction. The complete 

 boundary value problem is well known. 



The Laplace equation, 



V^({) = 0, (E-1) 



applies throughout the fluid below the free surface. 



The boundary condition, 



11. ^ g |1 + 2V<^ • V |{ + i V4> • V (V(}>-V*) = 0, (E-2) 



ot 



must be satisfied on the free surface, y = n. The boundary condition on 

 the bottom is : 



lim 3<j) _ p. .p ,. 



y^-oo 37 - (E-3) 



for an infinitely deep fluid. In addition a radiation condition re- 

 quiring the generated waves to travel away from the body is needed to 

 ensure uniqueness of the solution. 



In this formulation the x axis lies in the direction of incident 

 wave propagation. 



The difficulty in solving this boundary value problem stems from the 

 nonlinearity of the free-surface boundary condition. 



In order to "linearize" the free-surface boundary condition, expand 

 the velocity potential, cfi, in a Taylor series about the undisturbed free 

 surface: 



<})(x,n,t) = <()(x,0,t) 



. n[^il^l] ^ \ n' [^^%^^] ^ O(n^) . (E-4) 

 ^ y=0 ■' y=0 



Also expand n and '^ in power series: 



n(x,t) = eri^^-'(x,t) + e^n^^-'(x,t) + O(e^), 



(^(x,y,t) = e<f^ -^(x,y,t) + z"-^^ ^(x,y,t) + Q{zn . (E-5) 



148 



