p 



2 2 ^^1^ 2 . 2 '^^i^ 



P(t) = - J {w^ A. e + 0)2 ^2 ^ 



- 2a) u)„A-A„e cos [(k. - k„)x 



where. 



- (0)^ - ^^t + 6^ - 62]}, (11) 



= fluid density. 



0) ,co = incident wave frequencies, 



A,,A„ = incident wave amplitudes, „ „ 



k,,k_ = incident wave numbers = , , 



12 g g 



6 ,6 = incident wave phase angles. 



Combining this pressure with the pressure obtained from the linear theory 

 and integrating over the hull would provide additional exciting- force 

 terms at zero frequency and at the difference frequency. Carrying the 

 nonlinear exciting-force terms back through the linear response analysis 

 should provide a quasi- linear approach. While there is no reason to 

 expect this to provide exact correlation with measured data, the quasi- 

 linear approach would at least permit the natural phenomena to enter into 

 the mathematical analysis. 



One would expect terms to appear in the second-order pressure (eq. 

 11) at twice the incident wave frequency and at the sum of the inci- 

 dent wave frequencies. Terms at twice each of the incident wave fre- 

 quencies can be derived by applying the trigonometric relationships to 

 the terms at zero frequency. While a term at the sum of the incident 

 wave frequencies does not appear in the second-order incident wave po- 

 tential, this term may result when the second-order potentials repre- 

 senting diffraction or forced oscillation in calm water are included, 



A great deal more effort is required in this area to complete the 

 analysis. There is also one other area where a nonlinear, or quasi- 

 linear, analysis should be investigated. This is in the roll-damping- 

 coefficient. Here, viscous effects seem to be important, and while the 

 problem has not been dealt with within the present study, investigators 

 have included a. term proportional to velocity squared in the equation 

 for roll motion. 



Results . 



The computer program given in Appendix D has been developed to 



22 



