equation for surface elevation. The second-order pressure may be reduced 

 to: 



(2) p r 2, 2 2k, y 2. 2 2k^- 



^ ■' = - J {to. A, e I-' + 0)2 A2 e 2 



- 2a)^uj2A;^A2e^'^l*'^2^^cos[(k^ - k2)x - (co^ -a^)t + 6^ - 62]} (E-23) 



which indicates that the second-order pressure is composed of a component 

 independent of time and at the "difference frequency". 



This is surprising since the equation for the free-surface elevation 

 (eq. 18) includes terms at twice the incident wave frequencies and at the 

 sum of these two frequencies. Using trigonometric relationships the 

 first two terms in equation (E-23) could be expanded to yield terms at 

 twice the incident wave frequency. A term at the sum of the two inci- 

 dent wave frequencies may appear in the pressure computed using the 

 velocity potentials representing wave diffraction or forced oscillation. 

 It might also appear if the present analysis were carried to the third 

 order. The derivation included here was intended to reveal the presence 

 of a low- frequency component in the exciting force and has not been used 

 to determine the other velocity potentials or carried beyond the second 

 order. 



3. List of Special Symbols for Appendix E . 



A, ,A^ = Wave amplitudes 



g = Acceleration of gravity 



k, ,k„ = Wave numbers, , , respectively 



1' 2 g g ^ 



x,y = Cartesian coordinates (x-directed parallel to the 



direction of wave propagation, y-directed vertically 

 upward) 



6,, 6^ = Wave phase angles 



ri(x,t) = Free-surface elevation 



<l'(x,y,t) = Velocity potential 



oj, ,to„ = Wave circular frequencies 



155 



