64 



the potential function at each order. Nonlinear frequency modulation was not taken 

 into account, so the frequencies and wave numbers of each wave independently satisfy 

 the linear dispersion relation for water of infinite depth: 



u^l^gKl (4.8) 



/i and /2 are independent of each other, and are the familiar linear wave solution: 

 h = —, — e 1 smSi /2 = — — e ^ smS2 (4.9) 



fcl K2 



The higher order terms are all functions of both waves, and thus include the interac- 

 tion of the two waves. 



A number of investigators subsequently expanded upon this work. Hsu and Chen 

 (1992) presented a detailed examination of Longuet-Higgins (1962), pointing out diffi- 

 culties that arise from the assumption of the linear dispersion relation. They presented 

 a more mature analysis, including higher order modulation of the wave frequencies, 

 and higher order self interactions of the individual waves. This resulted in a com- 

 plete theory up to third order for two intersecting waves in deep water. Hsu and 

 Chen also proposed a systematic ordering in the phase relationships generated by the 

 interactions of two steady waves to arbitrary order. 



Expanding upon this work, Ohyama, Jeng and Hsu (1995a) extended the per- 

 turbation expansion method in a number of ways. The most recent version of the 

 method accommodates any number of waves, allows for water of finite depth, and is 

 accurate to fourth order. This last method can compute the water surface and full 

 kinematics of a highly irregular sea, as produced by a large number of intersecting 

 waves. A more detailed discussion of the method is given in Appendix A. 



Ohyama, Jeng and Hsu's work suggests a form for the potential function that can 

 be used for a local method for the recreation of three dimensional kinematics from 

 arrays of wave measurement devices, including water surface arrays, pressure arrays, 

 directional current meters, or any combination of these. 



The general form for the potential function representing A'^ intersecting steady 



