From a spectral matrix of the form in Equation (4.13), M = N(N-l) +1 

 different equations can be set up using Equation (7.27). This allows 

 us to get a system of equations for any m of the unknown coefficients 

 aQ, ai, a2, ... ; bj^j ^2' ^3» *•* ^^'^^^ assuming the rest of the 

 coefficients are negligible. We can then solve for the m desired 

 coefficient values . This has not worked well in practice for two 

 reasons. The inverse of the matrix of constants obtained is sparse 

 and often ill-conditioned. Further, if the wave energy is from a nar- 

 row beam width (30 degrees or less), the first 100 harmonics in the 

 Fourier series expansion can be significant. There is perhaps a 

 more efficient orthogonal set of functions than the sines and cosines 

 of the standard Fourier expansion. Search for such an orthogonal set 

 should prove fruitful. One might start with Walsh or Haar functions. 

 See Hammond and Johnson (February 1960) . 



8. SUMMARY 



It is believed that the least square method of using the informa- 

 tion in a spectral matrix is the best method presently available. 

 Examples of such analysis can be found in several of the papers in the 

 bibliography. A collection of ocean-wave induced, bottom pressure 

 directional spectra from these papers is given in Appendix A. 



An iterative extension of the least square method can be found in 

 an excellent paper by Munk et al (April 1963) . Some details of this 

 method are given in Appendix C along with an example result and a 

 FORTRAN program for the method. 



There is merit to using the coherency, 



R it) = _iRiiiilL 



to form the weights b. . in Equation (4.16). One idea being explored is 





39 



