INTRODUCTION 



The report presents an introductory discussion of the mathematics 

 pertaining to the directional analysis of ocean waves. The presenta- 

 tion is tutorial in form but does require a reasonably complete mathe- 

 matical background; a background equivalent to that required in reading 

 Kinsman's textbook Wind Waves (1965). 



The level of the presentation is moderate at the beginning. The 

 level picks up rapidly toward the middle but there should be sufficient 

 detail and redundancy in the mathematics to allow the reader to follow 

 the development without having to rediscover too many omitted steps. 

 It is in this sense that the report is tutorial. In some places the 

 mathematical development is intuitive rather than rigorous. This is 

 deliberate in order to provide insight and understanding. In most such 

 cases, references to rigorous reports are given. 



The development is reasonably detailed so that the interested 

 reader may apply the methods presented and use the report as an entry 

 point into the rigorous theory of the directional analysis of ocean 

 waves. In this respect, if the report serves as a bridge across the 

 gap between a handbook and a rigorous and sparse theory on the subject 

 then the objective of the report will have been fulfilled. 



The report first presents an intuitive development of a sea sur- 

 face model that assumes the sea surface to be a two-dimensional random 

 process definable in terms of a directional power spectrum. A discus- 

 sion of the space and time covariance function and its relationship to 

 the directional power spectrum follows . Both one- and two-sided power 

 spectra are discussed; however, the main development is in terms of 

 the two-sided spectrum. Next, the relationship between the power and 

 cross power spectrum for two fixed locations and the sea surface 

 directional spectrum is developed. Explicit relationships for the 

 special cases of an isotropic sea and a single wave of a given direc- 

 tion and frequency are then obtained. The related topic of the direc- 

 tional resolving power of an array of wave transducers is then 

 presented. 



Using the preliminary developments as a basis, several methods for 

 the directional analysis of ocean waves based on the information 

 obtainable from an array of wave transducers are presented. The methods 



