V. WAVES 



15. ANALYSIS AND STATISTICS 



D. E. Cart WRIGHT 



1. Introduction 



The mathematics of periodic and other analytically-defined waves in a heavy 

 fluid bounded by a free surface has been developed over more than a century ; 

 the simpler properties are well known and the theory can be found in works on 

 hydrodynamics (e.g. Lamb, 1932; Stoker, 1957; Coulson, 1944). Nevertheless, 

 methods of analysing the ever-changing random pattern of humps and hollows 

 continually present in the sea have been evolved only during the last fifteen 

 years or so. Previously it was thought that the randomness was beyond analysis, 

 that the best one could do was to assign average values for wavelengths, 

 directions, etc. and to apply to these the classical laws derived for periodic 

 motions, usually with but moderate success. Even Chapter XIV of "The 

 Oceans" (1942) does not go far beyond this stage. Apart from the beneficial use 

 of modern recording instruments, recent progress owes a great deal to modern 

 research on the statistical analysis of random noise, developed by telecommuni- 

 cation engineers. The study of sea waves has thereby developed into a combina- 

 tion of time-series analysis with a sort of statistical geometry, though tied as 

 firmly as possible to the basic laws of hydrodynamics. The fundamental 

 newness of this approach is that the variables are treated not as analytical 

 functions but as stochastic processes, definable only in terms of probabilities. 

 Despite its apparent vagueness, this has been found to be the only way of 

 coherently expressing the disordered oscillations of the sea ; it has not only 

 explained many observed features but has already been successfully applied to 

 important engineering problems connected with the sea, such as the motions 

 of ships. As is to be expected, the material is scattered among a number of 

 scientific papers, and the following pages are an attempt to give a connected 

 account of the most relevant results. 



If we defer for a moment an exact definition of what we mean by a "spectrum" 

 of sea waves, and accept intuitively that it represents the distribution of 

 "energy", or mean-square oscillation, over a scale of frequency. Fig. 1 can be 

 taken as a typical overall picture of the average distribution of energy present 

 in an unfiltered recording of sea-surface level above a fixed point. The spectrum 

 is seen to extend over a wide range of frequencies from the smallest ripples 

 (capillaries) of some cycles per second to meteorological changes of a cycle in 

 several hours. There are also long-term variations in sea-level extending over 

 years, due to climatic and geological changes. Various zones of interest are 



[MS received December, 1959] 567 



