SECT. 5] 



ANALYSIS AND STATISTICS 



573 



It should here be remarked that although statisticians regard the Gaussian 

 distribution as intuitive, there exist very few published demonstrations of its 

 fitness. BirkhoflF and Kotik (1952) actually concluded from the available 

 evidence that the normal distribution was inappropriate, and later acceptance 

 is apparently based on some rather inconclusive examples shown by Pierson 

 (1952) (see also Putz, 1954). The writer has attempted to fit normal distributions 

 to sets of 2000 digital ordinates taken from records of wave height and slopes. 

 Three examples are shown in Fig. 2. There is no obvious systematic departure 

 from the normal curve, but, in fact, the y^ test for goodness of fit fails signifi- 

 cantly in each case. However, satisfactory fitting has been found for distribu- 

 tions which depend on the normal law (e.g. Cartwright and Longuet-Higgins, 

 1956), and there exist several other cases of agreement with average values 

 calculated with the same assumption. On the whole, it appears safe to regard 

 the normal distribution as a useful if not wholly accurate working rule. 



Fig. 2. Gaussian probability distributions fitted to sets of 2000 ordinates from ocean-wave 

 records. Left to right; wave height (feet), up-wind wave slope (radians), cross-wind 

 wave slope (radians). The vertical ordinates are of percentage of population per step. 



Of the various formulations described above for a linear statistical model of 

 sea waves, we shall adopt the notation (6), at the risk of offending pure statisti- 

 cians. Our reasons are that its physical meaning is clear, it embodies all the 

 observed properties of stationarity, ergodicity (i.e. equivalence of time and 

 space averages), and differentiability, and also because its mathematics is 

 basically simple and easily handled by the average mathematical oceano- 

 grapher without requiring knowledge of very specialized statistical techniques. 



4. Properties of a Wave System in Terms of its Directional Energy Spectrum 



It is well known (Lamb, 1932, p. 369) that the mean wave energy per unit 

 area of water surface, which in deep water is half kinetic and half potential, is 

 given by the mean value of p^^2 oyer all x, y, t. On squaring the series (6), all 

 the cross products average out to zero, and each squared term contributes a 

 mean value Ipgan^ to the total energy. Thus, on comparison with (7), we see 

 that pgE{k, 6) represents the mean energy per unit area contributed by wave 

 components {k, 6) per unit increment of k and 6. This justifies the term "energy 



