SECT. 5] ANALYSIS AND STATISTICS 571 



them to bear any phase relation with components in another band. In fact one 

 would expect the phases to be random. Moreover these considerations are not 

 restricted to large distances from the generating area, since it can be proved 

 (Longuet-Higgins, 1950) that any free configuration of the surface can be 

 resolved uniquely into a continuum of straight, long-crested waves, and if these 

 are due to an uncorrelated distribution of sources, the phases again would be 

 expected to be uncorrelated. 



This leads us to consider properties of the surface which are common to a 

 family of surfaces consisting of a sum of sinusoids with various relative phases 

 and amplitudes. In this family the phases are distributed "at random" and such 

 that, when averages are taken with respect to the phase distribution, they are 

 the same as averages with respect to x, y or t. We may write such a surface as 



00 



i{x,y,t) = 2 CLn^iO^ixknGOsdn + yhn^indn-ant+en), (6) 



n=\ 



where for deep water every arn^^gkn, and kn and dn are densely distributed 

 over (0 < A:« < 00, 0^6n^ ^n), and in any interval {k, k + Sk), {6, 6 + W), 



k + Sk e + d9 



k e 



E{k, 6) being a finite spectral energy function which we shall call the "direc- 

 tional energy spectrum" of the wave system. The phases en are randomly 

 distributed in (0, 2'n.) Strictly, the series should first be considered finite, 

 0^n<^N , say, and N taken to infinity at a suitable stage in the argument to 

 produce any required limiting result, but we hope that such procedures will be 

 taken for granted by the reader. Further E{k, 6) should ideally be expressed as 

 E{k, d; x,y,t), a, function varying so slowly with x, y and t that, over a limited 

 range, E can be regarded as independent of them. 



The model given above is based on that due to Longuet-Higgins (1957) and 

 is similar to a model in one dimension considered by Rice (1945) in his theory 

 of random noise. Two other formulations should be mentioned, essentially 

 similar to the above, but differing in minor details. Birkhoff and Kotik (1952) 

 expressed waves recorded at a point as 



ni 



Ut) = lim {clm)y^ 2 ^o« M-^i), (8) 



where, for any (large) value of w, the at are selected at random with probability 

 density Q{a), and the €i are random in (0, £77). The spectrum of variance, 

 equivalent to our E{k{a), 6] for constant 6, is |cQ(cr). 



A more widely known formulation due to Pierson (1952, 1955) is 



i{x,y,t)= PI" GO^[—{xcoQe + y^\nd)-at+€{(j,d)\V{A{a,d)YdGdd, (9) 

 a stochastic integral, also expressible as the limit sum of a number of increments 



