SECT. 5] ANALYSIS AND STATISTICS 587 



2 of this Chapter). Higher approximations to a purely periodic wave, such as 

 those of Stokes and Gerstner, are not much help in dealing with random wave 

 patterns, but the basic method of taking the linear spectrum as first approxima- 

 tion and substituting this into the neglected terms of the equation of motion, 

 to obtain a better approximation, is still valid. This procedure is worked out 

 by Tick (1959) for a unidirectional wave pattern recorded at fixed points. If we 

 drop the terms involving y in equations (1), (2) and (3), and eliminate ^, the 

 following surface condition is obtained for 2 = 0, correct to second order : 



d^ dcf> _ 8^ dcf> 8(f> 8^ 8cf> 8^cf> 1 / 8^ 8^ 8^ 8(t>\ 

 'W~^~8z ~ '82^~8i~ 'dxdxdt~'8z'8zJt~g\8zJtW^8z8t^^j ^ ' 



With the right-hand side put equal to zero, we can develop a linear model of 

 random waves as described in section 3, with an energy spectrum Ei{a) of 

 surface height measured at a point. Substitution of the latter into equation (32) 

 then yields a spectrum Ei{a) + E2{(y), where 



^2(0-) = — {G + 2aj)^Ei{G+co)Ei{aj)dco 



+ —((J^-2cTco + 2co^)Ei{a-cjo)Ei{co)daj. (33) 

 Jo r 



The first of these two integrals is usually much smaller than the second, which 

 is greatest for frequencies a little over twice the peak frequency of Ei{a), while 

 both are usually small compared with Ei{a). 



Longuet-Higgins (1950) also obtained a double-frequency effect for the 

 second-order pressure variations in deep water due to standing-wave com- 

 ponents in the spectrum. The importance of this effect is that, though as a rule 

 negligible at the surface compared with the first-order pressure, the second- 

 order pressure variations persist to great depths while the former decrease 

 exponentially and ultimately become negligible. It was shown that the principal 

 effect consists of an integral of the product of spectral densities at wave numbers 

 opposite in sign, with twice their associated frequency. A spectrum was worked 

 out by Kierstead (1952). 



If we disregard second-order effects in waves at the surface, non-linear terms 

 may still be introduced by the recording apparatus. An important example 

 studied by Tucker (1959) is the signal from an accelerometer moving with the 

 wave surface but aligned normally to it and therefore not truly vertical. The 

 spectrum of the doubly-integrated signal is to first order the true spectrum of 

 wave height E\{a), but when second-order terms are included, an error spectrum 

 is introduced, given by 



M<^)=^Vt- co\a-ojYE^{ay)E^{a-co)doi, [^i( - co) = ^1(0;)], (34) 



where R2^ = {A2^ + B2^)IAq'^^\ is a parameter depending on the angular 

 spread in the directional spectrum E{k, 6), as defined in section 5 of this Chapter. 



