SEAWAY 



73 



The constant by which ^E is to be multipHcd in order 

 to obtain the average of \/n highest waves (or ampli- 

 tudes) is shown in Fig. 84 as a function of e. Cartwright 

 and Longuet-Higgins apphed their calculations to five 

 sample records; a pressure record of swell, two records of 

 waves measured by a ship-borne wave recorder, a re- 

 cord of a ship's pitching, and a record of a shijj's rolling. 

 The values of t in these five examples were found to be: 



Swell pressure 0.41 



Wave-height records 0.57 and . 67 



Ship's pitching record 0.48 



I's rolling record . 20 



It is seen that in all of these cases the narrow-spectrum 

 calculations are se\erelj^ reduced by the broad-spectrum 

 corrections. This is strange, in view of the fact that 

 Jasper (1956) found excellent agreement between height 

 distributions of spectra of all shapes (and different pa- 

 rameters) and the Raylcigh distribution, without any cor- 

 rective measures. This may be explained by consider- 

 ing that, in the case of swell, where high frequencies are 

 not prominent, the height distribution approaches the 

 Rayleigh law. While in wind-generated sea (wide band 

 spectra), high freciuencies are very prominent and the 

 distribution approaches the Gaussian law. If the high 

 frequencies are ignored, the Rayleigh law* will apply. 



Since rrii grows faster than vu and im, as higher fre- 

 quencies are added, and since e — »■ as nii -* ???.2^/»io, it 

 is seen that the correction factor for an ideally narrow- 

 spectrum will not be unity unless all the high frequencies 

 in the spectrum are resolved. The correction term of 

 Cartwright and Longuet-Higgins is well defined; aliasing 

 of energy and filtering of the signal seem to defeat its 

 purpose. 



Rice (K), derived the expression for the a\erage 

 number A'^o of zero up-crosses per second. In terms of 

 moments of the spectrum function 



iVo 



1 



{mJnuyi- 



(129) 



and by definition the average apparent period 

 1^ 1 



N, 



= 2ir{m(,/m2)^'' 



(i;i0) 



The total number of maxima N„,:,^ is expected to be 



Nr, 



T^ (wu/m,)"- 

 2ir 



(131) 



The proportion, r, of negative maxima to the total 

 number of maxima (from Cartwright and Longuet-Hig- 

 gins, 1956) is 



1 



9 



1 - 



(mowu 



[1 - (1 - e=)' 



(132) 



For a narrow spectrum, r ^ as e -^ 0; i.e., all the 

 maxima occur above the mean level. The sea surface is 

 represented in this case by a imiform, regular swell with 

 gradually varying amplitude and length. For a broad 



Fig. 84 Graphs of 7)(l/«), the mean height of the 1/wth 

 highest maxima as a function of e, and h = 1,2,3,5 and 10 



spectrum /■ -^ 0.5 as t —>■ 1.0. In this case, ripples ride 

 crests and troughs of larger wa\'es so that there are ecjual 

 numbers of maxima above and below the mean level. 

 Pierson (19546) applied Equation (130) to Neumann's 



spectrum and expressed f as a function of wind speed, 

 [', appropriate to the fully de\'eloped sea state 



f = 0M6{2TU/g) (Neumann) (133) 



and 



r„,ex = V2T (Neumann) (134) 



where T^ai denotes the period at which the spectral 

 density is a maximum. 



Pierson (1954/)) also considered the case of a sea de- 

 fined by a Neumann scalar spectrum modified by inclu- 

 sion of the factor cos-0, an estimate of angular dispersion 

 of wave energy which is assumed to be boimded by 

 (— ir/2 < 9 < 7r/2). For this case, Pierson derived the 

 expression 



9 



Lo = 



g 



(135) 



which states that the average distance between the suc- 

 cessive dominant crests of the sea surface, when obser\'ed 

 at a fixed time as a function of space, is equal to two 

 thirds of the value computed from the classical formula 



for the harmonic wave period T = T. 



Some of the statistics just given are only of general in- 

 terest and others may be computed directly from the 



