I. One-Dimensional Power Spectra 



The concepts leading to simple wave spectra can be illustrated by 

 assuming that a train of pure sine waves is propagated over the water sur- 

 face such that the water surface elevation n varies with distance x and 

 time t : 



n = igH cos (kx - at) ( I ) 



where k = 27r/L is the wave number, o = 2tt/T is the wave frequency, and H, 

 L, and T are the wave height, length, and period, respectively. The mean- 

 square elevation of the water surface <^n^ )>at any time is given by: 



<^^> " r 7- cos2 (kx - at) dx \ <=^ (2) ^\L 

 ^ Jo 4 '", ...L.xMtA 



(rf - z "^ 



hoi d I ng >&f^ constant, this becomes ((n^N = H^/S. The mean energy (including 

 both kinetic and potential) per unit area of water surface is equal to the 

 product of the mean-square elevation and the weight per unit volume pg of 

 the fluid, then 



E = pg <n2> - ^ pgH2 (3) 



Since pg is nearly constant for sea water, it is customary to use the mean- 

 square elevation <^n2^ as a measure of wave energy. 



If the sine wave is amplitude-modulated in a random manner, the wave 

 train can no longer be represented by a single height, but Includes a range 

 of heights. In this case, the appropriate wave height statistic in equation 

 (3) becomes the root-mean-square wave, height Hprns ( Longuet-HIgg I ns, 1952). 

 The significant wave height, H|/3, is related to the rms wave height of 

 simple narrow band spectra by the relation 



^h2 = h2 = 8 <n2> (4) 



1/3 rms 



It should be noted that the root-mean-square amplitude becomes a^^g = /2 • 

 Eroot-mean-square elevatTonH = ^ • t\^^^. 



a. Real Waves . Real ocean waves vary In both frequency and am- 

 plitude In a far more complex manner than the simplified waves illustrated 

 above, and their spectral analysis requires a more sophisticated technique. 

 Also, the presence of higher harmonics associated with the asymmetry of waves 

 in shallow water further complicates interpretation in terms of simple waves. 



The computation techniques developed In communication engineering 

 (Blackman and Tukey, 1959) for determining the spectra of signals in the 

 presence of noise have been found to be appi icable to the analysis of ocean 

 waves (Munk, Snodgrass and Tucker, 1959). The general procedure is to ob- 

 tain a digital time series of wave elevation, n(t), or a related parameter 

 such as pressure, spaced at equal sampling intervals At. The data is then 

 processed by a digital computer. After checking for errors and correcting 



2J 



