It is clear that the application of a weighting function will reduce the total 

 energy in the record. This effect is partly compensated for by the following 

 equation: 



/<P Z > 



P"(j) = / 5- P'(j) (4) 



<p ,z > 



thereby ensuring the same total energy in the altered and original time 

 series, where <p 2 > is the mean square value of the original time series 

 and <p ,2 > the mean square value of the weighted time series. It is the 

 altered time series p"(j) that is subjected to FFT analysis. The primes 

 will be dropped hereafter for convenience. The average mean depth of water 

 overlying the pressure sensors, Ad, is obtained by averaging the m time 

 series to obtain a (0). For two separate time series records, m = 1, 2 (wave 

 gages 1 and 2) , 



Ad = 0.5 [ai(0) + a 2 (0)] (5) 



The total water depth, d, is the sum of Ad and the distance, B, of 

 the pressure sensors above the bottom (in later examples B = 0.76 meter) . 



Each FFT pressure coefficient is transformed to a water surface displace- 

 ment coefficient by the following linear wave theory relationship discussed in 

 the Shore Protection Manual ( SPM) (see Ch. 2, U.S. Army, Corps of Engineers, 

 Coastal Engineering Research Center, 1977): 



water surface coefficients dynamic pressure coefficients 



1 

 [a m (n), b(n)l [a m (n) , b m (n)l (6) 



in which the subscripts n and p denote water surface and dynamic pressure 

 coefficients, respectively. The factor 



cosh k(n) B 



K_(n) = (7) 



z cosh k(n) d 



where y i- s the specific weight of fluid (seawater) and is included when 

 pressure coefficients are in normal units of pressure (i.e., N/M 2 or equiva- 

 lent). In equation (7), B represents the distance of the pressure sensors 

 above the bottom and k(n) is the wave number associated with the angular 

 frequency, to(n) = (2TrnAf), as obtained from the linear wave theory dispersion 

 relationship 



a)(n) 2 = gk(n) tanh k(n) d (8) 



One of the disadvantages of measuring waves with near-bottom pressure sen- 

 sors is evident by examining equations (6) and (7). For the higher frequen- 

 cies (shorter wave periods) K (n) is very small which means that the higher 

 frequency waves result in very small pressure fluctuations near the sea floor. 

 Thus, to avoid contaminating the calculated water surface displacements, it is 



