I imits of an estimate are a function of the degrees of freedom Y = 2 n/m, 

 where n = T-j-/ At is the number of data points, T^ is the total record 



length, and m, the number of lags. The root-mean-square err or o f the 



estimate, for 95 percent confidence limits, is found to be /TfT - /m/n. 

 This can be i nt^r pret ed as the probability that 95 percent of the samples 

 will fall with ±/m/n of the estimate. Graphs illustrating the effects 

 of record length (total number of samples) and sampling interval on the 

 expected error are shown in Figure 13. 



The frequency range of the spectrum obtained from digital computation 

 techniques is limited in both directions. No real Information is obtained 

 for frequencies lower than the resolution frequency f^ = I /Tp = l/Atm, 

 where m is the total number of lags and Tp Is the maximum lag in real time 

 between comparisons of two time series. Also oscillations having frequencies 

 greater than the Nyqulst (or folding) frequency, fn = l/2At. will give 

 aliased or ghost signatures that may appear in the frequency interval to 

 fn (Blackman and Tukey, 1959, p. 31). 



The discrete samp! i ng at equal ly spaced intervals At produces sum and 

 difference frequencies 



fg = (k/At) ± fi = 2 kfn ± fi (8) 



where f ;^ is the frequency of a real oscillation and k is an Integer. If 

 fi is less than f , both sum and difference frequencies f fall outside of 

 the frequency interval to f^. However, oscillations with frequencies 

 greater than fn will have aliased frequencies fg that fall within the In- 

 terval to fn. These aliased or ghost frequencies will be Indistinguish- 

 able from real oscillations of the same frequency. 



To prevent the occurrence at the sensor of energy In the frequency 

 range above fp, it is customary in wave pressure measurements to place 

 the pressure sensor at sufficient depth to filter out high frequency 

 "noise". For wave staffs and other water level sensors, It is necessary 

 to decrease the sampling interval At until energy in frequencies above 

 1/2 At is decreased to acceptable levels. 



The above considerations are il lustrated by two spectra of waves 

 measured in a laboratory channel (Figure 14). 'The spectra are computed 

 from synoptic measurements using two mini-digit wave staffs as sensors. 

 One sensor (Staff B) was placed in the constant depth portion of the 

 channel (still water depth 50 cm) and the other sensor (Staff C) at the 

 point where the waves broke after shoaling over an impermeable 1:12 slope. 

 The wave maker generated simple harmonic motion of 0.88 cps; and the wave 

 heights, as measured by pointer gages, were found to be 100 mm at the 

 constant depth portion of the channel and 120 mm at the breaker point. 

 The wave staffs were sampled 8 times per second for 15 minutes, giving 

 7,200 data poi nts for each channel. The data was analyzed using 200 lags 

 which gives a root-mean-square error of ±0.176 of the estimate of energy 

 density. The energy density for the wave spectra is expressed In mm^/Af, 



24 



