Frequency is often expressed in terms of hertz units where one hertz is equal 
to one cycle per second. One hertz is also equivalent to 21 radians per 
second. If the symbol f; denotes frequency in hertz, then anf = Ws. 
The amplitudes, aj, computed for a time series, give an indication of im- 
portance of each frequency, f;. The sum of the squared amplitudes is related 
to the variance of sea-surface elevations in the original time series (eq. 3-12 
in the SPM) and hence to the potential energy contained in the wavy sea surface. 
Because of this relationship, the distribution of squared amplitudes as a func- 
tion of frequency can be used to estimate the distribution of wave energy as a 
function of frequency. This distribution is called the energy spectrum and is 
often expressed as ; 
ag 
(Ej) (Af) 3 = $= S3 (2) 
where Ej = E(fj) is the energy density in jth component of energy spectrum, 
(Af)j the frequency bandwidth in hertz (difference between successive fj), 
and Sj = S(fj) energy in jth component of energy spectrum. Equation (2) is 
similar to equation (3-15) in the SPM. 
An energy spectrum computed from an ocean wave record is plotted in Fig- 
ure 1. Frequencies associated with large values of energy density (or large 
values of a%/[2(Af)-]; see eq. 2) represent dominant periodicities in the orig- 
inal time series. Frequencies associated with small values of a4/[2 (Af) 4] are ~ 
usually unimportant. It is common for ocean wave spectra to show two or more 
dominant periodicities (Fig. 1). 
25,000 
[__] Region used to Compute Spy 
20,000 ZA Region used to Compute Spe 
= 
Py 
= 
‘S 15,000 
= 
e 
[=< 
@ 
© 10,000 
~ 
mo 
o 
(=< 
ty 
5,000 
0 y 0.2 0.3 I 0.5 
fi i Frequency (Hz) 
Figure 1. Spectrum for Wrightsville Beach, North Carolina, 0700 e.s.t., 
12 February 1972; Hg = 4.2 feet (128 centimeters), Af = 0.01074 
hertz, and depth = 17.7 feet (5.4 meters). 
