One of the key concepts in seakeeping analysis is that of wave spectra.
The average energy, 
, over the wavelength is given by the equation below. It may
be seen that the energy is dependent only on the square of the wave amplitude.
(This is effectively the mean square of the signal multiplied by a constant – rg;
this draws certain parallels with AC electrical circuits.)
As has been discussed in the introduction, irregular waves are often described by a spectrum that indicates the amount of wave energy at different wave frequencies. A spectrum is shown by plotting spectral density against frequency; a typical wave spectrum is shown below:
These spectral representations of sea conditions are central to determining the response of a vessel in the seaway. This will be discussed more fully in the following sections.
There are several spectral characteristics that may be directly related to the time series representation. The key to calculating these characteristics is the spectral moment, mn. The nth spectral moment is calculated using the equation below; n may take any positive integer value (n = 0,1,2,…).
From these spectral moments it is possible to calculate many of the time series characteristics noted in the Introduction Section.
Of particular importance is the zeroth spectral moment, m0, this is equivalent to the area under the wave spectrum curve, which is also the variance of the wave time history.
The RMS or standard deviation,
, is
given by:
The average period, 
, may
be found by calculating the “centre of area” of the energy spectrum; thus the
average period is given by:
Note that the modal period, 
, is
the wave period at which the maximum wave energy occurs. For spectra that are
defined by continuous mathematical expressions, this may be found by
differentiation.
It may be shown that the mean period of the peaks, 
, is given by:
and the mean zero crossing period, 
, is given by:
The bandwidth parameter, e, may be found using the equation below. If e = 0 the spectrum is a narrow band spectrum, and if e = 1 the spectrum is a wide band spectrum. In general it is probably fair to say that most sea spectra are relatively narrow banded. Even if this is simply due to the fact that the very small, high frequency ripples are of no interest in the field of ship motions.
Finally, it may be shown that the significant wave height, 
, is dependent on the bandwidth of the spectrum and may be
calculated from the equation below:
In general, it is assumed that wave spectra are narrow banded (e = 0) and that the significant wave height is given by:
However, if the spectrum is wide band (e = 1):
It is often useful to define idealised wave spectra which broadly represent the characteristics of real wave energy spectra. Several such idealised spectra are available in Seakeeper and are described below:
Note: SI units have been used throughout for the constants in the equations below.
Bretschneider or ITTC two parameter spectrum
The Bretschneider or ITTC two parameter spectrum is defined below:
;
where:
and 
The two parameters are the characteristic wave height, 
, and the average period, 
.
By calculating the various spectral moments it may be shown that:
Thus the Bretschneider or ITTC two parameter spectrum
is a broad band spectrum and contains all wave frequencies up to infinity. This
is why the average period between peaks is zero since there will be
infinitesimally small ripples with adjacent peaks. However, in practice the
high frequency ripples are neglected and the spectrum will effectively be
narrow banded in which case 
hence 
.
The modal period may be found by differentiating the wave energy spectrum and finding the maximum (slope = 0), see below:
,
where 
The one-parameter Bretschneider spectrum is similar to the two parameter spectrum but is defined in terms of wave height only. The spectrum is defined as follows:
;
where: A is the Philp's constant, given by:
This spectrum may be used when only the significant wave height is known. The associated period is typical of fully developed seas.
The JONSWAP (JOint North Sea WAve Project) spectrum is often used to describe coastal waters where the fetch is limited. It is based on the ITTC spectrum and defined below. In general both spectra do not contain the same energy for specified significant wave height and characteristic period, however the JONSWAP always has a taller, narrower peak than the ITTC.
;
where:
with:
A more generalised spectrum formulation is used by DNV. Special cases of this spectrum include the Bretschneider spectrum when the peak enhancement factor is 1.0 and the JONSWAP spectrum when the peak enhancement factor is 3.3. The peak enhancement factor, g, used by DNV is determined from the significant wave height and the modal period:
The spectrum itself is defined as follows:
where:
The Pierson Moskowitz spectrum may be used to define a spectrum by a nominal wind speed, Uwind in m/s at a height of 19.5m above the sea surface.
;
where: A is the Philp's constant, given by:
