146
STATISTICAL DESCRIPTIONS OF OFFSHORE WAVES
any periodic function of time represented by équation (6.1) where the mean is
zéro (ao = 0), then the variance of r)(t) is
>r0
n/2
1 / [r)(t)\2dt = 2
+ b2) = a2
(6.3)
T° Jo
„=i
for which the root-mean-square (rms) value for 7?(t) is
It is observed that this spectral function is the resolution of the variance of
into its frequency components. That is, the value of S-^uJn) at each discrète
frequency un can be approximated by the sum of the Foncier coefficients, or
2
5„(wn) ~ — (a^+&2)
(6.5a)
ZAuJ
where
2n7T
a
2?r
/e cki
un —----- ;
Au = —
(6.5b)
TO
T0
Here, Au is the spacing between adjacent harmonies. It is emphasized that
the sampling interval At must satisfy the Nyquist sampling theorem to avoid
falsely distorting the Fourier coefficients. In practice, an experimentally-based
discrète energy spectrum is usually fit to a continuons function by smoothing.
Examples of smoothed spectra are discussed later in this chapter.
The character of the function S^(u) dépends on the surface wave élévation
record. For example, S^(u) for unidirectional swell is a narrow-banded function
centered about the dominant swell frequency. On the other hand, sea waves
in general are most often characterized by a broader-banded spectral function
than that for swell waves.
The interest in wave spectra arises from the need to select a représentative
design-loading history acting on the proposed offshore structure. The design
wave spectrum, usually furnished by a specialist in hindcasting or forecasting,
is chosen to represent the worst possible combination of waves which would
lead to the highest loading for the structure. To complicate matters, there
may be many different surface water wave élévation historiés characterized by
.ntially the same spectrum. In other words, a one-to-one correspondence
between the surface water wave élévation record and the spectrum does not
generally exist. However, ail records that yield identical spectra do hâve the
same statistical properties, although details in the records may vary widely.
1 lie suggested procedure, then, is to select a number of water wave élévation
records corresponding to the design wave spectrum and to analyze the proposed
offfshore structure under these alternative conditions.
STATISTICAL DESCRIPTIONS OF OFFSHORE WAVES
any periodic function of time represented by équation (6.1) where the mean is
zéro (ao = 0), then the variance of r)(t) is
>r0
n/2
1 / [r)(t)\2dt = 2
+ b2) = a2
(6.3)
T° Jo
„=i
for which the root-mean-square (rms) value for 7?(t) is
It is observed that this spectral function is the resolution of the variance of
into its frequency components. That is, the value of S-^uJn) at each discrète
frequency un can be approximated by the sum of the Foncier coefficients, or
2
5„(wn) ~ — (a^+&2)
(6.5a)
ZAuJ
where
2n7T
a
2?r
/e cki
un —----- ;
Au = —
(6.5b)
TO
T0
Here, Au is the spacing between adjacent harmonies. It is emphasized that
the sampling interval At must satisfy the Nyquist sampling theorem to avoid
falsely distorting the Fourier coefficients. In practice, an experimentally-based
discrète energy spectrum is usually fit to a continuons function by smoothing.
Examples of smoothed spectra are discussed later in this chapter.
The character of the function S^(u) dépends on the surface wave élévation
record. For example, S^(u) for unidirectional swell is a narrow-banded function
centered about the dominant swell frequency. On the other hand, sea waves
in general are most often characterized by a broader-banded spectral function
than that for swell waves.
The interest in wave spectra arises from the need to select a représentative
design-loading history acting on the proposed offshore structure. The design
wave spectrum, usually furnished by a specialist in hindcasting or forecasting,
is chosen to represent the worst possible combination of waves which would
lead to the highest loading for the structure. To complicate matters, there
may be many different surface water wave élévation historiés characterized by
.ntially the same spectrum. In other words, a one-to-one correspondence
between the surface water wave élévation record and the spectrum does not
generally exist. However, ail records that yield identical spectra do hâve the
same statistical properties, although details in the records may vary widely.
1 lie suggested procedure, then, is to select a number of water wave élévation
records corresponding to the design wave spectrum and to analyze the proposed
offfshore structure under these alternative conditions.