Page 212 - Dynamic Loading and Design of Structures

Page 185

spectral density (Sigbørnsson, 1979)

(5.22)

where ζand ζare the wave amplitudes at points m and n with co-ordinates (x , y ) and (x ,
n
m
m
n
m
y ), respectively, and
n
The probabilistic description of the wave kinematics in terms of the particle velocities and
accelerations is commonly achieved by applying the principle of superposition of independent
and arbitrarily distributed disturbances and the Airy wave theory. Then the frequency cross-
spectral density of, for example, the water particle velocity v may be expressed as follows,
x
applying eqns (5.8) and (5.22)

(5.23)

Analogous expressions hold for the frequency cross-spectral densities of acceleration, and
acceleration and velocity.

Higher order irregular wave theory
To reduce the deficiencies of the linear theory, especially in predicting extreme values, a
consistent second-order or higher order irregular wave theory, analogous to the higher order
regular wave theories mentioned, may be established. However, in current engineering
practice, improved kinematics is obtained by modification of the linear theory (e.g.
Gudmestad, 1993). It should be noted that this formulation does not represent the asymmetry
in wave elevation nor the non-linear interaction between individual waves in an irregular
wave process.
The sea surface elevation is not a perfect Gaussian process (see e.g. LonguetHiggins, 1963;
Haver and Moan, 1983; Vinje and Haver, 1994). In the same way as a finite regular wave is
not perfectly sinusoidal (i.e. the crest is larger than the trough), the random sea elevation is
skewed and has more kurtosis than a Gaussian process. Vinje and Haver (1994) found that the
skewness depends upon H m0 and T according to and kurtosis
p
. Non-Gaussian surface elevation may be generated by a second order
(irregular) wave model for instance based on Stokes’ expansion (see e.g. Longuet-Higgins,
1963), or by transformation of a Gaussian process by a Hermite expansion (see e.g.
Winterstein, 1988).

5.2.4 Wave kinematics of irregular waves in long-term periods

The non-stationary sea state in a long term period (i.e. of some years duration), can be
assumed to consist of a sequence of short term sea states (i.e. stationary zero