Page 44 - Marine Structural Design

Chapter 2 Wave Loads for Ship Design and Classijication 21

The parameters used to describe ocean waves are stochastic processes, which are continuous
functions of time. Thus the measurements of the same parameter taken at different times could
result in very dissimilar readings. The data regarding the parameters used to describe ocean
waves, is collected by taking different samples over a period of time. For the validity of this
data, it is essential to ensure that each sample is collected under similar conditions. In the case
of ocean waves, a parameter such as sea elevation is influenced by a number of different
variables, such as wind speed and wind direction. In order to be certain that these different
variables remain relatively constant from sample to sample, the data is collected within a short
observation period.
A random process is stationary if the statistical characteristics of the process do not change
with time t. This means that the averages and moments of the stationary process are invariant
over time. Ocean data is usually collected from samples spanning anywhere from 30 minutes
to 3 hours, because during this period the data is considered stationary.
There are two different methods for defining averages of samples of a random process: the
ensemble and the temporal. The ensemble average is the average taken over all of the samples
at one instant in time. The temporal average is the average of a particular sample over time. In
the case of random processes such as ocean waves, the time averages formed from a single
sample over a given time interval are equal to the ensemble averages. This situation is known
as an ergodic random process.
A random process may be characterized as a narrow-band or a wide-band process. In simple
terms, a narrow-band process is made up of waves with frequencies lying within a narrow
range, while a wide-band process consists of waves with widely varying frequencies. Ocean
wave data shows that a fully developed, wind-generated, mid-ocean sea-state (Le. with no
growth or decay, and no coastal effects), is essentially narrow-banded. Of course, there are
always wave components, which differ by having a high frequency, but these waves tend to be
small in both height and length and have little effect on the ship. It is also interesting to note
that a ship acts as a filter, only a narrow band of wave frequencies has an effect on the ship's
motion and hull girder loads. Thus the ship's response is even more narrow-banded than the
sea itself and this response is usually also characterized as a Gaussian and stationary process
just like the ocean waves.
Chapter 24 of this book contains more information on random variable definitions.
2.2.2 Statistical Representation of the Sea Surface

This Section deals with the representation of a complete sea surface. Of course, we know that
the sea surface is highly irregular and random under all sorts of conditions, calm or stormy
weather. However, it has been found that this random process may be accurately represented
by a series of different regular waves of varying heights, lengths, directions and phase that all
superimposed on each other.
Three papers, which paved the way for further work on statistical representations of the sea
surface, were published by Pierson (1952), St. Denis and Pierson (1953), and Pierson,
Neumann, and James (1955). These papers proved that the sea surface could be represented by
the superposition of a large number of regular sinusoidal waves of varying frequencies. A
typical sinusoidal wave may be represented by the following:
&,t) = asin(- lo^ - o + 8) (2.3)
t
where,