Page 107 - Introduction to Naval Architecture

94 THE ENVIRONMENT

therefore, of the nth component will be proportional to , and the
total energy of the composite system given by:

Total energy

Within a small interval, daj, the energy in the waves can be represented by
half the square of the mean surface elevation in that interval. Plotting
this against co, Figure 5.6, gives what is termed an energy spectrum. The
ordinate of the spectrum is usually denoted by S(a>). Since the ordinate
represents the energy in an interval whose units are 1/s its units will be
2
(height) (seconds). S(a)) is called the spectral density.

Figure 5,6 Energy spectrum

Some interesting general wave characteristics can be deduced from
the area under the spectrum. If this is m 0, and the distribution of wave
amplitude is Gaussian, then the probability that the magnitude of the
wave amplitude at a random instant, will exceed some value £ is:

In this expression erf is the error function which will be found in
standard mathematical tables.
However, wave observations show that generally the Rayleigh
distribution is better at representing the sea surface. In this case it can
be shown that:

The most frequent wave amplitude = 0.707
Average wave amplitude
Average amplitude off highest waves
Average amplitude of ^ highest waves