GUIDE WAVE ANALYSIS AND FORECASTING - WMO

8
1.3.2 Wave groups and group velocity
We have seen how waves on the ocean are combinations
of simple waves. In an irregular sea the number of differing
wavelengths may be quite large. Even in regular
swell, there are many different wavelengths present but
they tend to be grouped together. In Figure 1.12 we see
how simple waves with close wavelengths combine to
form groups of waves. This phenomenon is common.
Anyone who has carefully observed the waves of the sea
will have noticed that in nature also the larger waves
tend to come in groups.
Although various crests in a group are never
equidistant, one may speak of an average distance and
thus of an average wavelength. Despite the fact that
individual crests or wave tops advance at a speed corresponding
to their wavelength, the group, as a coherent
unit, advances at its own velocity — the group velocity.
For deep water this has magnitude (group speed):
c
cg
= .
(1.14)
2
A more general expression also valid in finite depth
water is:
(1.15)
The general form for the group speed can be shown
to be:
d
c g =
d k .
c kh
c +
kh
ω
.
2
g = ( 1
)
2 sinh 2
Derivations may be found in most fluid dynamics texts
(e.g. Crapper, 1984).
We can also show that the group velocity is the
velocity at which wave energy moves. If we consider the
energy flow (flux) due to a wave train, the kinetic energy
is associated with the movement of water particles in
nearly closed orbits and is not significantly propagated.
The potential energy however is associated with the net
displacement of water particles and this moves along
with the wave at the phase speed. Hence, in deep water,
the effect is as if half of the energy moves at the phase
speed, which is the same as the overall energy moving at
half the phase speed. The integrity of the wave is maintained
by a continuous balancing act between kinetic and
potential energy. As a wave moves into previously undisturbed
water potential energy at the front of the wave
train is converted into kinetic energy resulting in a loss
of amplitude. This leads to waves dying out as they
outrun their energy. At the rear of the wave train kinetic
energy is left behind and is converted into potential
energy with the result that new waves grow there.
One classical example of a wave group is the band
of ripples which expands outwards from the disturbance
created when a stone is cast into a still pond. If you fix
your attention on a particular wave crest then you will
notice that your wave creeps towards the outside of the
GUIDE TO WAVE ANALYSIS AND FORECASTING
band of ripples and disappears. Stating this slightly
differently, if we move along with waves at the phase
velocity we will stay with a wave crest, but the waves
ahead of us gradually disappear. Since the band of
ripples is made up of waves with components from a
narrow range of wavelengths the wavelength of our wave
will also increase a little (and there will be fewer waves
immediately around us). However, if we travel at the
group velocity, the waves ahead of us may lengthen and
those behind us shorten but the total number of waves
near us will be conserved.
Thus, the wave groups can be considered as carriers
of the wave energy (see also Section 1.3.7), and the
group velocity is also the velocity with which the wave
energy is propagated. This is an important consideration
in wave modelling.
1.3.3 Statistical description of wave records
The rather confusing pattern seen in Figure 1.13 can also
be viewed in terms of Equation 1.4 as the motion of the
water surface at a fixed point. A typical wave record for
this displacement is shown in Figure 1.14, in which the
vertical scale is expressed in metres and the horizontal
scale in seconds. Wave crests are indicated with dashes
and all zero-downcrossings are circled. The wave period
T is the time distance between two consecutive downcrossings
(or upcrossings*), whereas the wave height H
is the vertical distance from a trough to the next crest as
it appears on the wave record. Another and more
commonly used kind of wave height is the zero-crossing
wave height Hz, being the vertical distance between the
highest and the lowest value of the wave record between
two zero-downcrossings (or upcrossings). When the
wave record contains a great variety of wave periods, the
number of crests becomes greater than the number of
zero-downcrossings. In that case, there will be some
difference between the crest-to-trough wave height and
Hz. In this chapter, however, this difference will be
neglected and Hz will be used implicitly. A simple and
commonly used method for analysing wave records by
hand is the Tucker-Draper method which gives good
approximate results (see Section 8.7.2).
A measured wave record never repeats itself
exactly, due to the random appearance of the sea surface.
But if the sea state is “stationary”, the statistical properties
of the distribution of periods and heights will be
similar from one record to another. The most appropriate
parameters to describe the sea state from a measured
wave record are therefore statistical. The following are
frequently used:
_________
* There is no clear convention on the use of either zeroupcrossings
or downcrossings for determining the wave
height and period of zero-crossing waves. Generally, if the
record in sufficiently long, no measurable differences will
be found among mean values.