Page 131 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

122 Fundamentals of Ocean Renewable Energy

TABLE 5.2 Wave Transformation From
Deep to Shallow Water, for T = 8s
Environment L (m) C (m/s) C g (m/s)
Deep 100 12.5 6.2
Shallow 25 3.1 3.1

water of gradually decreasing depth (shoaling water), their phase speed, and
consequently their group velocity, reduces (Eq. 5.13), but their frequency tends
to remain constant—in other words, waves do not catch one another up in the
same way that they would if they were ‘dispersing’. By way of illustration, for
a wave period T = 8 s we can calculate the wave properties shown in Fig. 5.6
and Table 5.2 between ‘deep’ and ‘shallow’ water.
Therefore, a fourfold reduction in wavelength and phase speed is accom-
2
panied by a halving of the group velocity. Since E = 1/2ρga , when C g is
halved, it follows that E is doubled (Eq. 5.46), and so wave amplitude increases
√
by a factor of 2 (Fig. 5.6C). This is known as wave shoaling. Note that this
illustration is for a 1D case (Fig. 5.6A), but for a more generalized case, another
process can complicate wave transformation in shoaling water—refraction.

5.2.2 Wave Refraction
In the earlier shoaling example, the wave crests were parallel to the depth
contours, and the wave propagated normal to the coastline. In most situations,
waves propagate obliquely to the depth contours, and are observed to slowly
change direction as they approach the coast. This gradual change in wave
direction is known as refraction.
Examining Fig. 5.7, which shows wave crests propagating at an oblique
angle to the depth contours, there is a variation in water depth experienced
along each of the wave crests. Whereas further offshore the wave crest is
propagating in relatively deep water, the part of the wave crest closer to the
coast is propagating in relatively shallow water. Since wave celerity is related to
water depth through the dispersion equation (5.13), this means that the portion
of the wave crest in deeper water is travelling faster than the portion of the wave
crest that is travelling in shallower water. As a result of this change in phase
speed along each wave crest, in a given time interval the crest moves over a
larger distance in deeper water than it does in shallower water, and so the waves
tend to turn (i.e. refract) towards the depth contours as they propagate.
Variation of ψ (the angle of incidence of the wave orthogonal to the outward
normal of the coastline; see Fig. 5.7) is governed by Snell’s law. This is the