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COASTAL PROCESSES
with Engineering Applications
ROBERT G. DEAN
University of Florida
ROBERT A. DALRYMPLE
University of Delaware
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CHAPTER FIVE
Waves and Wave-Induced Hydrodynamics
Freak waves are waves with heights that far exceed the average. Freak waves
in storms have been blamed for damages to shipping and coastal structures and
for loss of life. The reasons for these waves likely include wave reflection from
shorelines or currents, wave focusing by refraction, and wave–current interaction.
Lighthouses, built to warn navigators of shallow water and the presence of
land, are often the target of such waves. For example, at the Unst Light in
Scotland, a door 70 m above sea level was stove in by waves, and at the Flannan
Light, a mystery has grown up about the disappearance of three lighthouse
keepers during a storm in 1900 presumably by a freak wave, for a lifeboat, fixed
at 70 m above the water, had been torn from its mounts. A poem by Wilfred W.
Gibson has fueled the legend of a supernatural event.
The (unofficial) world’s record for a water wave appears to be the earthquake
and landslide-created wave in Lituya Bay, Alaska (Miller 1960). A wave created
by a large landslide into the north arm of the bay sheared off trees on the side of
a mountain over 525 m above sea level!
5.1 INTRODUCTION
Waves are the prime movers for the littoral processes at the shoreline. For the most
part, they are generated by the action of the wind over water but also by moving
objects such as passing boats and ships. These waves transport the energy imparted
to them over vast distances, for dissipative effects, such as viscosity, play only a small
role. Waves are almost always present at coastal sites owing to the vastness of the
ocean’s surface area, which serves as a generating site for waves, and the relative
smallness of the surf zone, that thin ribbon of area around the ocean basins where
the waves break and the wind-derived energy is dissipated.
Energies dissipated within the surf zone can be quite large. The energy of a wave
is related to the square of its height. Often it is measured in terms of energy per unit
water surface area, but it could also be the energy per unit wave length, or energy
flux per unit width of crest. The first definition will serve to be more useful for our
purposes. If we define the wave height as H , the energy per unit surface area is
E = 1
8
ρgH 2,
88
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5.2 WATER WAVE MECHANICS 89
where thewater density ρ and the acceleration of gravity g are important parameters.
(Gravity is, in fact, the restoring mechanism for the waves, for it is constantly trying
to smooth the water surface into a flat plane, serving a role similar to that of the
tension within the head of a drum.) For a 1-m high wave, the energy per unit area
is approximately 1250 N-m/m2, or 1250 J/m2. If this wave has a 6-s period (that is,
there are 10 waves per minute), then about 4000 watts enter the surf zone per meter
of shoreline.∗ For waves that are 2 m in height, the rate at which the energy enters
the surf zone goes up by a factor of four. Now think about the power in a 6-m high
wave driven by 200 km/hr winds. That plenty of wave energy is available to make
major changes in the shoreline should not be in doubt.
The dynamics and kinematics of water waves are discussed in several textbooks,
including the authors’ highly recommended text (Dean andDalrymple 1991),Wiegel
(1964), and for the more advanced reader Mei (1983). Here, a brief overview is
provided. The generation of waves by wind is a topic unto itself and is not discussed
here. A difficult, but excellent, text on the subject is Phillips’ (1980) The Dynamics
of the Upper Ocean.
5.2 WATER WAVE MECHANICS
The shape, velocity, and the associated water motions of a single water wave train are
very complex; they are even more so in a realistic sea state when numerous waves
of different sizes, frequencies, and propagation directions are present. The simplest
theory to describe the wave motion is the Airy wave theory, often referred to as the
linear wave theory, because of its simplifying assumptions. This theory is predicated
on the following conditions: an incompressible fluid (a good assumption), irrotational
fluid motion (implying that there is no viscosity in the water, which would seem to be
a bad assumption, but works out fairly well), an impermeable flat bottom (not too
true in nature), and very small amplitude waves (not a good assumption, but again
it seems to work pretty well unless the waves become large).
The simplest form of a wave is given by the linear wave theory (Airy 1845),
illustrated inFigure 5.1,which showsawaveassumed tobepropagating in thepositive
x direction (left to right in the figure). The equation governing the displacement of
the water surface η(x, t) from the mean water level is
η(x, t) = H
2
cos k(x − Ct) = H
2
cos(kx − σ t) (5.1)
Because the wave motion is assumed to be periodic (repeating identically every
wavelength) in the wave direction (+x), the factor k, thewave number, is used to en-
sure that the cosine (a periodic mathematical function) repeats itself over a distance
L , the wavelength. This forces the definition of k to be k = 2π/L . For periodicity in
time, which requires that the wave repeat itself every T seconds, we have σ = 2π/T ,
where T denotes the wave period. We refer to σ as the angular frequency of the
waves. Finally, C is the speed at which the wave form travels, C = L/T = σ/k. The
C is referred to as the wave celerity, from the Latin. As a requirement of the Airy
∗ If we could capture 100 percent of this energy, we could power 40 100-watt lightbulbs.
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90 WAVES AND WAVE-INDUCED HYDRODYNAMICS
Figure 5.1 Schematic of a water wave (from Dean and Dalrymple 1991).
wave theory, the wavelength and period of the wave are related to the water depth
by the dispersion relationship
σ 2 = gk tanh kh (5.2)
The “dispersion” meant by this relationship is the frequency dispersion of the waves:
longerperiodwaves travel faster than shorterperiodwaves, and thus if one startedout
with a packet of waves of different frequencies and then allowed them to propagate
(which is easily done by throwing a rock into a pond, for example), then at some
distance away, the longer period waves would arrive first and the shorter ones last.
This frequency dispersion applies also to waves arriving at a shoreline from a distant
storm.
The dispersion relationship can be rewritten using the definitions of angular fre-
quency and wave number
L = g
2π
T 2 tanh kh = L0 tanh kh, (5.3)
defining the deep water wavelength L0, which is dependent only on the square of
the wave period. This equation shows that the wavelength monotonically decreases
as the water depth decreases owing to the behavior of the tanh kh function, which
increases linearly with small kh but then becomes asymptotic to unity in deep water
(taken arbitrarily as kh > π or, equivalently, h > L/2).
The dispersion relationship is difficult to solve