798 OCEANOGRAPHY
WIND WAVESIn this section, we will treat the generation and characteris-
tics of wind-induced surface gravity waves. These are waves
formed at the air–sea interface by the action of a surface wind
stress. The term “gravity wave” infers that the restoring force,
or that force which seeks to restore the water surface to its
equilibrium position, is the force due to gravity. An example
of a different type of restoring force is the force due to surface
tension. This force is actually the result of molecular attraction
among water molecules, which, at the air—water interface,
creates a net, adhesive force retaining the water molecules
at the water surface. As one would expect, this force is only
important in cases of very small scale motions, such as capil-
lary waves, and is therefore unimportant for our purposes.
Early theories of wind-wave generation include the work
of Kelvin (1871) and Helmholtz (1888), who examined the
onset and growth of wave forms arising from instabilities at the
interface of two fluids, each with different density and moving
at different velocities. Although providing the basis for much
of our present knowledge of flow instabilities and turbulence,
the Kelvin–Helmholtz mechanism proved inadequate as a
description of the onset of wind wave generation. The theory
predicts wave generation only at surface wind speeds exceed-
ing approximately 14 miles/hour, far greater than the minimum
wind speeds required for wave generation observed in nature.
Jeffreys (1924, 1925) introduced what is commonly
referred to as the “sheltering hypothesis” of wind wave
generation. He proposed that the airflow over a previously
rough water surface separates on the downwind sides of
crests much the same as the flow separation observed in the
lee of a cylinder. The resulting asymmetry in wind velocity
leads to an asymmetric pressure distribution along the water
surface, giving rise to a resultant force in the direction of the
airflow. Theoretically, this pressure forcing will continue to
add energy to the waveform as long as the velocity of the
airflow exceeds the phase velocity of the wave.
Although providing valuable insights into the growth
of wind waves, the sheltering hypothesis failed to address
the critical question of the onset of wave generation because
of its assumption of a previously rough water surface. This
problem of wave generation on an initially smooth water sur-
face was examined by Phillips (1957), who proposed that the
initiation of wave formation is due to the presence of turbu-
lence in the airflow overlying the water surface. Associated
with this turbulence are random fluctuations in velocity, and
hence, pressure. One can imagine that in areas of high pres-
sure, the water surface will be depressed, whereas in regions
of low pressure, the interface will, relatively speaking, rise,
thereby creating a waveform at the air–water interface. Phillips
proposed that the pressure fluctuations will continue to act on
those waveforms having a phase speed equal to the speed of
the turbulent pressure fluctuations, so that selective growth
of only certain wave frequency components occurs.
Phillips’ mechanisms was found to provide an excellent
description of the initial stages of wind-wave formation. The
problem of continued wave growth was addressed by Miles
(1957). Using a model of shear flow instability similar to the
Kelvin–Helmholtz theory mentioned earlier, Miles illustratedthat a coupling exists between the airflow and the wave motion.
In addition to the airflow doing work on the water surface, the
perturbations in the water surface (i.e., the waveforms) can
induce instabilities in the airflow. The energy transfer due to
these instabilities in what is essentially a two-fluid shear flow
is responsible for continued wave growth.
Although the Phillips–Miles mechanism for the initiation
and growth of surface wind waves was proven quite accu-
rate, measurements of the long-term evolution of wave fields
indicated the importance of a third factor, the interaction
among the individual wave components themselves. Recall
from our previous discussion that selective generation and
growth of different wave frequencies occurs, depending on
the relative speeds of the wave forms and the turbulent pres-
sure fluctuations overhead. The resulting water surface is
therefore not characterized by a single, uniform wave form,
but is rather composed of a multitude of wave components,
each with a different frequency, amplitude, and wavelength.
For this reason, wave data are typically analysed with the
use of a wave spectrum (analogous to the color spectrum of
optics), separating the observed sea into its various frequency
or wavelength components. Phillips (1960) and Hasselmann
(1962) illustrated that these individual components interact
nonlinearly, resulting in a transfer of energy from the central
frequencies of the spectrum to the high and low frequencies.
As measurement and modeling techniques have improved
over the last 25 years, functional relations for all of the forc-
ing mechanisms mentioned here have been developed, and
have led to the creation of quite accurate computer algorithms
for the prediction of surface wind waves.
In seeking to describe the characteristics of a specific
wave field, perhaps the most important relationship is the
“dispersion relation”, which defines the relationship between
the wave frequency, v, and the wavenumber, k:
v 2 gk tanh(kh),
where g is the acceleration due to gravity, h is the water depth,
v 2 p / T, T is the wave period (time of travel of one wave-
form), k 2 p /L, and L is the wavelength. Figure 6 illustrates
the most commonly used surface wave parameters.
The term, “dispersive,” signifies that waves of differ-
ent frequencies move at different phase speeds, C v /k.
Clearly, therefore, longer waves have higher phase speeds.
Note also the dependence on water depth, h, with a given
wave in deep water having a higher phase speed than the
corresponding were in shallow water. Simply speaking, this
depth dependence is responsible for the refraction of wave as
they approach at an angle to the shoreline, with the portion
of the wavetrain in deeper water leading that portion located
in shallow water.
Several interesting observations can be obtained by
examining the dispersion relation in the two limits of very
deep and very shallow water depths. Note that “deep” and
“shallow” water waves are defined by the water depth relative
to the wavelength, as illustrated in the dependence on (kh).
For this reason, a deep water wave is often termed a “short”
wave and a shallow water wave, a long wave. In the case of
short waves, we have a very large value for (kh) so that the
quantity, tanh(kh) goes to unity and the dispersion relationC015_001_r03.indd 798C015_001_r03.indd 798 11/18/2005 11:10:05 AM11/18/2005 11:10:05 AM