) Wind waves are mechanical
waves that propagate along the interface between
water and
air; the restoring force is provided by gravity, and so they are often referred to as
surface gravity waves. As the
wind blows, pressure and friction perturb the equilibrium of the water surface and transfer energy from the air to the water, forming waves. The initial formation of waves by the wind is described in the theory of Phillips from 1957, and the subsequent growth of the small waves has been modeled by
Miles, also in 1957. ) in a
wave flume. The wave conditions are: mean water depth
d = ,
wave height H = , wavelength λ = ,
period T = 1.12 s. In linear plane waves of one wavelength in deep water,
parcels near the surface move not plainly up and down but in circular orbits: forward above and backward below (compared to the wave propagation direction). As a result, the surface of the water forms not an exact
sine wave, but more a
trochoid with the sharper curves upwards—as modeled in
trochoidal wave theory. Wind waves are thus a combination of
transversal and
longitudinal waves. When waves propagate in
shallow water, (where the depth is less than half the wavelength) the particle trajectories are compressed into
ellipses. In reality, for
finite values of the wave amplitude (height), the particle paths do not form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as
Stokes drift. As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the
wavelength λ, the orbital movement has decayed to less than 5% of its value at the surface. The
phase speed (also called the celerity) of a surface gravity wave is—for pure
periodic wave motion of small-
amplitude waves—well approximated by :c=\sqrt{\frac{g \lambda}{2\pi} \tanh \left(\frac{2\pi d}{\lambda}\right)} where :
c =
phase speed; :
λ =
wavelength; :
d = water depth; :
g =
acceleration due to gravity at the Earth's surface. In deep water, where d \ge \frac{1}{2}\lambda, so \frac{2\pi d}{\lambda} \ge \pi and the hyperbolic tangent approaches 1, the speed c approximates :c_\text{deep}=\sqrt{\frac{g\lambda}{2\pi}}. In SI units, with c_\text{deep} in m/s, c_\text{deep} \approx 1.25\sqrt\lambda, when \lambda is measured in metres. This expression tells us that waves of different wavelengths travel at different speeds. The fastest waves in a storm are the ones with the longest wavelength. As a result, after a storm, the first waves to arrive on the coast are the long-wavelength swells. For intermediate and shallow water, the
Boussinesq equations are applicable, combining
frequency dispersion and nonlinear effects. And in very shallow water, the
shallow water equations can be used. If the wavelength is very long compared to the water depth, the phase speed (by taking the
limit of when the wavelength approaches infinity) can be approximated by :c_\text{shallow} = \lim_{\lambda\rightarrow\infty} c = \sqrt{gd}. On the other hand, for very short wavelengths,
surface tension plays an important role and the phase speed of these
gravity-capillary waves can (in deep water) be approximated by :c_\text{gravity-capillary}=\sqrt{\frac{g \lambda}{2\pi} + \frac{2\pi S}{\rho\lambda}} where :
S =
surface tension of the air-water interface; :\rho =
density of the water. When several wave trains are present, as is always the case in nature, the waves form groups. In deep water, the groups travel at a
group velocity which is half of the
phase speed. Following a single wave in a group one can see the wave appearing at the back of the group, growing, and finally disappearing at the front of the group. As the water depth d decreases towards the
coast, this will have an effect: wave height changes due to
wave shoaling and
refraction. As the wave height increases, the wave may become unstable when the
crest of the wave moves faster than the
trough. This causes
surf, a breaking of the waves. The movement of wind waves can be captured by
wave energy devices. The energy density (per unit area) of regular sinusoidal waves depends on the water
density \rho, gravity acceleration g and the wave height H (which, for regular waves, is equal to twice the
amplitude, a): :E=\frac{1}{8}\rho g H^2=\frac{1}{2}\rho g a^2. The velocity of propagation of this energy is the
group velocity. ==Models==