Wind wave

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Science of waves[edit]

Stokes drift in shallow water waves (Animation)

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.[1][2]

Stokes drift in a deeper water wave (Animation)
Photograph of the water particle orbits under a – progressive and periodic – surface gravity wave in a wave flume. The wave conditions are: mean water depth d = 2.50 ft (0.76 m), wave height H = 0.339 ft (0.103 m), wavelength λ = 6.42 ft (1.96 m), period T = 1.12 s.[3]

In linear plane waves of one wavelength in deep water, particles near the surface move not plainly up and down but in vertical circles, forwards above and backwards below. As a result, the surface of the water forms not an exact sine wave, but a curtate cycloid with the sharper curves upwards. As (wave height) / wavelength increases, the wave shape becomes more like a cycloid, and when (wave height) = wavelength / π, the wave shape becomes a cycloid, with the cusps upwards. If something such as wind tries to make the wave any higher at that wavelength, the wave shape tries to become a prolate cycloid, which has a loop at its cusps, and the wave's crest breaks up into a line of foam commonly called a "whitecap" or "white horse". Likewise, in a mixture of waves of various lengths moving in various directions and long waves overtaking short waves, as often seen at sea, if at any time and place the resulting wave motion "goes prolate" and tries to make water go in a raised loop through other water, it cannot, and some of that wave's energy is used up in throwing up spray and foam.

When waves propagate in shallow water, (where the depth is less than half the wavelength) the particle trajectories are compressed into ellipses.[4][5]

As the wave amplitude (height) increases, the particle paths no longer form closed orbits; rather, after the passage of each crest, particles are displaced slightly from their previous positions, a phenomenon known as Stokes drift.[6][7]

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 c 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.[8]

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.[9] 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.

  1. ^ Phillips, O. M. (1957), "On the generation of waves by turbulent wind", Journal of Fluid Mechanics 2 (5): 417–445, Bibcode:1957JFM.....2..417P, doi:10.1017/S0022112057000233 
  2. ^ Miles, J. W. (1957), "On the generation of surface waves by shear flows", Journal of Fluid Mechanics 3 (2): 185–204, Bibcode:1957JFM.....3..185M, doi:10.1017/S0022112057000567 
  3. ^ Figure 6 from: Wiegel, R.L.; Johnson, J.W. (1950), "Elements of wave theory", Proceedings 1st International Conference on Coastal Engineering, Long Beach, California: ASCE, pp. 5–21 
  4. ^ For the particle trajectories within the framework of linear wave theory, see for instance:
    Phillips (1977), page 44.
    Lamb, H. (1994). Hydrodynamics (6th edition ed.). Cambridge University Press. ISBN 978-0-521-45868-9.  Originally published in 1879, the 6th extended edition appeared first in 1932. See §229, page 367.
    L. D. Landau and E. M. Lifshitz (1986). Fluid mechanics. Course of Theoretical Physics 6 (Second revised edition ed.). Pergamon Press. ISBN 0-08-033932-8.  See page 33.
  5. ^ A good illustration of the wave motion according to linear theory is given by Prof. Robert Dalrymple's Java applet.
  6. ^ For nonlinear waves, the particle paths are not closed, as found by George Gabriel Stokes in 1847, see the original paper by Stokes. Or in Phillips (1977), page 44: "To this order, it is evident that the particle paths are not exactly closed … pointed out by Stokes (1847) in his classical investigation".
  7. ^ Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:
    J.M. Williams (1981). "Limiting gravity waves in water of finite depth". Philosophical Transactions of the Royal Society A 302 (1466): 139–188. Bibcode:1981RSPTA.302..139W. doi:10.1098/rsta.1981.0159. 
    J.M. Williams (1985). Tables of progressive gravity waves. Pitman. ISBN 978-0-273-08733-5. 
  8. ^ Carl Nordling, Jonny Östermalm (2006). Physics Handbook for Science and Engineering (Eight edition ed.). Studentliteratur. p. 263. ISBN 978-91-44-04453-8. 
  9. ^ In deep water, the group velocity is half the phase velocity, as is shown here. Another reference is [1].