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In oceanography, **sea state** is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterized by statistics, including the wave height, period, and power spectrum. The sea state varies with time, as the wind conditions or swell conditions change. The sea state can either be assessed by an experienced observer, like a trained mariner, or through instruments like weather buoys, wave radar or remote sensing satellites.

**Oceanography**, also known as **oceanology**, is the study of the physical and biological aspects of the ocean. It is an important Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynamics; plate tectonics and the geology of the sea floor; and fluxes of various chemical substances and physical properties within the ocean and across its boundaries. These diverse topics reflect multiple disciplines that oceanographers blend to further knowledge of the world ocean and understanding of processes within: astronomy, biology, chemistry, climatology, geography, geology, hydrology, meteorology and physics. Paleoceanography studies the history of the oceans in the geologic past.

In physics, a **free surface** is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids, for example liquid water and the air in the Earth's atmosphere. Unlike liquids, gases cannot form a free surface on their own. Fluidized/liquified solids, including slurries, granular materials, and powders may form a free surface.

In fluid dynamics, **wind waves**, or **wind-generated waves**, are surface waves that occur on the free surface of bodies of water. They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.

In case of buoy measurements, the statistics are determined for a time interval in which the sea state can be considered to be constant. This duration has to be much longer than the individual wave period, but smaller than the period in which the wind and swell conditions vary significantly. Typically, records of one hundred to one thousand wave-periods are used to determine the wave statistics.

The large number of variables involved in creating the sea state cannot be quickly and easily summarized, so simpler scales are used to give an approximate but concise description of conditions for reporting in a ship's log or similar record.

The World Meteorological Organization (WMO) sea state code largely adopts the 'wind sea' definition of the Douglas Sea Scale.

WMO Sea State Code | Wave height | Characteristics |
---|---|---|

0 | 0 metres (0 ft) | Calm (glassy) |

1 | 0 to 0.1 metres (0.00 to 0.33 ft) | Calm (rippled) |

2 | 0.1 to 0.5 metres (3.9 in to 1 ft 7.7 in) | Smooth (wavelets) |

3 | 0.5 to 1.25 metres (1 ft 8 in to 4 ft 1 in) | Slight |

4 | 1.25 to 2.5 metres (4 ft 1 in to 8 ft 2 in) | Moderate |

5 | 2.5 to 4 metres (8 ft 2 in to 13 ft 1 in) | Rough |

6 | 4 to 6 metres (13 to 20 ft) | Very rough |

7 | 6 to 9 metres (20 to 30 ft) | High |

8 | 9 to 14 metres (30 to 46 ft) | Very high |

9 | Over 14 metres (46 ft) | Phenomenal |

0. None | |

Low | 1. Short or average 2. Long |

Moderate | 3. Short 4. Average 5. Long |

High | 6. Short 7. Average 8. Long |

9. Confused |

- The direction from which the swell is coming should be recorded.

In engineering applications, sea states are often characterized by the following two parameters:

- The significant wave height H
_{1/3}— the mean wave height of the one third highest waves. - The mean wave period, T
_{1}.

In physical oceanography, the **significant wave height** is defined traditionally as the mean wave height of the highest third of the waves (*H*_{1/3}). Nowadays it is usually defined as four times the standard deviation of the surface elevation – or equivalently as four times the square root of the zeroth-order moment (area) of the wave spectrum. The symbol *H*_{m0} is usually used for that latter definition. The significant wave height may thus refer to *H*_{m0} or *H*_{1/3}; the difference in magnitude between the two definitions is only a few percent.

In fluid dynamics, the **wave height** of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean and naval engineering.

The sea state is in addition to these two parameters (or variation of the two) also described by the wave spectrum which is a function of a wave height spectrum and a wave direction spectrum . Some wave height spectra are listed below. The dimension of the wave spectrum is , and many interesting properties about the sea state can be found from the spectrum.

The relationship between the spectrum and the wave amplitude for a wave component is:

- ITTC
^{ [1] }recommended spectrum model for fully developed sea (ISSC^{ [2] }spectrum/modified Pierson-Moskowitz spectrum):^{ [3] }

- ITTC recommended spectrum model for limited fetch (JONSWAP spectrum)

The **fetch**, also called the **fetch length**, is the length of water over which a given wind has blown. Fetch is used in geography and meteorology and its effects are usually associated with sea state and when it reaches shore it is the main factor that creates storm surge which leads to coastal erosion and flooding. It also plays a large part in longshore drift as well.

where

and

(The latter model has since its creation improved based on the work of Phillips and Kitaigorodskii to better model the wave height spectrum for high wavenumbers.^{ [4] })

An example function might be:

Thus the sea state is fully determined and can be recreated by the following function where is the wave elevation, is uniformly distributed between 0 and , and is randomly drawn from the directional distribution function ^{ [5] }

In addition to the short term wave statistics presented above, long term sea state statistics are often given as a joint frequency table of the significant wave height and the mean wave period. From the long and short term statistical distributions it is possible to find the extreme values expected in the operating life of a ship. A ship designer can find the most extreme sea states (extreme values of H_{1/3} and T_{1}) from the joint frequency table, and from the wave spectrum the designer can find the most likely highest wave elevation in the most extreme sea states and predict the most likely highest loads on individual parts of the ship from the response amplitude operators of the ship. Surviving the once in 100 years or once in 1000 years sea state is a normal demand for design of ships and offshore structures.

- ↑
*International Towing Tank Conference (ITTC)*, retrieved 11 November 2010 - ↑ International Ship and Offshore Structures Congress
- ↑ Pierson, W. J.; Moscowitz, L. (1964), "A proposed spectral form for fully developed wind seas based on the similarity theory of S A Kitaigorodskii",
*Journal of Geophysical Research*,**69**(24): 5181–5190, Bibcode:1964JGR....69.5181P, doi:10.1029/JZ069i024p05181 - ↑ Elfouhaily, T.; Chapron, B.; Katsaros, K.; Vandemark, D. (July 15, 1997). "A unified directional spectrum for long and short wind-driven waves" (PDF).
*Journal of Geophysical Research*.**102**(C7): 15781–15796. Bibcode:1997JGR...10215781E. doi:10.1029/97jc00467. - ↑ Jefferys, E. R. (1987), "Directional seas should be ergodic",
*Applied Ocean Research*,**9**(4): 186–191, doi:10.1016/0141-1187(87)90001-0

**Frequency modulation synthesis** is a form of sound synthesis where the frequency of a waveform, called the carrier, is changed by modulating its frequency with a modulator. The frequency of an oscillator is altered "in accordance with the amplitude of a modulating signal."

**Double-sideband suppressed-carrier transmission** (**DSB-SC**) is transmission in which frequencies produced by amplitude modulation (AM) are symmetrically spaced above and below the carrier frequency and the carrier level is reduced to the lowest practical level, ideally being completely suppressed.

In optics, **Lambert's cosine law** says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle *θ* between the direction of the incident light and the surface normal. The law is also known as the **cosine emission law** or **Lambert's emission law**. It is named after Johann Heinrich Lambert, from his *Photometria*, published in 1760.

**Bremsstrahlung**, from *bremsen* "to brake" and *Strahlung* "radiation"; i.e., "braking radiation" or "deceleration radiation", is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic energy, which is converted into radiation, thus satisfying the law of conservation of energy. The term is also used to refer to the process of producing the radiation. Bremsstrahlung has a continuous spectrum, which becomes more intense and whose peak intensity shifts toward higher frequencies as the change of the energy of the decelerated particles increases.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, i.e., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

A **tautochrone** or **isochrone curve** is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is the same as the brachistochrone curve for any given starting point.

In physics, the **Rabi cycle** is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In physics and engineering, a **phasor**, is a complex number representing a sinusoidal function whose amplitude (*A*), angular frequency (*ω*), and initial phase (*θ*) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as **phasor**, **complex amplitude**, and **sinor** or even **complexor**.

The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

**Sinusoidal plane-wave solutions** are particular solutions to the electromagnetic wave equation.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

**Zero sound** is the name given by Lev Landau to the unique quantum vibrations in quantum Fermi liquids.

In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

The **Appleton–Hartree equation**, sometimes also referred to as the **Appleton–Lassen equation** is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton–Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and German radio physicist H. K. Lassen. Lassen's work, completed two years prior to Appleton and five years prior to Hartree, included a more thorough treatment of collisional plasma; but, published only in German, it has not been widely read in the English speaking world of radio physics. Further, regarding the derivation by Appleton, it was noted in the historical study by Gilmore that Wilhelm Altar first calculated the dispersion relation in 1926.

In mathematics, an **analytic semigroup** is particular kind of strongly continuous semigroup. Analytic semigroups are used in the solution of partial differential equations; compared to strongly continuous semigroups, analytic semigroups provide better regularity of solutions to initial value problems, better results concerning perturbations of the infinitesimal generator, and a relationship between the type of the semigroup and the spectrum of the infinitesimal generator.

**Surface-extended X-ray absorption fine structure** (SEXAFS) is the surface-sensitive equivalent of the EXAFS technique. This technique involves the illumination of the sample by high-intensity X-ray beams from a synchrotron and monitoring their photoabsorption by detecting in the intensity of Auger electrons as a function of the incident photon energy. Surface sensitivity is achieved by the interpretation of data depending on the intensity of the Auger electrons instead of looking at the relative absorption of the X-rays as in the parent method, EXAFS.

In optics, the **Fraunhofer diffraction equation** is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

**Partial wave analysis**, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular momentum components and solving using boundary conditions.

The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform.

The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.

The simplest way to derive the spectrum of a chirp, now that computers are widely available, is to sample the time-domain waveform at a frequency well above the Nyquist limit and call up an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, where possible, or to graphical or approximation methods, otherwise. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps.

- Bowditch, Nathaniel (1938),
*American Practical Navigator*, H.O. pub No. 9 (revised ed.), United States Hydrographic Office, OCLC 31033357 - Faltinsen, O. M. (1990),
*Sea Loads on Ships and Offshore Structures*, [Cambridge University Press], ISBN 0-521-45870-6

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