1.
Wave
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In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium. Frequency refers to the addition of time, wave motion transfers energy from one point to another, which displace particles of the transmission medium–that is, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations, around almost fixed locations, there are two main types of waves. Mechanical waves propagate through a medium, and the substance of this medium is deformed, restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors, when the molecules collide, they also bounce away from each other. This keeps the molecules from continuing to travel in the direction of the wave, the second main type, electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves, in addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. While mechanical waves can be transverse and longitudinal, all electromagnetic waves are transverse in free space. A single, all-encompassing definition for the wave is not straightforward. A vibration can be defined as a back-and-forth motion around a reference value, however, a vibration is not necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify a phenomenon as a results in a blurred line. The term wave is often understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium and it may appear that the description of waves is closely related to their physical origin for each specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a rather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, or elasticity and this difference in origin introduces certain wave characteristics particular to the properties of the medium involved
2.
Coriolis force
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In physics, the Coriolis force is an inertial force that acts on objects that are in motion relative to a rotating reference frame. In a reference frame with clockwise rotation, the acts to the left of the motion of the object. In one with anticlockwise rotation, the acts to the right. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology, deflection of an object due to the Coriolis force is called the Coriolis effect. Newtons laws of motion describe the motion of an object in a frame of reference. When Newtons laws are transformed to a frame of reference. Both forces are proportional to the mass of the object, the Coriolis force is proportional to the rotation rate and the centrifugal force is proportional to its square. The Coriolis force acts in a perpendicular to the rotation axis. The centrifugal force acts outwards in the direction and is proportional to the distance of the body from the axis of the rotating frame. These additional forces are termed inertial forces, fictitious forces or pseudo forces and they allow the application of Newtons laws to a rotating system. They are correction factors that do not exist in a non-accelerating or inertial reference frame, a commonly encountered rotating reference frame is the Earth. The Coriolis effect is caused by the rotation of the Earth, such motions are constrained by the surface of the Earth, so only the horizontal component of the Coriolis force is generally important. This force causes moving objects on the surface of the Earth to be deflected to the right in the Northern Hemisphere, the horizontal deflection effect is greater near the poles, since the effective rotation rate about a local vertical axis is largest there, and smallest at the equator. This effect is responsible for the rotation of large cyclones, riccioli, Grimaldi, and Dechales all described the effect as part of an argument against the heliocentric system of Copernicus. In other words, they argued that the Earths rotation should create the effect, the effect was described in the tidal equations of Pierre-Simon Laplace in 1778. Gaspard-Gustave Coriolis published a paper in 1835 on the yield of machines with rotating parts. That paper considered the forces that are detected in a rotating frame of reference. Coriolis divided these forces into two categories
3.
Topographic
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Topography is the study of the shape and features of the surface of the Earth and other observable astronomical objects including planets, moons, and asteroids. The topography of an area could refer to the shapes and features themselves. This field of geoscience and planetary science is concerned with detail in general, including not only relief but also natural and artificial features. This meaning is common in the United States, where topographic maps with elevation contours have made topography synonymous with relief. The older sense of topography as the study of place still has currency in Europe, topography in a narrow sense involves the recording of relief or terrain, the three-dimensional quality of the surface, and the identification of specific landforms. This is also known as geomorphometry, in modern usage, this involves generation of elevation data in digital form. It is often considered to include the representation of the landform on a map by a variety of techniques, including contour lines, hypsometric tints. The term topography originated in ancient Greece and continued in ancient Rome, the word comes from the Greek τόπος and -γραφία. In classical literature this refers to writing about a place or places, in Britain and in Europe in general, the word topography is still sometimes used in its original sense. Detailed military surveys in Britain were called Ordnance Surveys, and this term was used into the 20th century as generic for topographic surveys, the earliest scientific surveys in France were called the Cassini maps after the family who produced them over four generations. The term topographic surveys appears to be American in origin, the earliest detailed surveys in the United States were made by the “Topographical Bureau of the Army, ” formed during the War of 1812, which became the Corps of Topographical Engineers in 1838. In the 20th century, the term started to be used to describe surface description in other fields where mapping in a broader sense is used. An objective of topography is to determine the position of any feature or more generally any point in terms of both a horizontal coordinate system such as latitude, longitude, and altitude, identifying features, and recognizing typical landform patterns are also part of the field. There are a variety of approaches to studying topography, which method to use depend on the scale and size of the area under study, its accessibility, and the quality of existing surveys. Work on one of the first topographic maps was begun in France by Giovanni Domenico Cassini, in areas where there has been an extensive direct survey and mapping program, the compiled data forms the basis of basic digital elevation datasets such as USGS DEM data. This data must often be cleaned to eliminate discrepancies between surveys, but it forms a valuable set of information for large-scale analysis. The original American topographic surveys involved not only recording of relief, remote sensing is a general term for geodata collection at a distance from the subject area. Besides their role in photogrammetry, aerial and satellite imagery can be used to identify and delineate terrain features, certainly they have become more and more a part of geovisualization, whether maps or GIS systems
4.
Waveguide
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A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. This is an effect to waves of water constrained within a canal. Without the physical constraint of a waveguide, waves are decreasing according to the square law as they expand into three dimensional space. There are different types of waveguides for each type of wave, the original and most common meaning is a hollow conductive metal pipe used to carry high frequency radio waves, particularly microwaves. The geometry of a waveguide reflects its function, slab waveguides confine energy in one dimension, fiber or channel waveguides in two dimensions. The frequency of the wave also dictates the shape of a waveguide. As a rule of thumb, the width of a waveguide needs to be of the order of magnitude as the wavelength of the guided wave. Some naturally occurring structures can act as waveguides. The SOFAR channel layer in the ocean can guide the sound of whale song across enormous distances, waves propagate in all directions in open space as spherical waves. The power of the falls with the distance R from the source as the square of the distance. A waveguide confines the wave to propagate in one dimension, so that, under ideal conditions, due to total reflection at the walls, waves are confined to the interior of a waveguide. The first structure for guiding waves was proposed by J. J. Thomson in 1893, the first mathematical analysis of electromagnetic waves in a metal cylinder was performed by Lord Rayleigh in 1897. For sound waves, Lord Rayleigh published a mathematical analysis of propagation modes in his seminal work. The study of dielectric waveguides began as early as the 1920s, by people, most famous of which are Rayleigh, Sommerfeld. Optical fiber began to receive attention in the 1960s due to its importance to the communications industry. The development of radio communication initially occurred at the lower frequencies because these could be easily propagated over large distances. The long wavelengths made these frequencies unsuitable for use in hollow metal waveguides because of the large diameter tubes required. Consequently, research into hollow metal waveguides stalled and the work of Lord Rayleigh was forgotten for a time and had to be rediscovered by others, practical investigations resumed in the 1930s by George C
5.
Dispersion (water waves)
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In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in context, are waves propagating on the water surface, with gravity. As a result, water with a surface is generally considered to be a dispersive medium. For a certain depth, surface gravity waves – i. e. waves occurring at the air–water interface. On the other hand, for a wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves propagate faster for shorter wavelengths, besides frequency dispersion, water waves also exhibit amplitude dispersion. This is an effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves. This section is about frequency dispersion for waves on a fluid layer forced by gravity, for surface tension effects on frequency dispersion, see surface tension effects in Airy wave theory and capillary wave. The simplest propagating wave of unchanging form is a sine wave. Characteristic phases of a wave are, the upward zero-crossing at θ =0, the wave crest at θ = ½ π, the downward zero-crossing at θ = π. A certain phase repeats itself after an integer m multiple of 2π, the dispersion relation has two solutions, ω = +Ω and ω = −Ω, corresponding to waves travelling in the positive or negative x–direction. The dispersion relation will in general depend on other parameters in addition to the wavenumber k. For gravity waves, according to theory, these are the acceleration by gravity g. The dispersion relation for these waves is, an equation with tanh denoting the hyperbolic tangent function. An initial wave phase θ = θ0 propagates as a function of space and its subsequent position is given by, x = λ T t + λ2 π θ0 = ω k t + θ0 k. This shows that the moves with the velocity, c p = λ T = ω k = Ω k. A sinusoidal wave, of small amplitude and with a constant wavelength, propagates with the phase velocity. While the phase velocity is a vector and has an associated direction, according to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth
6.
Group velocity
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The group velocity of a wave is the velocity with which the overall shape of the waves amplitudes—known as the modulation or envelope of the wave—propagates through space. For example, if a stone is thrown into the middle of a very still pond, the expanding ring of waves is the wave group, within which one can discern individual wavelets of differing wavelengths traveling at different speeds. The longer waves travel faster than the group as a whole, the shorter waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. The group velocity vg is defined by the equation, v g ≡ ∂ ω ∂ k where ω is the angular frequency. The phase velocity is, vp = ω / k, the function ω, which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity, if ω is a linear function of k, but not directly proportional, then the group velocity and phase velocity are different. The envelope of a packet will travel at the group velocity, while the individual peaks. If ω is not a function of k, the envelope of a wave packet will become distorted as it travels. Since a wave packet contains a range of different frequencies, the group velocity ∂ω/∂k will be different for different values of k, therefore, the envelope does not move at a single velocity, but its wavenumber components move at different velocities, distorting the envelope. If the wavepacket has a range of frequencies, and ω is approximately linear over that narrow range. For example, for deep water gravity waves, ω=√gk, and this underlies the Kelvin wake pattern for the bow wave of all ships and swimming objects. Regardless of how fast they are moving, as long as their velocity is constant, one derivation of the formula for group velocity is as follows. Consider a wave packet as a function of x and time t, α. Let A be its Fourier transform at time t=0, α = ∫ − ∞ ∞ d k A e i k x. By the superposition principle, the wavepacket at any time t is α = ∫ − ∞ ∞ d k A e i, assume that the wave packet α is almost monochromatic, so that A is sharply peaked around a central wavenumber k0. Then, linearization gives ω ≈ ω0 + ω0 ′ where ω0 = ω and ω0 ′ = ∂ ω ∂ k | k = k 0, then, after some algebra, α = e i ∫ − ∞ ∞ d k A e i. There are two factors in this expression, the first factor, e i, describes a perfect monochromatic wave with wavevector k0, with peaks and troughs moving at the phase velocity ω0 / k 0 within the envelope of the wavepacket
7.
Fluid dynamics
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In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. They are expressed using the Reynolds transport theorem, in addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects, however, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of molecules is ignored. The unsimplified equations do not have a general solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve, some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation
8.
Vortex
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In fluid dynamics, a vortex is a region in a fluid in which the flow rotates around an axis line, which may be straight or curved. The plural of vortex is either vortices or vortexes, vortices form in stirred fluids, and may be observed in phenomena such as smoke rings, whirlpools in the wake of boat, or the winds surrounding a tornado or dust devil. Vortices are a component of turbulent flow. The distribution of velocity, vorticity, as well as the concept of circulation are used to characterize vortices, in most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis. Once formed, vortices can move, stretch, twist, a moving vortex carries with it some angular and linear momentum, energy, and mass. A key concept in the dynamics of vortices is the vorticity, conceptually, the vorticity could be observed by placing a tiny rough ball at the point in question, free to move with the fluid, and observing how it rotates about its center. The direction of the vorticity vector is defined to be the direction of the axis of rotation of this imaginary ball while its length is twice the angular velocity. The local rotation measured by the vorticity ω → must not be confused with the velocity vector of that portion of the fluid with respect to the external environment or to any fixed axis. In a vortex, in particular, ω → may be opposite to the angular velocity vector of the fluid relative to the vortexs axis. In theory, the u of the particles in a vortex may vary with the distance r from the axis in many ways. Ω → =, r → =, u → = Ω → × r → =, ω → = ∇ × u → = =2 Ω →. In this case the vorticity ω → is zero at any point not on that axis, Ω → =, r → =, u → = Ω → × r → =, ω → = ∇ × u → =0. In the absence of forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern. Irrotational vortices are also called free vortices, for an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis, and has a fixed value, Γ, for any contour that does enclose the axis once. The tangential component of the velocity is then u θ = Γ2 π r. The angular momentum per unit mass relative to the axis is therefore constant. However, the ideal irrotational vortex flow is not physically realizable, indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational, the vorticity ω → becomes non-zero, the Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r0, and irrotational flow outside that core regions
9.
Superfluidity
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Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without loss of kinetic energy. When stirred a superfluid forms cellular vortices that continue to rotate indefinitely, superfluidity occurs in two isotopes of helium when they are liquified by cooling to cryogenic temperatures. It is also a property of other exotic states of matter theorized to exist in astrophysics, high-energy physics. Superfluidity was originally discovered in liquid helium, by Pyotr Kapitsa and it has since been described through phenomenology and microscopic theories. In liquid helium-4, the superfluidity occurs at far higher temperatures than it does in helium-3, each atom of helium-4 is a boson particle, by virtue of its integer spin. A helium-3 atom is a particle, it can form bosons only by pairing with itself at much lower temperatures. The discovery of superfluidity in helium-3 was the basis for the award of the 1996 Nobel Prize in Physics and this process is similar to the electron pairing in superconductivity. Superfluidity in an ultracold fermionic gas was experimentally proven by Wolfgang Ketterle, such vortices had previously been observed in an ultracold bosonic gas using 87Rb in 2000, and more recently in two-dimensional gases. As early as 1999 Lene Hau created such a condensate using sodium atoms for the purpose of slowing light, with a double light-roadblock setup, we can generate controlled collisions between shock waves resulting in completely unexpected, nonlinear excitations. We have observed hybrid structures consisting of vortex rings embedded in dark solitonic shells, the vortex rings act as phantom propellers leading to very rich excitation dynamics. The idea that superfluidity exists inside neutron stars was first proposed by Arkady Migdal, superfluid vacuum theory is an approach in theoretical physics and quantum mechanics where the physical vacuum is viewed as superfluid. The ultimate goal of the approach is to develop scientific models that unify quantum mechanics with gravity and this makes SVT a candidate for the theory of quantum gravity and an extension of the Standard Model. Boojum Condensed matter physics Macroscopic quantum phenomena Quantum hydrodynamics Slow light Supersolid Guénault, annett, James F. Superconductivity, superfluids, and condensates. The Universe in a helium droplet
10.
Momentum
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In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
11.
Continuity equation
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A continuity equation in physics is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, continuity equations are a stronger, local form of conservation laws. For example, a version of the law of conservation of energy states that energy can neither be created nor destroyed—i. e. The total amount of energy is fixed and this statement does not immediately rule out the possibility that energy could disappear from a field in Canada while simultaneously appearing in a room in Indonesia. A stronger statement is that energy is conserved, Energy can neither be created nor destroyed. A continuity equation is the way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of charge at any point can only change by the amount of electric current flowing into or out of that point. In an everyday example, there is a continuity equation for the number of people alive, it has a term to account for people being born. Any continuity equation can be expressed in a form, which applies to any finite region. Continuity equations underlie more specific transport equations such as the equation, Boltzmann transport equation. Before we can write down the continuity equation, we must first define flux, the continuity equation is useful when there is some quantity q which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let ρ be the density of this property, i. e. the amount of q per unit volume. The way that this quantity q is flowing is described by its flux, the flux of q is a vector field, which we denote as j. Here are some examples and properties of flux, The dimension of flux is amount of q flowing per unit time, outside the pipe, where there is no water, the flux is zero. In a well-known example, the flux of electric charge is the current density. In a simple example, V could be a building, and q could be the number of people in the building, the surface S would consist of the walls, doors, roof, and foundation of the building. Terms that generate or remove q are referred to as a sources and this general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation, mathematically it is an automatic consequence of Maxwells equations, although charge conservation is more fundamental than Maxwells equations
12.
William Thomson, 1st Baron Kelvin
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William Thomson, 1st Baron Kelvin, OM, GCVO, PC, FRS, FRSE was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824. He worked closely with mathematics professor Hugh Blackburn in his work and he also had a career as an electric telegraph engineer and inventor, which propelled him into the public eye and ensured his wealth, fame and honour. For his work on the telegraph project he was knighted in 1866 by Queen Victoria. He had extensive maritime interests and was most noted for his work on the mariners compass, absolute temperatures are stated in units of kelvin in his honour. He was ennobled in 1892 in recognition of his achievements in thermodynamics and he was the first British scientist to be elevated to the House of Lords. The title refers to the River Kelvin, which close by his laboratory at the University of Glasgow. His home was the red sandstone mansion Netherhall, in Largs. William Thomsons father, James Thomson, was a teacher of mathematics and engineering at Royal Belfast Academical Institution, James Thomson married Margaret Gardner in 1817 and, of their children, four boys and two girls survived infancy. Margaret Thomson died in 1830 when William was six years old, William and his elder brother James were tutored at home by their father while the younger boys were tutored by their elder sisters. James was intended to benefit from the share of his fathers encouragement, affection. In 1832, his father was appointed professor of mathematics at Glasgow, the Thomson children were introduced to a broader cosmopolitan experience than their fathers rural upbringing, spending mid-1839 in London and the boys were tutored in French in Paris. Mid-1840 was spent in Germany and the Netherlands, language study was given a high priority. His sister, Anna Thomson, was the mother of James Thomson Bottomley FRSE, Thomson had heart problems and nearly died when he was 9 years old. In school, Thomson showed a keen interest in the classics along with his natural interest in the sciences, at the age of 12 he won a prize for translating Lucian of Samosatas Dialogues of the Gods from Latin to English. In the academic year 1839/1840, Thomson won the prize in astronomy for his Essay on the figure of the Earth which showed an early facility for mathematical analysis. Throughout his life, he would work on the problems raised in the essay as a strategy during times of personal stress. On the title page of this essay Thomson wrote the lines from Alexander Popes Essay on Man. These lines inspired Thomson to understand the world using the power and method of science, Go
13.
Lamb waves
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Lamb waves propagate in solid plates. They are elastic waves whose particle motion lies in the plane contains the direction of wave propagation. In 1917, the English mathematician Horace Lamb published his classic analysis and their properties turned out to be quite complex. Since the 1990s, the understanding and utilization of Lamb waves has advanced greatly, Lambs theoretical formulations have found substantial practical application, especially in the field of nondestructive testing. The term Rayleigh–Lamb waves embraces the Rayleigh wave, a type of wave propagates along a single surface. Both Rayleigh and Lamb waves are constrained by the properties of the surface that guide them. In general, elastic waves in solid materials are guided by the boundaries of the media in which they propagate and this is a classic eigenvalue problem. Waves in plates were among the first guided waves to be analyzed in this way, the analysis was developed and published in 1917 by Horace Lamb, a leader in the mathematical physics of his day. Lambs equations were derived by setting up formalism for a solid plate having infinite extent in the x and y directions, and thickness d in the z direction. Displacement is a function of x, z, t only, there is no displacement in the y direction, the physical boundary condition for the free surfaces of the plate is that the component of stress in the z direction at z = +/- d/2 is zero. Applying these two conditions to the solutions to the wave equation, a pair of characteristic equations can be found. Inherent in these equations is a relationship between the angular frequency ω and the number k. The solution of these equations also reveals the precise form of the particle motion, figure 1 illustrates a member of each family. Lamb’s characteristic equations were established for waves propagating in an infinite plate - a homogeneous, in formulating his problem, Lamb confined the components of particle motion to the direction of the plate normal and the direction of wave propagation. By definition, Lamb waves have no motion in the y-direction. Motion in the y-direction in plates is found in the so-called SH or shear-horizontal wave modes and these have no motion in the x- or z-directions, and are thus complementary to the Lamb wave modes. These two are the only wave types which can propagate with straight, infinite wave fronts in a plate as defined above. Lamb waves exhibit velocity dispersion, that is, their velocity of propagation c depends on the frequency, as well as on the elastic constants and this phenomenon is central to the study and understanding of wave behavior in plates
14.
Linearization
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In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in such as engineering, physics, economics. Linearizations of a function are lines—usually lines that can be used for purposes of calculation, in short, linearization approximates the output of a function near x = a. However, what would be an approximation of 4.001 =4 +.001. For any given function y = f, f can be approximated if it is near a known differentiable point, the most basic requisite is that L a = f, where L a is the linearization of f at x = a. The point-slope form of an equation forms an equation of a line, given a point, the general form of this equation is, y − K = M. Using the point, L a becomes y = f + M, because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f at x = a. While the concept of local linearity applies the most to points arbitrarily close to x = a, the slope M should be, most accurately, the slope of the tangent line at x = a. Visually, the diagram shows the tangent line of f at x. At f, where h is any positive or negative value. The final equation for the linearization of a function at x = a is, the derivative of f is f ′, and the slope of f at a is f ′. To find 4.001, we can use the fact that 4 =2. The linearization of f = x at x = a is y = a +12 a, substituting in a =4, the linearization at 4 is y =2 + x −44. In this case x =4.001, so 4.001 is approximately 2 +4.001 −44 =2.00025. The true value is close to 2.00024998, so the linearization approximation has an error of less than 1 millionth of a percent. Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest, in stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium
15.
Baroclinity
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In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In atmospheric terms, the zones of the Earth are generally found in the central latitudes, or tropics. Baroclinity is proportional to ∇ p × ∇ ρ which is proportional to the sine of the angle between surfaces of constant pressure and surfaces of constant density, thus, in a barotropic fluid, these surfaces are parallel. Areas of high atmospheric baroclinity are characterized by the frequent formation of cyclones, baroclinic instability is a fluid dynamical instability of fundamental importance in the atmosphere and in the oceans. In the atmosphere it is the dominant mechanism shaping the cyclones and anticyclones that dominate weather in mid-latitudes, in the ocean it generates a field of mesoscale eddies that play various roles in oceanic dynamics and the transport of tracers. Baroclinic instability is a relevant to rapidly rotating, strongly stratified fluids. Whether a fluid counts as rapidly rotating is determined in this context by the Rossby number, more precisely, a flow in solid body rotation has vorticity that is proportional to its angular velocity. The Rossby number is a measure of the departure of the vorticity from that of body rotation. The Rossby number must be small for the concept of baroclinic instability to be relevant, when the Rossby number is large, other kinds of instabilities, often referred to as inertial, become more relevant. The simplest example of a stably stratified flow is a flow with density decreasing with height. In a compressible gas such as the atmosphere, the relevant measure is the gradient of the entropy. This measure is the Richardson number, when the Richardson number is large, the stratification is strong enough to prevent this shear instability. The most important feature of baroclinic instability is that it even in the situation of rapid rotation. The energy source for baroclinic instability is the energy in the environmental flow. As the instability grows, the center of mass of the fluid is lowered, in growing waves in the atmosphere, cold air moving downwards and equatorwards displaces the warmer air moving polewards and upwards. Baroclinic instability can be investigated in the using a rotating. The annulus is heated at the wall and cooled at the inner wall. The term baroclinic refers to the mechanism by which vorticity is generated, vorticity is the curl of the velocity field
16.
Rossby wave
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Rossby waves, also known as planetary waves, are a natural phenomenon in the atmosphere and oceans of planets that largely owe their properties to rotation of the planet. Rossby waves are a subset of inertial waves, atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. These waves are associated with systems and the jet stream. Oceanic Rossby waves move along the thermocline, the boundary between the upper layer and the cold deeper part of the ocean. Atmospheric Rossby waves result from the conservation of vorticity and are influenced by the Coriolis force. A fluid, on the Earth, that moves toward the pole will deviate toward the east, the deviations are caused by the Coriolis force and conservation of potential vorticity which leads to changes of relative vorticity. This is analogous to conservation of momentum in mechanics. In planetary atmospheres, including Earth, Rossby waves are due to the variation in the Coriolis effect with latitude, carl-Gustaf Arvid Rossby first identified such waves in the Earths atmosphere in 1939 and went on to explain their motion. One can identify a terrestrial Rossby wave as its velocity, marked by its wave crest. However, the set of Rossby waves may appear to move in either direction with what is known as its group velocity. In general, shorter waves have a group velocity and long waves a westward group velocity. The terms barotropic and baroclinic are used to distinguish the structure of Rossby waves. Barotropic Rossby waves do not vary in the vertical, and have the fastest propagation speeds, the baroclinic wave modes, on the other hand, do vary in the vertical. They are also slower, with speeds of only a few centimeters per second or less, most investigations of Rossby waves has been done on those in Earths atmosphere. Rossby waves in the Earths atmosphere are easy to observe as large-scale meanders of the jet stream, the action of Rossby waves partially explains why eastern continental edges, such as the Northeast United States and Eastern Canada, are colder than Western Europe at the same latitudes. Deep convection to the troposphere is enhanced over very warm sea surfaces in the tropics and this tropical forcing generates atmospheric Rossby waves that have a poleward and eastward migration. Poleward-propagating Rossby waves explain many of the observed statistical connections between low- and high-latitude climates, one such phenomenon is sudden stratospheric warming. Poleward-propagating Rossby waves are an important and unambiguous part of the variability in the Northern Hemisphere, similar mechanisms apply in the Southern Hemisphere and partly explain the strong variability in the Amundsen Sea region of Antarctica
17.
Rossby-gravity waves
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Rossby-gravity waves are equatorially trapped waves, meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more known as the equatorial Rossby deformation radius. They always carry energy eastward, but their crests and troughs may propagate westward if their periods are long enough, the eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter vanishes at 0 degrees latitude, the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and β is the variation of the coriolis parameter with latitude, ∂ f ∂ y = β. Once the frequency relation is formulated in terms of ω, the angular frequency and these three solutions correspond to the equatorially trapped gravity wave, the equatorially trapped Rossby wave and the mixed Rossby-gravity wave. Equatorial gravity waves can be either westward- or eastward-propagating, and correspond to n=1 on a dispersion relation diagram, as mentioned earlier, the group velocity is always directed toward the east with a maximum for short waves. If this Brunt–Vaisala frequency does not change, then these waves become vertically propagating solutions. On a typical m, k dispersion diagram, the velocity would be directed at right angles to the n =0 and n =1 curves. Typical group velocities for each component are the following,1 cm/s for gravity waves and 2 mm/s for planetary waves and these vertically propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai. They had the characteristics, 4–5 days, horizontal wavenumbers of 4, vertical wavelengths of 4–8 km. Also, the vertically propagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400 km, and vertical wavelengths of 5 km
18.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
19.
Physical oceanography
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Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters. Physical oceanography is one of several sub-domains into which oceanography is divided, others include biological, chemical and geological oceanography. Physical oceanography may be subdivided into descriptive and dynamical physical oceanography, descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precise as possible. Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and these are part of the large field of Geophysical Fluid Dynamics that is shared together with meteorology. The fundamental role of the oceans in shaping Earth is acknowledged by ecologists, geologists, meteorologists, climatologists, an Earth without oceans would truly be unrecognizable. Roughly 97% of the water is in its oceans. The tremendous heat capacity of the oceans moderates the planets climate, the oceans influence extends even to the composition of volcanic rocks through seafloor metamorphism, as well as to that of volcanic gases and magmas created at subduction zones. Though this apparent discrepancy is great, for land and sea, the respective extremes such as mountains and trenches are rare. Because the vast majority of the oceans volume is deep water. The same percentage falls in a salinity range between 34–35 ppt, there is still quite a bit of variation, however. Surface temperatures can range from below freezing near the poles to 35 °C in restricted tropical seas, in terms of temperature, the oceans layers are highly latitude-dependent, the thermocline is pronounced in the tropics, but nonexistent in polar waters. The halocline usually lies near the surface, where evaporation raises salinity in the tropics and these variations of salinity and temperature with depth change the density of the seawater, creating the pycnocline. Energy for the ocean circulation comes from solar radiation and gravitational energy from the sun, perhaps three quarters of this heat is carried in the atmosphere, the rest is carried in the ocean. The atmosphere is heated from below, which leads to convection, by contrast the ocean is heated from above, which tends to suppress convection. Instead ocean deep water is formed in regions where cold salty waters sink in fairly restricted areas. This is the beginning of the thermohaline circulation, oceanic currents are largely driven by the surface wind stress, hence the large-scale atmospheric circulation is important to understanding the ocean circulation. The Hadley circulation leads to Easterly winds in the tropics and Westerlies in mid-latitudes and this leads to slow equatorward flow throughout most of a subtropical ocean basin. The return flow occurs in an intense, narrow, poleward western boundary current, like the atmosphere, the ocean is far wider than it is deep, and hence horizontal motion is in general much faster than vertical motion
20.
Wind wave
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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 high, when directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells, more generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago, wind waves in the ocean are called ocean surface waves. Wind waves have an amount of randomness, subsequent waves differ in height, duration. The key statistics of wind waves in evolving sea states can be predicted with wind wave models, although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves. The great majority of large breakers seen at a result from distant winds. Water depth All of these work together to determine the size of wind waves. Further exposure to that wind could only cause a dissipation of energy due to the breaking of wave tops. Waves in an area typically have a range of heights. For weather reporting and for analysis of wind wave statistics. This figure represents an average height of the highest one-third of the waves in a time period. The significant wave height is also the value a trained observer would estimate from visual observation of a sea state, given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm. Wave formation on a flat water surface by wind is started by a random distribution of normal pressure of turbulent wind flow over the water. This pressure fluctuation produces normal and tangential stresses in the surface water and it is assumed that, The water is originally at rest. There is a distribution of normal pressure to the water surface from the turbulent wind. Correlations between air and water motions are neglected, the second mechanism involves wind shear forces on the water surface
21.
Airy wave theory
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In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the layer has a uniform mean depth. This theory was first published, in form, by George Biddell Airy in the 19th century. Further, several second-order nonlinear properties of gravity waves, and their propagation. Airy wave theory is also a good approximation for tsunami waves in the ocean and this linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the height to water depth. Airy wave theory uses a potential approach to describe the motion of gravity waves on a fluid surface. This is due to the fact that for the part of the fluid motion. Airy wave theory is used in ocean engineering and coastal engineering. Diffraction is one of the effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoaling and refraction can be predicted, earlier attempts to describe surface gravity waves using potential flow were made by, among others, Laplace, Poisson, Cauchy and Kelland. But Airy was the first to publish the correct derivation and formulation in 1841, soon after, in 1847, the linear theory of Airy was extended by Stokes for non-linear wave motion – known as Stokes wave theory – correct up to third order in the wave steepness. Even before Airys linear theory, Gerstner derived a nonlinear wave theory in 1802. Airy wave theory is a theory for the propagation of waves on the surface of a potential flow. The waves propagate along the surface with the phase speed cp. The angular wavenumber k and frequency ω are not independent parameters, surface gravity waves on a fluid are dispersive waves – exhibiting frequency dispersion – meaning that each wavenumber has its own frequency and phase speed. Note that in engineering the wave height H – the difference in elevation between crest and trough – is often used, H =2 a and a =12 H, underneath the surface, there is a fluid motion associated with the free surface motion. While the surface shows a propagating wave, the fluid particles are in an orbital motion
22.
Ballantine scale
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The Ballantine scale is a biologically defined scale for measuring the degree of exposure level of wave action on a rocky shore. The species present in the littoral zone therefore indicate the degree of the shores exposure, an abbreviated summary of the scale is given below. The scale runs from 1) an extremely exposed shore, to 8) an extremely sheltered shore, the littoral zone generally is the zone between low and high tides. The supra-littoral is above the barnacle line, the eulittoral zone is dominated by barnacles and limpets with a coralline belt in the very low littoral along with other Rhodophyta and Alaria in the upper sublittoral. Exposed shores show a Verrucaria belt mainly above the tide, with Porphyra. The mid shore is dominated by barnacles, limpets and some Fucus, Himanthalia and some Rhodophyta such as Mastocarpus and Corallina are found in the low littorral with Himanthalia, Alaria and Laminaria digitata dominant in the upper sublittoral. Semi-exposed shores show a Verrucaria belt just above high tide with clear Pelvetia in the upper-littoral, limpets, barnacles and short Fucus vesiculosus midshore. Laminaria and Saccorhiza polyschides and small algae common in the sublittoral, sheltered shores show a narrow Verrucaria zone at high water and a full sequence of fucoids, Pelvetia, Fucus spiralis, Fucus vesiculosus, Fucus serratus, Ascophyllum nodosum. Laminaria digitata is dominant the upper sublittoral, very sheltered shores show a very narrow zone of Verrucaria, the dominance of the littoral by a full sequence of the fucoids and Ascophyllum covering the rocks. Laminaria saccharina, Halidrys, Chondrus and or Furcellaria, a Biologically-defined Exposure Scale for the Comparative Description of Rocky Shores
23.
Modulational instability
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The phenomenon was first discovered − and modelled − for periodic surface gravity waves on deep water by T. Brooke Benjamin and Jim E. Feir, in 1967. Therefore, it is known as the Benjamin−Feir instability. It is a mechanism for the generation of rogue waves. Modulation instability only happens under certain circumstances, the most important condition is anomalous group velocity dispersion, whereby pulses with shorter wavelengths travel with higher group velocity than pulses with longer wavelength. The instability is strongly dependent on the frequency of the perturbation, at certain frequencies, a perturbation will have little effect, whilst at other frequencies, a perturbation will grow exponentially. The overall gain spectrum can be derived analytically, as is shown below, random perturbations will generally contain a broad range of frequency components, and so will cause the generation of spectral sidebands which reflect the underlying gain spectrum. The tendency of a signal to grow makes modulation instability a form of amplification. By tuning an input signal to a peak of the gain spectrum, the imaginary unit i satisfies i 2 = −1. The model includes group velocity dispersion described by the parameter β2, a periodic waveform of constant power P is assumed. The beginning of instability can be investigated by perturbing this solution as A = e i γ P z, the complex conjugate of ε is denoted as ε ∗. Instability can now be discovered by searching for solutions of the equation which grow exponentially. The nonlinear Schrödinger equation is constructed by removing the carrier wave of the light being modelled, therefore, ω m and k m dont represent absolute frequencies and wavenumbers, but the difference between these and those of the initial beam of light. It can be shown that the function is valid, provided c 2 = c 1 ∗. Therefore, instability will occur when β22 ω m 2 +2 γ P β2 <0 and this condition describes the requirement for anomalous dispersion. The gain spectrum can be described by defining a gain parameter as g ≡2 | ℑ |, the growth rate is maximum for ω2 = − γ P / β2. Modulation instability of optical fields has been observed in systems, namely. Modulation instability occurs owing to inherent optical nonlinearity of the due to photoreaction-induced changes in the refractive index. Ostrovsky, L. A. Modulation instability, The beginning
24.
Boussinesq approximation (water waves)
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In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation, the 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations. The Boussinesq approximation for water waves takes into account the structure of the horizontal and vertical flow velocity. This results in partial differential equations, called Boussinesq-type equations. In coastal engineering, Boussinesq-type equations are used in computer models for the simulation of water waves in shallow seas. While the Boussinesq approximation is applicable to fairly long waves – that is and this is useful because the waves propagate in the horizontal plane and have a different behaviour in the vertical direction. Often, as in Boussinesqs case, the interest is primarily in the wave propagation and this elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave. Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations, thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the partial differential equations are in terms of functions of the horizontal coordinates. As an example, consider potential flow over a bed in the plane, with x the horizontal. The bed is located at z = −h, where h is the water depth. Invoking Laplaces equation for φ, as valid for incompressible flow, gives, φ = + = and this series may subsequently be truncated to a finite number of terms. Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions, further, in the resulting equations only the linear and quadratic terms with respect to η and ub are retained. The cubic and higher terms are assumed to be negligible. This set of equations has been derived for a horizontal bed. When the right-hand sides of the equations are set to zero. From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number, for the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion. The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation
25.
Breaking wave
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At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behaviour. The most generally familiar sort of breaking wave is the breaking of surface waves on a coastline. Wave breaking generally occurs where the amplitude reaches the point that the crest of the wave actually overturns—though the types of breaking water waves are discussed in more detail below. Certain other effects in fluid dynamics have also been termed breaking waves, wave breaking also occurs in plasmas, when the particle velocities exceed the waves phase speed. Breaking of water surface waves may occur anywhere that the amplitude is sufficient, however, it is particularly common on beaches because wave heights are amplified in the region of shallower water. See also waves and shallow water, there are four basic types of breaking water waves. They are spilling, plunging, collapsing, and surging, when the ocean floor has a gradual slope, the wave will steepen until the crest becomes unstable, resulting in turbulent whitewater spilling down the face of the wave. This continues as the approaches the shore, and the waves energy is slowly dissipated in the whitewater. Because of this, spilling waves break for a time than other waves. Onshore wind conditions make spillers more likely, a plunging wave occurs when the ocean floor is steep or has sudden depth changes, such as from a reef or sandbar. A plunging wave breaks with more energy than a significantly larger spilling wave, the wave can trap and compress the air under the lip, which creates the crashing sound associated with waves. With large waves, this crash can be felt by beachgoers on land, offshore wind conditions can make plungers more likely. This is the tube that is so highly sought after by surfers, the surfer tries to stay near or under the crashing lip, often trying to stay as deep in the tube as possible while still being able to shoot forward and exit the barrel before it closes. A plunging wave that is parallel to the beach can break along its length at once, rendering it unrideable. Surfers refer to waves as closed out. Collapsing waves are a cross between plunging and surging, in which the crest never fully breaks, yet the bottom face of the wave gets steeper and collapses, surging breakers originate from long period, low steepness waves and/or steep beach profiles. The outcome is the movement of the base of the wave up the swash slope. The front face and crest of the wave remain relatively smooth with little foam or bubbles, resulting in a very narrow surf zone, or no breaking waves at all
26.
Clapotis
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The resulting clapotic wave does not travel horizontally, but has a fixed pattern of nodes and antinodes. These waves promote erosion at the toe of the wall, the term was coined in 1877 by French mathematician and physicist Joseph Valentin Boussinesq who called these waves ‘le clapotis’ meaning ‘’the lapping. The standing waves alternately rise and fall in a mirror image pattern, as energy is converted to potential energy. This may also occur at sea between two different wave trains of near equal wavelength moving in opposite directions, but with unequal amplitudes, in partial clapotis the wave envelope contains some vertical motion at the nodes. When a wave strikes a wall at an oblique angle. In this situation, the individual crests formed at the intersection of the incident and this wave motion, when combined with the resultant vortices, can erode material from the seabed and transport it along the wall, undermining the structure until it fails. Clapotic waves on the sea surface also radiate infrasonic microbaroms into the atmosphere, clapotis has been called the bane and the pleasure of Sea kayaking. Rogue wave Boussinesq, J. Théorie des ondes liquides périodiques, mémoires présentés par divers savants à lAcadémie des Sciences. Boussinesq, J. Essai sur la théorie des eaux courantes, mémoires présentés par divers savants à lAcadémie des Sciences. Clapotis and Wave Reflection, With an Application to Vertical Breakwater Design, clapotis Wave Action – via YouTube
27.
Cnoidal wave
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In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn and they are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth. The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, in the limit of infinite wavelength, the cnoidal wave becomes a solitary wave. The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics. The KdV equation is a wave equation, including both frequency dispersion and amplitude dispersion effects. Shallow water equations — are also nonlinear and do have amplitude dispersion, Boussinesq equations — have the same range of validity as the KdV equation, but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are more difficult to solve than the KdV equation. Airy wave theory — has full frequency dispersion, so valid for arbitrary depth and wavelength, however, for long waves the Boussinesq approach—as also applied in the KdV equation—is often preferred. This is because in shallow water the Stokes perturbation series needs many terms before convergence towards the solution, due to the peaked crests, while the KdV or Boussinesq models give good approximations for these long nonlinear waves. The KdV equation can be derived from the Boussinesq equations, further improvements in short-wave performance can be obtained by starting to derive a one-way wave equation from a modern improved Boussinesq model, valid for even shorter wavelengths. The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, solitary wave solutions for nonlinear and dispersive long waves had been found earlier by Boussinesq in 1872, and Rayleigh in 1876. The search for these solutions was triggered by the observations of this solitary wave by Russell, cnoidal wave solutions of the KdV equation are stable with respect to small perturbations. Further cn is one of the Jacobi elliptic functions and K is the elliptic integral of the first kind. The latter, m, determines the shape of the cnoidal wave, for m equal to zero the cnoidal wave becomes a cosine function, while for values close to one the cnoidal wave gets peaked crests and flat troughs. For values of m less than 0.95, the function can be approximated with trigonometric functions. An important dimensionless parameter for nonlinear long waves is the Ursell parameter, for small values of U, say U <5, a linear theory can be used, and at higher values nonlinear theories have to be used, like cnoidal wave theory. The demarcation zone between—third or fifth order—Stokes and cnoidal wave theories is in the range 10–25 of the Ursell parameter. Based on the analysis of the nonlinear problem of surface gravity waves within potential flow theory
28.
Cross sea
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In surface navigation, a cross sea is a sea state with two wave systems traveling at oblique angles. This may occur when waves from one weather system continue despite a shift in wind. Waves generated by the new wind run at an angle to the old, creating a shifting, two weather systems that are far from each other may create a cross sea when the waves from the systems meet, usually at a place far from either weather system. Until the older waves have dissipated, they create a sea hazard among the most perilous and this sea state is fairly common and a larger percentage of ship accidents were found to have occurred in this state. A cross swell is generated when the systems are longer period swell
29.
Equatorial wave
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Equatorial waves are ocean waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earths rotation and its shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Equatorial waves may be separated into a series of subclasses depending on their fundamental dynamics, at shortest periods are the equatorial gravity waves while the longest periods are associated with the equatorial Rossby waves. In addition to these two subclasses, there are two special subclasses of equatorial waves known as the mixed Rossby-gravity wave and the equatorial Kelvin wave. The latter two share the characteristics that they can have any period and also that they may carry only in an eastward direction. The remainder of this article discusses the relationship between the period of waves, their wavelength in the zonal direction and their speeds for a simplified ocean. Rossby-gravity waves, first observed in the stratosphere by M. Yanai, but, oddly, their crests and troughs may propagate westward if their periods are long enough. The eastward speed of propagation of waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter vanishes at 0 degrees latitude, the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and β is the variation of the coriolis parameter with latitude, ∂ f ∂ y = β. Once the frequency relation is formulated in terms of ω, the angular frequency and these three solutions correspond to the equatorial gravity waves, the equatorially trapped Rossby waves and the mixed Rossby-gravity wave. Equatorial gravity waves can be either westward- or eastward-propagating, the governing equations for these equatorial waves are similar to those presented above, except that there is no meridional velocity component. The continuity equation, ∂ ϕ ∂ t + c 2 ∂ u ∂ x =0 the u-momentum equation, ∂ u ∂ t = − ∂ ϕ ∂ x the v-momentum equation, u β y = − ∂ ϕ ∂ y. The solution to these equations yields the following phase speed, c2 = gH, also, these Kelvin waves only propagate towards the east. Kelvin waves have been connected to El Niño in recent years in terms of precursors to this atmospheric and oceanic phenomenon, the weak low pressure in the Indian Ocean typically propagates eastward into the North Pacific Ocean and can produce easterly winds. This wave can be observed at the surface by a rise in sea surface height of about 8 cm. If the Kelvin wave hits the South American coast, its water gets transferred upward
30.
Fetch (geography)
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The fetch, also called the fetch length, is the length of water over which a given wind has blown. It also plays a part in longshore drift as well. Fetch length, along with the speed, determines the size of waves produced. The wind direction is considered constant, the longer the fetch and the faster the wind speed, the more wind energy is imparted to the water surface and the larger the resulting sea state will be. Sea state Ocean surface wave Storm surge
31.
Gravity wave
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In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media when the force of gravity or buoyancy tries to restore equilibrium. An example of such an interface is that between the atmosphere and the ocean, which rise to wind waves. A gravity wave results when fluid is displaced from a position of equilibrium, the restoration of the fluid to equilibrium will produce a movement of the fluid back and forth, called a wave orbit. Gravity waves on an interface of the ocean are called surface gravity waves or surface waves. Wind-generated waves on the surface are examples of gravity waves, as are tsunamis. Wind-generated gravity waves on the surface of the Earths ponds, lakes, seas. Shorter waves are affected by surface tension and are called gravity–capillary waves. Alternatively, so-called infragravity waves, which are due to nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves. In the Earths atmosphere, gravity waves are a mechanism that produce the transfer of momentum from the troposphere to the stratosphere and mesosphere, Gravity waves are generated in the troposphere by frontal systems or by airflow over mountains. At first, waves propagate through the atmosphere without appreciable change in mean velocity, but as the waves reach more rarefied air at higher altitudes, their amplitude increases, and nonlinear effects cause the waves to break, transferring their momentum to the mean flow. This transfer of momentum is responsible for the forcing of the many large-scale dynamical features of the atmosphere, thus, this process plays a key role in the dynamics of the middle atmosphere. The effect of gravity waves in clouds can look like altostratus undulatus clouds, and are confused with them. The phase velocity c of a gravity wave with wavenumber k is given by the formula c = g k. When surface tension is important, this is modified to c = g k + σ k ρ, where σ is the surface tension coefficient and ρ is the density. Since c = ω / k is the speed in terms of the angular frequency ω and the wavenumber. The group velocity of a wave is given by c g = d ω d k, the group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive, Gravity waves traveling in shallow water, are nondispersive, the phase and group velocities are identical and independent of wavelength and frequency. When the water depth is h, c p = c g = g h, wind waves, as their name suggests, are generated by wind transferring energy from the atmosphere to the oceans surface, and capillary-gravity waves play an essential role in this effect
32.
Infragravity wave
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Infragravity waves are ocean surface gravity waves generated by ocean waves of shorter periods. The amplitude of infragravity waves is most relevant in shallow water, in particular along coastlines hit by high amplitude and long period wind waves, wind waves and ocean swells are shorter, with typical dominant periods of 1 to 25 s. This distinguishes infragravity waves from normal oceanic gravity waves, which are created by wind acting on the surface of the sea, whatever the details of their generation mechanism, discussed below, infragravity waves are these subharmonics of the impinging gravity waves. Technically infragravity waves are simply a subcategory of gravity waves and refer to all gravity waves with greater than 30 s. This could include such as tides and oceanic Rossby waves. The term infragravity wave appears to have coined by Walter Munk in 1950. Two main processes can explain the transfer of energy from the wind waves to the long infragravity waves. The most common process is the interaction of trains of wind waves which was first observed by Munk and Tucker and explained by Longuet-Higgins. Because wind waves are not monochromatic they form groups, the Stokes drift induced by these groupy waves transports more water where the waves are highest. The waves also push the water around in a way that can be interpreted as a force, combining mass and momentum conservation, Longuet-Higgins and Stewart give, with three different methods, the now well-known result. Namely, the sea level oscillates with a wavelength that is equal to the length of the group, with a low level where the wind waves are highest. This oscillation of the sea surface is proportional to the square of the wave amplitude. Another process was proposed later by Graham Symonds and his collaborators and it appears that this is probably a good explanation for infragravity wave generation on a reef. In the case of coral reefs, the infragravity periods are established by resonances with the reef itself, infragravity waves generated along the Pacific coast of North America have been observed to propagate transoceanically to Antarctica and there to impinge on the Ross Ice Shelf. Their frequencies more closely couple with the ice shelf natural frequencies, further, they are not damped by sea ice as normal ocean swell is. As a result, they flex floating ice shelves such as the Ross Ice Shelf, media related to Gravity waves at Wikimedia Commons
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Internal wave
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Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified, the density must decrease continuously or discontinuously with depth/height due to changes, for example, If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid. Internal waves, also called gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude. If propagating horizontally along an interface where the density decreases with height. If the interfacial waves are large amplitude they are called solitary waves or internal solitons. If moving vertically through the atmosphere where substantial changes in air density influences their dynamics, If generated by flow over topography, they are called Lee waves or mountain waves. If the mountain waves break aloft, they can result in strong winds at the ground known as Chinook winds or Foehn winds. If generated in the ocean by tidal flow over submarine ridges or the continental shelf, If they evolve slowly compared to the Earths rotational frequency so that their dynamics are influenced by the Coriolis effect, they are called inertia gravity waves or, simply, inertial waves. Internal waves are usually distinguished from Rossby waves, which are influenced by the change of Coriolis frequency with latitude. An internal wave can readily be observed in the kitchen by slowly tilting back, clouds that reveal internal waves launched by flow over hills are called lenticular clouds because of their lens-like appearance. Less dramatically, a train of waves can be visualized by rippled cloud patterns described as herringbone sky or mackerel sky. The outflow of air from a thunderstorm can launch large amplitude internal solitary waves at an atmospheric inversion. In northern Australia, these result in Morning Glory clouds, used by some daredevils to glide along like a surfer riding an ocean wave, satellites over Australia and elsewhere reveal these waves can span many hundreds of kilometers. According to Archimedes principle, the weight of an object is reduced by the weight of fluid it displaces. This holds for a parcel of density ρ surrounded by an ambient fluid of density ρ0. Its weight per volume is g, in which g is the acceleration of gravity. Dividing by a density, ρ00, gives the definition of the reduced gravity. Because water is more dense than air, the displacement of water by air from a surface gravity wave feels nearly the full force of gravity
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Kinematic wave
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These waves are also applied to model the motion of highway traffic flows. In these flows, mass and momentum equations can be combined to yield a kinematic wave equation, depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave celerity is a constant or a variable. In general, the wave can be advecting and diffusing, however, in simple situation, the kinematic wave is mainly advecting. For F = h 2 /2, this reduces to the Burgers equation
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Longshore drift
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Longshore drift is a geological process that consists of the transportation of sediments along a coast parallel to the shoreline, which is dependent on oblique incoming wind direction. Oblique incoming wind squeezes water along the coast, and so generates a current which moves parallel to the coast. Longshore drift is simply the sediment moved by the longshore current and this current and sediment movement occurs within the surf zone. Beach sand is moved on such oblique wind days, due to the swash and backwash of water on the beach. Breaking surf sends water up the beach at an oblique angle, thus beach sand can move downbeach in a zig zag fashion many tens of meters per day. This process is called beach drift but some regard it as simply part of longshore drift because of the overall movement of sand parallel to the coast. Longshore drift affects numerous sediment sizes as it works in different ways depending on the sediment. Sand is largely affected by the force of breaking waves. There are numerous calculations that take into consideration the factors that produce longshore drift, some of these are, Geological changes, e. g. erosion, backshore changes and emergence of headlands. Change in hydrodynamic forces, e. g. change in wave diffraction in headland, change to hydrodynamic influences, e. g. the influence of new tidal inlets and deltas on drift. Alterations of the sediment budget, e. g. switch of shorelines from drift to swash alignment, the intervention of humans, e. g. cliff protection, groynes, detached breakwaters. The sediment budget takes into consideration sediment sources and sinks within a system, a good example of the sediment budget and longshore drift working together in the coastal system is inlet ebb-tidal shoals, which store sand that has been transported by long shore transport. As well as storing sand these systems may also transfer or by pass sand into other systems, therefore inlet ebb-tidal systems provide a good sources. Long shore occurs in a 90 to 80 degree backwash so it would be presented as an angle with the wave line. This section consists of features of long shore drift that occur on a coast where long shore drift occurs uninterrupted by man-made structures, spits are formed when longshore drift travels past a point where the dominant drift direction and shoreline do not veer in the same direction. As well as dominant drift direction, spits are affected by the strength of wave driven current, wave angle, spits are landforms that have two important features, with the first feature being the region at the up-drift end or proximal end. The proximal end is attached to land and may form a slight “barrier” between the sea and an estuary or lagoon. As an example, the New Brighton spit in Canterbury, New Zealand, was created by longshore drift of sediment from the Waimakariri River to the north and this spit system is currently in equilibrium but undergoes phases of deposition and erosion
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Luke's variational principle
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In fluid dynamics, Lukes variational principle is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J. C. Luke, who published it in 1967, Lukes Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface. This is often used when modelling the spectral density evolution of the free-surface in a sea state, both the Lagrangian and Hamiltonian formulations can be extended to include surface tension effects, and by using Clebsch potentials to include vorticity. Lukes Lagrangian formulation is for surface gravity waves on an—incompressible. The Lagrangian L, as given by Luke, is, L = − ∫ t 0 t 1 d t, from Bernoullis principle, this Lagrangian can be seen to be the integral of the fluid pressure over the whole time-dependent fluid domain V. This is in agreement with the principles for inviscid flow without a free surface. This may also include moving wavemaker walls and ship motion. For the case of an unbounded domain with the free fluid surface at z=η. The bed-level term proportional to h2 in the energy has been neglected, since it is a constant. Below, Lukes variational principle is used to arrive at the equations for non-linear surface gravity waves on a potential flow. The variation δ L =0 in the Lagrangian with respect to variations in the velocity potential Φ, consider a small variation δΦ in the velocity potential Φ. Then the resulting variation in the Lagrangian is, δ Φ L = L − L = − ∫ t 0 t 1 ∬ d x d t. The first integral on the right-hand side integrates out to the boundaries, in x and t, of the integration domain and is zero since the variations δΦ are taken to be zero at these boundaries. Similarly, variations δΦ only non-zero at the bottom z = -h result in the kinematic bed condition, ∇ Φ ⋅ ∇ h + ∂ Φ ∂ z =0 at z = − h. Considering the variation of the Lagrangian with respect to small changes δη gives, the Hamiltonian structure of surface gravity waves on a potential flow was discovered by Vladimir E. The Hamiltonian H is the sum of the kinetic and potential energy of the fluid and this is expressed by the Dirichlet-to-Neumann operator D, acting linearly on φ. The Hamiltonian density can also be written as, H =12 ρ φ +12 ρ g η2, as a result, the Hamiltonian is a quadratic functional of the surface potential φ. Also the potential energy part of the Hamiltonian is quadratic, the source of non-linearity in surface gravity waves is through the kinetic energy depending non-linear on the free surface shape η
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Mild-slope equation
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It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is used in coastal engineering to compute the wave-field changes near harbours. As a result, it describes the variations in wave amplitude, from the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. Most often, the equation is solved by computer using methods from numerical analysis. Also parabolic approximations to the equation are often used, in order to reduce the computational cost. In case of a constant depth, the equation reduces to the Helmholtz equation for wave diffraction. For a given angular frequency ω, the k has to be solved from the dispersion equation. The last equation shows that energy is conserved in the mild-slope equation. The effective group speed | v g | is different from the speed c g. The first equation states that the effective wavenumber κ is irrotational, a consequence of the fact it is the derivative of the wave phase θ. The second equation is the eikonal equation, otherwise, κ2 can even become negative. When the diffraction effects are neglected, the effective wavenumber κ is equal to k. The mild-slope equation can be derived by the use of several methods, here, we will use a variational approach. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational and these assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers. Because the flow is irrotational, the motion can be described using potential flow theory. The time-dependent mild-slope equation can be used to model waves in a band of frequencies around ω0. Consider monochromatic waves with complex amplitude η and angular frequency ω, ζ = ℜ, with ω and ω0 chosen equal to each other, ω = ω0. Using this in the time-dependent form of the equation, recovers the classical mild-slope equation for time-harmonic wave motion
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Radiation stress
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The radiation stresses behave as a second-order tensor. The radiation stress tensor describes the additional forcing due to the presence of the waves, as a result, varying radiation stresses induce changes in the mean surface elevation and the mean flow. For the mean energy density in the part of the fluid motion. Radiation stress derives its name from the effect of radiation pressure for electromagnetic radiation. As a result, a long wave propagates together with the group. While, according to the relation, a long wave of this length should propagate at its own – higher – phase velocity. For uni-directional wave propagation – say in the x-coordinate direction – the component of the stress tensor of dynamical importance is Sxx. Further ρ is the density and g is the acceleration by gravity. The last term on the side, ½ρg2, is the integral of the hydrostatic pressure over the still-water depth. Further E is the mean depth-integrated wave energy density per unit of horizontal area, note this equation is for periodic waves, in random waves the root-mean-square wave height Hrms should be used with Hrms = Hm0 / √2, where Hm0 is the significant wave height. For wave propagation in two dimensions the radiation stress S is a second-order tensor with components, S =. The phase and group speeds, cp and cg respectively, are the lengths of the phase and group velocity vectors, cp = |cp|, the radiation stress tensor is an important quantity in the description of the phase-averaged dynamical interaction between waves and mean flows. Propagating waves induce a – relatively small – mean mass transport in the propagation direction. To lowest order, the wave momentum Mw is, per unit of area, M w = k k E c p. The difference k⋅v is the Doppler shift, the mean horizontal momentum M is also the mean of the depth-integrated horizontal mass flux, and consists of two contributions, one by the mean current and the other is due to the waves. Now the mass transport velocity u is defined as, u ¯ = M ρ = v ¯ + M w ρ, observe that first the depth-integrated horizontal momentum is averaged, before the division by the mean water depth is made. The equation of mass conservation is, in vector notation, ∂ ∂ t + ∇ ⋅ =0. Further I is the identity tensor, with components given by the Kronecker delta δij, note that the right hand side of the momentum equation provides the non-conservative contributions of the bed slope ∇h, as well the forcing by the wind and the bed friction