The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Korteweg–de Vries equation is an approximation to the Boussinesq equations for the case of one-way wave propagation, cnoidal waves are approximate solutions to the Boussinesq equations.
Korteweg–de Vries, and Benjamin–Bona–Mahony equations
Validity of several theories for periodic water waves, according to Le Méhauté (1976). The light-blue area gives the range of validity of cnoidal wave theory; light-yellow for Airy wave theory; and the dashed blue lines demarcate between the required order in Stokes' wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves (H > ¼ Hbreaking).
The Korteweg–de Vries equation (KdV equation) can be used to describe the uni-directional propagation of weakly nonlinear and long waves—where long wave means: having long wavelengths as compared with the mean water depth—of surface gravity waves on a fluid layer. The KdV equation is a dispersive wave equation, including both frequency dispersion and amplitude dispersion effects. In its classical use, the KdV equation is applicable for wavelengths λ in excess of about five times the average water depth h, so for λ > 5 h; and for the periodτ greater than with g the strength of the gravitational acceleration. To envisage the position of the KdV equation within the scope of classical wave approximations, it distinguishes itself in the following ways:
Korteweg–de Vries equation — describes the forward propagation of weakly nonlinear and dispersive waves, for long waves with λ > 7 h.
Shallow water equations — are also nonlinear and do have amplitude dispersion, but no frequency dispersion; they are valid for very long waves, λ > 20 h.
Boussinesq equations — have the same range of validity as the KdV equation (in their classical form), but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are often more difficult to solve than the KdV equation; and in many applications wave reflections are small and may be neglected.
Airy wave theory — has full frequency dispersion, so valid for arbitrary depth and wavelength, but is a linear theory without amplitude dispersion, limited to low-amplitude waves.
Stokes' wave theory — a perturbation-series approach to the description of weakly nonlinear and dispersive waves, especially successful in deeper water for relative short wavelengths, as compared to the water depth. 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 and long flat troughs of the nonlinear waves. While the KdV or Boussinesq models give good approximations for these long nonlinear waves.
The KdV equation can be derived from the Boussinesq equations, but additional assumptions are needed to be able to split off the forward wave propagation. For practical applications, the Benjamin–Bona–Mahony equation (BBM equation) is preferable over the KdV equation, a forward-propagating model similar to KdV but with much better frequency-dispersion behaviour at shorter wavelengths. 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.
Cnoidal wave profiles for three values of the elliptic parameter m.
: m = 0,
: m = 0.9 and
: m = 0.99999.
The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, which article is based on the PhD thesis by de Vries in 1894. 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 (or "wave of translation") by Russell, both in nature and in laboratory experiments. Cnoidal wave solutions of the KdV equation are stable with respect to small perturbations.
The surface elevation η(x,t), as a function of horizontal position x and time t, for a cnoidal wave is given by:
An important dimensionless parameter for nonlinear long waves (λ ≫ h) 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. As can be seen from the formula for the Ursell parameter, for a given relative wave height H/h the Ursell parameter—and thus also the nonlinearity—grows quickly with increasing relative wavelength λ/h.
Based on the analysis of the full nonlinear problem of surface gravity waves within potential flow theory, the above cnoidal waves can be considered the lowest-order term in a perturbation series. Higher-order cnoidal wave theories remain valid for shorter and more nonlinear waves. A fifth-order cnoidal wave theory was developed by Fenton in 1979. A detailed description and comparison of fifth-order Stokes' and fifth-order cnoidal wave theories is given in the review article by Fenton.
Cnoidal wave descriptions, through a renormalisation, are also well suited to waves on deep water, even infinite water depth; as found by Clamond. A description of the interactions of cnoidal waves in shallow water, as found in real seas, has been provided by Osborne in 1994.
with s another integration constant. This is written in the form
The cubic polynomial f(η) becomes negative for large positive values of η, and positive for large negative values of η. Since the surface elevation η is real valued, also the integration constants r and s are real. The polynomial f can be expressed in terms of its rootsη1, η2 and η3:
Because f(η) is real valued, the three roots η1, η2 and η3 are either all three real, or otherwise one is real and the remaining two are a pair of complex conjugates. In the latter case, with only one real-valued root, there is only one elevation η at which f(η) is zero. And consequently also only one elevation at which the surface slopeη’ is zero. However, we are looking for wave like solutions, with two elevations—the wave crest and trough (physics)—where the surface slope is zero. The conclusion is that all three roots of f(η) have to be real valued.
Without loss of generality, it is assumed that the three real roots are ordered as:
Solution of the first-order ordinary-differential equation
Now, from equation (A) it can be seen that only real values for the slope exist if f(η) is positive. This corresponds with η2 ≤ η≤ η1, which therefore is the range between which the surface elevation oscillates, see also the graph of f(η). This condition is satisfied with the following representation of the elevation η(ξ):
in agreement with the periodic character of the sought wave solutions and with ψ(ξ) the phase of the trigonometric functions sin and cos. From this form, the following descriptions of various terms in equations (A) and (B) can be obtained:
Using these in equations (A) and (B), the following ordinary differential equation relating ψ and ξ is obtained, after some manipulations:
with the right hand side still positive, since η1 − η3 ≥ η1 − η2. Without loss of generality, we can assume that ψ(ξ) is a monotone function, since f(η) has no zeros in the interval η2 < η < η1. So the above ordinary differential equation can also be solved in terms of ξ(ψ) being a function of ψ:
where m is the so-called elliptic parameter, satisfying 0 ≤ m ≤ 1 (because η3 ≤ η2 ≤ η1).
If ξ = 0 is chosen at the wave crest η(0) = η1 integration gives
Fourth, from equations (A) and (B) a relationship can be established between the phase speedc and the roots η1, η2 and η3:
The relative phase-speed changes are depicted in the figure below. As can be seen, for m > 0.96 (so for 1 − m < 0.04) the phase speed increases with increasing wave height H. This corresponds with the longer and more nonlinear waves. The nonlinear change in the phase speed, for fixed m, is proportional to the wave height H. Note that the phase speed c is related to the wavelength λ and periodτ as:
Relative phase speed increase of cnoidal wave solutions for the Korteweg–de Vries equation as a function of 1−m, with m the elliptic parameter. The horizontal axis is on a logarithmic scale, from 10−6 to 100=1. The figure is for non-dimensional quantities, i.e. the phase speed c is made dimensionless with the shallow-water phase speed , and the wave height H is made dimensionless with the mean water depth h.
The cnoidal-wave solution of the KdV equation is:
Most often, the known wave parameters are the wave height H, mean water depth h, gravitational acceleration g, and either the wavelength λ or else the period τ. Then the above relations for λ, c and τ are used to find the elliptic parameter m. This requires numerical solution by some iterative method.
All quantities have the same meaning as for the KdV equation. The BBM equation is often preferred over the KdV equation because it has a better short-wave behaviour.
Details of the derivation
The derivation is analogous to the one for the KdV equation. The dimensionless BBM equation is, non-dimensionalised using mean water depth h and gravitational acceleration g:
This can be brought into the standard form
through the transformation:
but this standard form will not be used here.
Analogue to the drivation of the cnoidal wave solution for the KdV equation, periodic wave solutions η(ξ), with ξ = x−ct are considered Then the BBM equation becomes a third-order ordinary differential equation, which can be integrated twice, to obtain:
Which only differs from the equation for the KdV equation through the factor c in front of (η′)2 in the left hand side. Through a coordinate transformation β = ξ / the factor c may be removed, resulting in the same first-order ordinary differential equation for both the KdV and BBM equation. However, here the form given in the preceding equation is used. This results in a different formulation for Δ as found for the KdV equation:
The relation of the wavelength λ, as a function of H and m, is affected by this change in
For the rest, the derivation is analogous to the one for the KdV equation, and will not be repeated here.
The results are presented in dimensional form, for water waves on a fluid layer of depth h.
The cnoidal wave solution of the BBM equation, together with the associated relationships for the parameters is:
Parameter relations for cnoidal wave solutions of the Korteweg–de Vries equation. Shown is −log10 (1−m), with m the elliptic parameter of the complete elliptic integrals, as a function of dimensionless periodτ√g/h and relative wave heightH / h. The values along the contour lines are −log10 (1−m), so a value 1 corresponds with m = 1 − 10−1 = 0.9 and a value 40 with m = 1 − 10−40.
In this example, a cnoidal wave according to the Korteweg–de Vries (KdV) equation is considered. The following parameters of the wave are given:
Instead of the period τ, in other cases the wavelengthλ may occur as a quantity known beforehand.
First, the dimensionless period is computed:
which is larger than seven, so long enough for cnoidal theory to be valid. The main unknown is the elliptic parameter m. This has to be determined in such a way that the wave period τ, as computed from cnoidal wave theory for the KdV equation:
is consistent with the given value of τ; here λ is the wavelength and c is the phase speed of the wave. Further, K(m) and E(m) are complete elliptic integrals of the first and second kind, respectively. Searching for the elliptic parameter m can be done by trial and error, or by use of a numerical root-finding algorithm. In this case, starting from an initial guess minit = 0.99, by trial and error the answer
is found. Within the process, the wavelength λ and phase speed c have been computed:
showing a 3.8% increase due to the effect of nonlinear amplitudedispersion, which wins in this case from the reduction of phase speed by frequency dispersion.
Now the wavelength is known, the Ursell number can be computed as well:
which is not small, so linear wave theory is not applicable, but cnoidal wave theory is. Finally, the ratio of wavelength to depth is λ / h = 10.2 > 7, again indicating this wave is long enough to be considered as a cnoidal wave.
Further, for the same limit of m → 1, the complete elliptic integral of the first kind K(m) goes to infinity, while the complete elliptic integral of the second kind E(m) goes to one. This implies that the limiting values of the phase speed c and minimum elevelation η2 become:
Consequently, in terms of the width parameter Δ, the solitary wave solution to both the KdV and BBM equation is:
The width parameter, as found for the cnoidal waves and now in the limit m → 1, is different for the KdV and the BBM equation:
: KdV equation, and
: BBM equation.
But the phase speed of the solitary wave in both equations is the same, for a certain combination of height H and depth h.
For infinitesimal wave height the results of cnoidal wave theory are expected to converge towards those of Airy wave theory for the limit of long waves λ ≫ h. First the surface elevation, and thereafter the phase speed, of the cnoidal waves for infinitesimal wave height will be examined.
Then the hyperbolic-cosine terms, appearing in the Fourier series, can be expanded for small m ≪ 1 as follows:
with the nome q given by
The nome q has the following behaviour for small m:
Consequently, the amplitudes of the first terms in the Fourier series are:
So, for m ≪ 1 the Jacobi elliptic function has the first Fourier series terms:
And its square is
The free surface η(x,t) of the cnoidal wave will be expressed in its Fourier series, for small values of the elliptic parameter m. First, note that the argument of the cn function is ξ/Δ, and that the wavelength λ = 2 ΔK(m), so:
Further, the mean free-surface elevation is zero. Therefore, the surface elevation of small amplitude waves is
Also the wavelength λ can be expanded into a Maclaurin series of the elliptic parameter m, differently for the KdV and the BBM equation, but this is not necessary for the present purpose.
Note: The limiting behaviour for zero m—at infinitesimal wave height—can also be seen from:
but the higher-order term proportional to m in this approximation contains a secular term, due to the mismatch between the period of cn(z|m), which is 4 K(m), and the period 2π for the cosine cos(z). The above Fourier series for small m does not have this drawback, and is consistent with forms as found using the Lindstedt–Poincaré method in perturbation theory.
For infinitesimal wave height, in the limit m → 0, the free-surface elevation becomes:
So the wave amplitude is ½H, half the wave height. This is of the same form as studied in Airy wave theory, but note that cnoidal wave theory is only valid for long waves with their wavelength much longer than the average water depth.
The phase speed of a cnoidal wave, both for the KdV and BBM equation, is given by:
In this formulation the phase speed is a function of wave heightH and parameter m. However, for the determination of wave propagation for waves of infinitesimal height, it is necessary to determine the behaviour of the phase speed at constant wavelengthλ in the limit that the parameter m approaches zero. This can be done by using the equation for the wavelength, which is different for the KdV and BBM equation:
and using the above equations for the phase speed and wavelength, the factor H / m in the phase speed can be replaced by κh and m. The resulting phase speeds are:
The limiting behaviour for small m can be analysed through the use of the Maclaurin series for K(m) and E(m), resulting in the following expression for the common factor in both formulas for c:
so in the limit m → 0, the factor γ → −1⁄6. The limiting value of the phase speed for m ≪ 1 directly results.
The phase speeds for infinitesimal wave height, according to the cnoidal wave theories for the KdV equation and BBM equation, are
with κ = 2π / λ the wavenumber and κh the relative wavenumber. These phase speeds are in full agreement with the result obtained by directly searching for sine-wave solutions of the linearised KdV and BBM equations. As is evident from these equations, the linearised BBM equation has a positive phase speed for all κh. On the other hand, the phase speed of the linearised KdV equation changes sign for short waves with κh > . This is in conflict with the derivation of the KdV equation as a one-way wave equation.
Direct derivation from the full inviscid-flow equations
Undular bore and whelps near the mouth of Araguari River in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude.
Cnoidal waves can be derived directly from the inviscid, irrotational and incompressible flow equations, and expressed in terms of three invariants of the flow, as shown by Benjamin & Lighthill (1954) in their research on undular bores. In a frame of reference moving with the phase speed, in which reference frame the flow becomes a steady flow, the cnoidal wave solutions can directly be related to the mass flux, momentum flux and energy head of the flow. Following Benjamin & Lighthill (1954)—using a stream function description of this incompressible flow—the horizontal and vertical components of the flow velocity are the spatial derivatives of the stream function Ψ(ξ,z): +∂zΨ and −∂ξΨ, in the ξ and z direction respectively (ξ = x−ct). The vertical coordinate z is positive in the upward direction, opposite to the direction of the gravitational acceleration, and the zero level of z is at the impermeable lower boundary of the fluid domain. While the free surface is at z = ζ(ξ); note that ζ is the local water depth, related to the surface elevation η(ξ) as ζ = h + η with h the mean water depth.
In this steady flow, the dischargeQ through each vertical cross section is a constant independent of ξ, and because of the horizontal bed also the horizontal momentum flux S, divided by the densityρ, through each vertical cross section is conserved. Further, for this inviscid and irrotational flow, Bernoulli's principle can be applied and has the same Bernoulli constant R everywhere in the flow domain. They are defined as:
For fairly long waves, assuming the water depth ζ is small compared to the wavelength λ, the following relation is obtained between the water depth ζ(ξ) and the three invariants Q, R and S:
with ρ the fluid density, is one of the infinite number of invariants of the KdV equation. This can be seen by multiplying the KdV equation with the surface elevation η(x,t); after repeated use of the chain rule the result is:
which is in conservation form, and is an invariant after integration over the interval of periodicity—the wavelength for a cnoidal wave. The potential energy is not an invariant of the BBM equation, but ½ρg [η2 + 1⁄6h2 (∂xη)2] is.
First the variance of the surface elevation in a cnoidal wave is computed. Note that η2 = −(1/λ) 0∫λH cn2(ξ/Δ|m) dx, cn(ξ/Δ|m) = cos ψ(ξ) and λ = 2 ΔK(m), so
The potential energy, both for the KdV and the BBM equation, is subsequently found to be
The infinitesimal wave-height limit (m → 0) of the potential energy is Epot = 1⁄16ρgH2, which is in agreement with Airy wave theory. The wave height is twice the amplitude, H = 2a, in the infinitesimal wave limit.
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The United States Armed Forces are the federal armed forces of the United States. They consist of the Army, Marine Corps, Navy, Air Force, from the time of its inception, the military played a decisive role in the history of the United States. A sense of unity and identity was forged as a result of victory in the First Barbary War. Even so, the Founders were suspicious of a permanent military force and it played an important role in the American Civil War, where leading generals on both sides were picked from members of the United States military. Not until the outbreak of World War II did a standing army become officially established. The National Security Act of 1947, adopted following World War II and during the Cold Wars onset, the U. S. military is one of the largest militaries in terms of number of personnel. It draws its personnel from a pool of paid volunteers. As of 2016, the United States spends about $580.3 billion annually to fund its military forces, put together, the United States constitutes roughly 40 percent of the worlds military expenditures. For the period 2010–14, the Stockholm International Peace Research Institute found that the United States was the worlds largest exporter of major arms, the United States was also the worlds eighth largest importer of major weapons for the same period. The history of the U. S. military dates to 1775 and these forces demobilized in 1784 after the Treaty of Paris ended the War for Independence. All three services trace their origins to the founding of the Continental Army, the Continental Navy, the United States President is the U. S. militarys commander-in-chief. Rising tensions at various times with Britain and France and the ensuing Quasi-War and War of 1812 quickened the development of the U. S. Navy, the reserve branches formed a military strategic reserve during the Cold War, to be called into service in case of war. Time magazines Mark Thompson has suggested that with the War on Terror, Command over the armed forces is established in the United States Constitution. The sole power of command is vested in the President by Article II as Commander-in-Chief, the Constitution also allows for the creation of executive Departments headed principal officers whose opinion the President can require. This allowance in the Constitution formed the basis for creation of the Department of Defense in 1947 by the National Security Act, the Defense Department is headed by the Secretary of Defense, who is a civilian and member of the Cabinet. The Defense Secretary is second in the chain of command, just below the President. Together, the President and the Secretary of Defense comprise the National Command Authority, to coordinate military strategy with political affairs, the President has a National Security Council headed by the National Security Advisor. The collective body has only power to the President
A swell, in the context of an ocean, sea or lake, is a series of mechanical waves that propagate along the interface between water and air and so they are often referred to as surface gravity waves. These series of gravity waves are not generated by the immediate local wind, instead by distant weather systems. This is the definition of a swell as opposed to a locally generated wind wave. More generally, a swell consists of wind-generated waves that are not—or are hardly—affected by the wind at that time. Swell wavelength, also, varies from event to event, occasionally, swells which are longer than 700 m occur as a result of the most severe storms. Swell direction is the direction from which the swell is coming and it is measured in degrees, and often referred to in general directions, such as a NNW or SW swell. Large breakers one observes on a beach may result from distant weather systems over a fetch of ocean, further exposure to that specific wind could only cause a loss of energy due to the breaking of wave tops and formation of whitecaps. 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. Sea water wave is generated by many kinds of such as Seismic events, gravity. The generation of wave is initiated by the disturbances of cross wind field on the surface of the sea water. Two major mechanisms of surface wave formation by winds and other sources of wave formation can explain the generation of wind waves. 1) Starts from Fluctuations of wind, the wind wave formation on water surface by wind is started by a distribution of normal pressure acting on the water from the wind. By the mechanism developed by O. M and this pressure fluctuation arise normal and tangential stresses to the surface water, and generates wave behavior on the water surface. Since the wind profile Ua is logarithmic to the water surface and this relations show the wind flow transferring its kinetic energy to the water surface at their interface, and arises wave speed, c. For example, If we suppose a very flat sea surface, turbulent wind flows form random pressure fluctuations at the sea surface
Panama, officially called the Republic of Panama, is a country usually considered to be entirely in North America or Central America. It is bordered by Costa Rica to the west, Colombia to the southeast, the Caribbean Sea to the north, the capital and largest city is Panama City, whose metropolitan area is home to nearly half of the countrys 4.1 million people. Panama was inhabited by indigenous tribes prior to settlement by the Spanish in the 16th century. Panama broke away from Spain in 1821 and joined a union of Nueva Granada, Ecuador, when Gran Colombia dissolved in 1831, Panama and Nueva Granada remained joined, eventually becoming the Republic of Colombia. With the backing of the United States, Panama seceded from Colombia in 1903, in 1977 an agreement was signed for the total transfer of the Canal from the United States to Panama by the end of the 20th century, which culminated on 31 December 1999. Revenue from canal tolls continues to represent a significant portion of Panamas GDP, although commerce, banking, in 2015 Panama ranked 60th in the world in terms of the Human Development Index. Since 2010, Panama remains the second most competitive economy in Latin America, covering around 40 percent of its land area, Panamas jungles are home to an abundance of tropical plants and animals – some of them to be found nowhere else on the planet. There are several theories about the origin of the name Panama, some believe that the country was named after a commonly found species of tree. Others believe that the first settlers arrived in Panama in August, when butterflies abound, the best-known version is that a fishing village and its nearby beach bore the name Panamá, which meant an abundance of fish. Captain Antonio Tello de Guzmán, while exploring the Pacific side in 1515, in 1517 Don Gaspar De Espinosa, a Spanish lieutenant, decided to settle a post there. In 1519 Pedrarias Dávila decided to establish the Empires Pacific city in this site, the new settlement replaced Santa María La Antigua del Darién, which had lost its function within the Crowns global plan after the beginning of the Spanish exploitation of the riches in the Pacific. Blending all of the above together, Panamanians believe in general that the word Panama means abundance of fish and this is the official definition given in social studies textbooks approved by the Ministry of Education in Panama. However, others believe the word Panama comes from the Kuna word bannaba which means distant or far away, at the time of the arrival of the Spanish in the 16th century, the known inhabitants of Panama included the Cuevas and the Coclé tribes. These people have disappeared, as they had no immunity from European infectious diseases. The earliest discovered artifacts of indigenous peoples in Panama include Paleo-Indian projectile points, later central Panama was home to some of the first pottery-making in the Americas, for example the cultures at Monagrillo, which date back to 2500–1700 BC. These evolved into significant populations best known through their spectacular burials at the Monagrillo archaeological site, the monumental monolithic sculptures at the Barriles site are also important traces of these ancient isthmian cultures. Before Europeans arrived Panama was widely settled by Chibchan, Chocoan, the largest group were the Cueva. The size of the population of the isthmus at the time of European colonization is uncertain
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
In mathematics and physical sciences, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, physicists and mathematicians, nonlinear systems may appear chaotic, unpredictable or counterintuitive, contrasting with the much simpler linear systems. In other words, in a system of equations, the equation to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as non-linear, regardless of whether or not known linear functions appear in the equations. In particular, an equation is linear if it is linear in terms of the unknown function and its derivatives. As nonlinear equations are difficult to solve, nonlinear systems are approximated by linear equations. This works well up to some accuracy and some range for the input values and it follows that some aspects of the behavior of a nonlinear system appear commonly to be counterintuitive, unpredictable or even chaotic. Although such chaotic behavior may resemble random behavior, it is not random. For example, some aspects of the weather are seen to be chaotic and this nonlinearity is one of the reasons why accurate long-term forecasts are impossible with current technology. Some authors use the term nonlinear science for the study of nonlinear systems and this is disputed by others, Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. In mathematics, a function f is one which satisfies both of the following properties, Additivity or superposition, f = f + f, Homogeneity. Additivity implies homogeneity for any rational α, and, for continuous functions, for a complex α, homogeneity does not follow from additivity. For example, a map is additive but not homogeneous. The equation is called homogeneous if C =0, if f contains differentiation with respect to x, the result will be a differential equation. Nonlinear algebraic equations, which are also called polynomial equations, are defined by equating polynomials to zero, for example, x 2 + x −1 =0. For a single equation, root-finding algorithms can be used to find solutions to the equation. However, systems of equations are more complicated, their study is one motivation for the field of algebraic geometry. It is even difficult to decide whether a given system has complex solutions
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, any function which is not periodic is called aperiodic. A function f is said to be periodic with period P if we have f = f for all values of x in the domain, geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P and this definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane. A function that is not periodic is called aperiodic, for example, the sine function is periodic with period 2 π, since sin = sin x for all values of x. This function repeats on intervals of length 2 π, everyday examples are seen when the variable is time, for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position of the system are expressible as periodic functions, for a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a function is the function f that gives the fractional part of its argument. In particular, f = f = f =, =0.5 The graph of the function f is the sawtooth wave. The trigonometric functions sine and cosine are periodic functions, with period 2π. The subject of Fourier series investigates the idea that a periodic function is a sum of trigonometric functions with matching periods. According to the definition above, some functions, for example the Dirichlet function, are also periodic, in the case of Dirichlet function. For example, f = sin has period 2 π therefore sin will have period 2 π5, a function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions, if L is the period of the function then, L =2 π / k One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f = −f for all x, for example, the sine or cosine function is π-antiperiodic and 2π-periodic. A further generalization appears in the context of Bloch waves and Floquet theory, in this context, the solution is typically a function of the form, f = e i k P f where k is a real or complex number. Functions of this form are sometimes called Bloch-periodic in this context, a periodic function is the special case k =0, and an antiperiodic function is the special case k = π/P