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The **wave equation** is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

In mathematics, a **partial differential equation** (**PDE**) is a differential equation that contains unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

In physics, mathematics, and related fields, a **wave** is a disturbance of a field in which a physical attribute oscillates repeatedly at each point or propagates from each point to neighboring points, or seems to move through space.

**Classical physics** refers to theories of physics that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the realm of "classical physics".

- Introduction
- Wave equation in one space dimension
- Derivation of the wave equation
- General solution
- Scalar wave equation in three space dimensions
- Spherical waves
- Solution of a general initial-value problem
- Scalar wave equation in two space dimensions
- Scalar wave equation in general dimension and Kirchhoff's formulae
- Odd dimensions
- Even dimensions
- Problems with boundaries
- One space dimension
- Several space dimensions
- Inhomogeneous wave equation in one dimension
- Other coordinate systems
- Further generalizations
- Elastic waves
- Dispersion relation
- See also
- Notes
- References
- External links

Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.^{ [1] }^{ [2] }^{ [3] }^{ [4] } In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.^{ [5] }

A **musical instrument** is an instrument created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates to the beginnings of human culture. Early musical instruments may have been used for ritual, such as a trumpet to signal success on the hunt, or a drum in a religious ceremony. Cultures eventually developed composition and performance of melodies for entertainment. Musical instruments evolved in step with changing applications.

**Jean-Baptiste le Rond d'Alembert** was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was co-editor with Denis Diderot of the *Encyclopédie*. D'Alembert's formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d'Alembert's equation.

**Leonhard Euler** was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.

The wave equation is a partial differential equation that may constrain some scalar function *u* = *u* (*x*_{1}, *x*_{2}, …, *x _{n}*;

A **scalar** is an element of a field which is used to define a vector space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a vector.

**Pressure** is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the pressure relative to the ambient pressure.

A **displacement** is a vector whose length is the shortest distance from the initial to the final position of a point P. It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point. A displacement may be identified with the translation that maps the initial position to the final position.

where *c* is a fixed non-negative real coefficient.

Using the notations of Newtonian mechanics and vector calculus, the wave equation can be written more compactly as

**Vector calculus**, or **vector analysis**, is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

where denotes double time derivative, ∇ is the nabla operator, and ∇^{2} = ∇ · ∇ is the (spatial) Laplacian operator:

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇^{2}. The Laplacian ∇·∇*f*(*p*) of a function *f* at a point *p*, is the rate at which the average value of *f* over spheres centered at *p* deviates from *f*(*p*) as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

A solution of this equation can be quite complicated, but it can be analyzed as a linear combination of simple solutions that are sinusoidal plane waves with various directions of propagation and wavelengths but all with the same propagation speed *c*. This analysis is possible because the wave equation is linear; so that any multiple of a solution is also a solution, and the sum of any two solutions is again a solution. This property is called the superposition principle in physics.

In physics, a **plane wave** is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.

In mathematics, a **linear equation** is an equation that may be put in the form

The **superposition principle**, also known as **superposition property**, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input *A* produces response *X* and input *B* produces response *Y* then input produces response.

The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments.

The wave equation is the simplest example of a hyperbolic differential equation. It, and its modifications, play fundamental roles in continuum mechanics, quantum mechanics, plasma physics, general relativity, geophysics, and many other scientific and technical disciplines.

The wave equation in one space dimension can be written as follows:

- .

This equation is typically described as having only one space dimension *x*, because the only other independent variable is the time *t*. Nevertheless, the dependent variable *u* may represent a second space dimension, if, for example, the displacement *u* takes place in y-direction, as in the case of a string that is located in the x–y plane.

The wave equation in one space dimension can be derived in a variety of different physical settings. Most famously, it can be derived for the case of a string that is vibrating in a two-dimensional plane, with each of its elements being pulled in opposite directions by the force of tension.^{ [6] }

Another physical setting for derivation of the wave equation in one space dimension utilizes Hooke's Law. In the theory of elasticity, Hooke's Law is an approximation for certain materials, stating that the amount by which a material body is deformed (the strain) is linearly related to the force causing the deformation (the stress).

The wave equation in the one-dimensional case can be derived from Hooke's Law in the following way: Imagine an array of little weights of mass *m* interconnected with massless springs of length *h*. The springs have a spring constant of *k*:

Here the dependent variable *u*(*x*) measures the distance from the equilibrium of the mass situated at *x*, so that *u(x)* essentially measures the magnitude of a disturbance (i.e. strain) that is traveling in an elastic material. The forces exerted on the mass *m* at the location *x* + *h* are:

The equation of motion for the weight at the location *x* + *h* is given by equating these two forces:

where the time-dependence of *u*(*x*) has been made explicit.

If the array of weights consists of *N* weights spaced evenly over the length *L* = *Nh* of total mass *M* = *Nm*, and the total spring constant of the array *K* = *k*/*N* we can write the above equation as:

Taking the limit *N* → ∞, *h* → 0 and assuming smoothness one gets:

which is from the definition of a second derivative. *KL*^{2}/*M* is the square of the propagation speed in this particular case.

In the case of a stress pulse propagating through a beam the beam acts much like an infinite number of springs in series and can be taken as an extension of the equation derived for Hooke's law. A beam of constant cross-section made from a linear elastic material has a stiffness *K* given by

Where *A* is the cross-sectional area and *E* is the Young's modulus of the material. The wave equation becomes

*AL* is equal to the volume of the beam and therefore

where is the density of the material. The wave equation reduces to

The speed of a stress wave in a bar is therefore .

The one-dimensional wave equation is unusual for a partial differential equation in that a relatively simple general solution may be found. Defining new variables:^{ [7] }

changes the wave equation into

which leads to the general solution

or equivalently:

In other words, solutions of the 1D wave equation are sums of a right traveling function *F* and a left traveling function *G*. "Traveling" means that the shape of these individual arbitrary functions with respect to *x* stays constant, however the functions are translated left and right with time at the speed *c*. This was derived by Jean le Rond d'Alembert.^{ [8] }

Another way to arrive at this result is to note that the wave equation may be "factored":

As a result, if we define *v* thus,

then

From this, *v* must have the form *G*(*x* + *ct*), and from this the correct form of the full solution *u* can be deduced.^{ [9] }

For an initial value problem, the arbitrary functions *F* and *G* can be determined to satisfy initial conditions:

The result is d'Alembert's formula:

In the classical sense if *f*(*x*) ∈ *C ^{k}* and

The basic wave equation is a linear differential equation and so it will adhere to the superposition principle. This means that the net displacement caused by two or more waves is the sum of the displacements which would have been caused by each wave individually. In addition, the behavior of a wave can be analyzed by breaking up the wave into components, e.g. the Fourier transform breaks up a wave into sinusoidal components.

Another way to solve for the solutions to the one-dimensional wave equation is to first analyze its frequency eigenmodes. A so-called eigenmode is a solution that oscillates in time with a well-defined *constant* angular frequency , with which the temporal part of the wave function for such eigenmode takes a specific form . The rest of the wave function is then only dependent on the spatial variable , hence amounting to separation of variables. Now writing the wave function as

we can obtain an ordinary differential equation for the spatial part

Therefore:

which is precisely an eigenvalue equation for , hence the name eigenmode. It has the well-known plane wave solutions

with wave number .

The total wave function for this eigenmode is then the linear combination

where complex numbers depend in general on any initial and boundary conditions of the problem.

Eigenmodes are useful in constructing a full solution to the wave equation, because each of them evolves in time trivially with the phase factor . so that a full solution can be decomposed into an eigenmode expansion

or in terms of the plane waves,

which is exactly in the same form as in the algebraic approach. Functions are known as the Fourier component and are determined by initial and boundary conditions. This is a so-called frequency-domain method, alternative to direct time-domain propagations, such as FDTD method, of the wave packet , which is complete for representing waves in absence of time dilations. Completeness of the Fourier expansion for representing waves in the presence of time dilations has been challenged by chirp wave solutions allowing for time variation of .^{ [10] } The chirp wave solutions seem particularly implied by very large but previously inexplicable radar residuals in the flyby anomaly, and differ from the sinusoidal solutions in being receivable at any distance only at proportionally shifted frequencies and time dilations, corresponding to past chirp states of the source.

A solution of the initial-value problem for the wave equation in three space dimensions can be obtained from the corresponding solution for a spherical wave. The result can then be also used to obtain the same solution in two space dimensions.

The wave equation can be solved using the technique of separation of variables. To obtain a solution with constant frequencies, let us first Fourier transform the wave equation in time as

So we get,

This is the Helmholtz equation and can be solved using separation of variables. If spherical coordinates are used to describe a problem, then the solution to the angular part of the Helmholtz equation is given by spherical harmonics and the radial equation now becomes ^{ [11] }

Here and the complete solution is now given by

where and are the spherical Hankel functions. To gain a better understanding of the nature of these spherical waves, let us go back and look at the case when . In this case, there is no angular dependence and the amplitude depends only on the radial distance i.e. . In this case, the wave equation reduces to

This equation can be rewritten as

where the quantity satisfies the one-dimensional wave equation. Therefore, there are solutions in the form

where *F* and *G* are general solutions to the one-dimensional wave equation, and can be interpreted as respectively an outgoing or incoming spherical wave. Such waves are generated by a point source, and they make possible sharp signals whose form is altered only by a decrease in amplitude as *r* increases (see an illustration of a spherical wave on the top right). Such waves exist only in cases of space with odd dimensions.^{[ citation needed ]}

For physical examples of non-spherical wave solutions to the 3D wave equation that do possess angular dependence, see dipole radiation.

Although the word "monochromatic" is not exactly accurate since it refers to light or electromagnetic radiation with well-defined frequency, the spirit is to discover the eigenmode of the wave equation in three dimensions. Following the derivation in the previous section on Plane wave eigenmodes, if we again restrict our solutions to spherical waves that oscillate in time with well-defined *constant* angular frequency , then the transformed function has simply plane wave solutions,

- ,

or

- .

From this we can observe that the peak intensity of the spherical wave oscillation, characterized as the squared wave amplitude

- .

drops at the rate proportional to , an example of the inverse-square law.

The wave equation is linear in *u* and it is left unaltered by translations in space and time. Therefore, we can generate a great variety of solutions by translating and summing spherical waves. Let *φ*(*ξ*, *η*, *ζ*) be an arbitrary function of three independent variables, and let the spherical wave form *F* be a delta function: that is, let *F* be a weak limit of continuous functions whose integral is unity, but whose support (the region where the function is non-zero) shrinks to the origin. Let a family of spherical waves have center at (*ξ*, *η*, *ζ*), and let *r* be the radial distance from that point. Thus

If *u* is a superposition of such waves with weighting function *φ*, then

the denominator 4*πc* is a convenience.

From the definition of the delta function, *u* may also be written as

where *α*, *β*, and *γ* are coordinates on the unit sphere *S*, and *ω* is the area element on *S*. This result has the interpretation that *u*(*t*, *x*) is *t* times the mean value of *φ* on a sphere of radius *ct* centered at *x*:

It follows that

The mean value is an even function of *t*, and hence if

then

These formulas provide the solution for the initial-value problem for the wave equation. They show that the solution at a given point *P*, given (*t*, *x*, *y*, *z*) depends only on the data on the sphere of radius *ct* that is intersected by the **light cone** drawn backwards from *P*. It does *not* depend upon data on the interior of this sphere. Thus the interior of the sphere is a lacuna for the solution. This phenomenon is called ** Huygens' principle **. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973).

In two space dimensions, the wave equation is

We can use the three-dimensional theory to solve this problem if we regard *u* as a function in three dimensions that is independent of the third dimension. If

then the three-dimensional solution formula becomes

where *α* and *β* are the first two coordinates on the unit sphere, and *dω* is the area element on the sphere. This integral may be rewritten as a double integral over the disc *D* with center (*x*,*y*) and radius *ct*:

It is apparent that the solution at (*t*, *x*, *y*) depends not only on the data on the light cone where

but also on data that are interior to that cone.

We want to find solutions to *u _{tt}* − Δ

Assume *n* ≥ 3 is an odd integer and *g* ∈ *C*^{m+1}(**R**^{n}), *h* ∈ *C ^{m}*(

then

*u*∈*C*^{2}(**R**^{n}× [0, ∞))*u*− Δ_{tt}*u*= 0 in**R**^{n}× (0, ∞)

Assume *n* ≥ 2 is an even integer and *g* ∈ *C*^{m+1}(**R**^{n}), *h* ∈ *C ^{m}*(

then

*u*∈*C*^{2}(**R**^{n}× [0, ∞))*u*− Δ_{tt}*u*= 0 in**R**^{n}× (0, ∞)

A flexible string that is stretched between two points *x* = 0 and *x* = *L* satisfies the wave equation for *t* > 0 and 0 < *x* < *L*. On the boundary points, *u* may satisfy a variety of boundary conditions. A general form that is appropriate for applications is

where *a* and *b* are non-negative. The case where u is required to vanish at an endpoint is the limit of this condition when the respective *a* or *b* approaches infinity. The method of separation of variables consists in looking for solutions of this problem in the special form

A consequence is that

The eigenvalue *λ* must be determined so that there is a non-trivial solution of the boundary-value problem

This is a special case of the general problem of Sturm–Liouville theory. If *a* and *b* are positive, the eigenvalues are all positive, and the solutions are trigonometric functions. A solution that satisfies square-integrable initial conditions for *u* and *u _{t}* can be obtained from expansion of these functions in the appropriate trigonometric series.

Approximating the continuous string with a finite number of equidistant mass points one gets the following physical model:

If each mass point has the mass *m*, the tension of the string is *f*, the separation between the mass points is Δ*x* and *u _{i}*,

**(1)**

and the vertical component of the force towards point *i* − 1 is

**(2)**

Taking the sum of these two forces and dividing with the mass *m* one gets for the vertical motion:

**(3)**

As the mass density is

this can be written

**(4)**

The wave equation is obtained by letting Δ*x* → 0 in which case *u _{i}*(

But the discrete formulation (** 3 **) of the equation of state with a finite number of mass point is just the suitable one for a numerical propagation of the string motion. The boundary condition

where *L* is the length of the string takes in the discrete formulation the form that for the outermost points *u*_{1} and *u _{n}* the equation of motion are

**(5)**

and

**(6)**

while for 1 < *i* < *n*

**(7)**

where

If the string is approximated with 100 discrete mass points one gets the 100 coupled second order differential equations (** 5 **), (** 6 **) and (** 7 **) or equivalently 200 coupled first order differential equations.

Propagating these up to the times

using an 8th order multistep method the 6 states displayed in figure 2 are found:

The red curve is the initial state at time zero at which the string is "let free" in a predefined shape^{ [12] } with all . The blue curve is the state at time i.e. after a time that corresponds to the time a wave that is moving with the nominal wave velocity would need for one fourth of the length of the string.

Figure 3 displays the shape of the string at the times . The wave travels in direction right with the speed without being actively constraint by the boundary conditions at the two extremes of the string. The shape of the wave is constant, i.e. the curve is indeed of the form *f*(*x* − *ct*).

Figure 4 displays the shape of the string at the times . The constraint on the right extreme starts to interfere with the motion preventing the wave to raise the end of the string.

Figure 5 displays the shape of the string at the times when the direction of motion is reversed. The red, green and blue curves are the states at the times while the 3 black curves correspond to the states at times with the wave starting to move back towards left.

Figure 6 and figure 7 finally display the shape of the string at the times and . The wave now travels towards left and the constraints at the end points are not active any more. When finally the other extreme of the string the direction will again be reversed in a way similar to what is displayed in figure 6.

The one-dimensional initial-boundary value theory may be extended to an arbitrary number of space dimensions. Consider a domain *D* in *m*-dimensional *x* space, with boundary *B*. Then the wave equation is to be satisfied if *x* is in *D* and *t* > 0. On the boundary of *D*, the solution *u* shall satisfy

where *n* is the unit outward normal to *B*, and *a* is a non-negative function defined on *B*. The case where *u* vanishes on *B* is a limiting case for *a* approaching infinity. The initial conditions are

where *f* and *g* are defined in *D*. This problem may be solved by expanding *f* and *g* in the eigenfunctions of the Laplacian in *D*, which satisfy the boundary conditions. Thus the eigenfunction *v* satisfies

in *D*, and

on *B*.

In the case of two space dimensions, the eigenfunctions may be interpreted as the modes of vibration of a drumhead stretched over the boundary *B*. If *B* is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle *θ*, multiplied by a Bessel function (of integer order) of the radial component. Further details are in Helmholtz equation.

If the boundary is a sphere in three space dimensions, the angular components of the eigenfunctions are spherical harmonics, and the radial components are Bessel functions of half-integer order.

The inhomogeneous wave equation in one dimension is the following:

with initial conditions given by

The function *s*(*x*, *t*) is often called the source function because in practice it describes the effects of the sources of waves on the medium carrying them. Physical examples of source functions include the force driving a wave on a string, or the charge or current density in the Lorenz gauge of electromagnetism.

One method to solve the initial value problem (with the initial values as posed above) is to take advantage of a special property of the wave equation in an odd number of space dimensions, namely that its solutions respect causality. That is, for any point (*x _{i}*,

In terms of finding a solution, this causality property means that for any given point on the line being considered, the only area that needs to be considered is the area encompassing all the points that could causally affect the point being considered. Denote the area that casually affects point (*x _{i}*,

To simplify this greatly, we can use Green's theorem to simplify the left side to get the following:

The left side is now the sum of three line integrals along the bounds of the causality region. These turn out to be fairly easy to compute

In the above, the term to be integrated with respect to time disappears because the time interval involved is zero, thus *dt* = 0.

For the other two sides of the region, it is worth noting that *x* ± *ct* is a constant, namely *x _{i}* ±

And similarly for the final boundary segment:

Adding the three results together and putting them back in the original integral:

Solving for *u*(*x _{i}*,

In the last equation of the sequence, the bounds of the integral over the source function have been made explicit. Looking at this solution, which is valid for all choices (*x _{i}*,

In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.

The elastic wave equation (also known as the Navier-Cauchy equation) in three dimensions describes the propagation of waves in an isotropic homogeneous elastic medium. Most solid materials are elastic, so this equation describes such phenomena as seismic waves in the Earth and ultrasonic waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:

where:

*λ*and*μ*are the so-called Lamé parameters describing the elastic properties of the medium,*ρ*is the density,**f**is the source function (driving force),- and
**u**is the displacement vector.

By using the elastic wave equation can be rewritten into the more common form of the Navier-Cauchy equation.

Note that in the elastic wave equation, both force and displacement are vector quantities. Thus, this equation is sometimes known as the vector wave equation. As an aid to understanding, the reader will observe that if **f** and ∇ ⋅ **u** are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field **E**, which has only transverse waves.

In dispersive wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a dispersion relation

where *ω* is the angular frequency and **k** is the wavevector describing plane wave solutions. For light waves, the dispersion relation is *ω* = ±*c*|**k**|, but in general, the constant speed *c* gets replaced by a variable phase velocity:

- Acoustic attenuation
- Acoustic wave equation
- Bateman transform
- Electromagnetic wave equation
- Helmholtz equation
- Inhomogeneous electromagnetic wave equation
- Laplace operator
- Mathematics of oscillation
- Maxwell's equations
- Schrödinger equation
- Standing wave
- Vibrations of a circular membrane
- Wheeler–Feynman absorber theory

- ↑ Cannon, John T.; Dostrovsky, Sigalia (1981).
*The evolution of dynamics, vibration theory from 1687 to 1742*. Studies in the History of Mathematics and Physical Sciences.**6**. New York: Springer-Verlag. pp. ix + 184 pp. ISBN 978-0-3879-0626-3.GRAY, JW (July 1983). "BOOK REVIEWS".*Bulletin (New Series) of the American Mathematical Society*.**9**(1). (retrieved 13 Nov 2012). - ↑ Gerard F Wheeler. The Vibrating String Controversy, (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33–37.
- ↑ For a special collection of the 9 groundbreaking papers by the three authors, see First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. – the controversy about vibrating strings (retrieved 13 Nov 2012). Herman HJ Lynge and Son.
- ↑ For de Lagrange's contributions to the acoustic wave equation, one can consult Acoustics: An Introduction to Its Physical Principles and Applications Allan D. Pierce, Acoustical Soc of America, 1989; page 18 (retrieved 9 Dec 2012).
- 1 2 3 Speiser, David.
*Discovering the Principles of Mechanics 1600–1800*, p. 191 (Basel: Birkhäuser, 2008). - ↑ Tipler, Paul and Mosca, Gene.
*Physics for Scientists and Engineers, Volume 1: Mechanics, Oscillations and Waves; Thermodynamics*, pp. 470–471 (Macmillan, 2004). - ↑ Eric W. Weisstein. "d'Alembert's Solution". MathWorld . Retrieved 2009-01-21.
- ↑ D'Alembert (1747) "Recherches sur la courbe que forme une corde tenduë mise en vibration" (Researches on the curve that a tense cord forms [when] set into vibration),
*Histoire de l'académie royale des sciences et belles lettres de Berlin*, vol. 3, pages 214–219.- See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further researches on the curve that a tense cord forms [when] set into vibration),
*Histoire de l'académie royale des sciences et belles lettres de Berlin*, vol. 3, pages 220–249. - See also: D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration,"
*Histoire de l'académie royale des sciences et belles lettres de Berlin*, vol. 6, pages 355–360.

- See also: D'Alembert (1747) "Suite des recherches sur la courbe que forme une corde tenduë mise en vibration" (Further researches on the curve that a tense cord forms [when] set into vibration),
- ↑ http://math.arizona.edu/~kglasner/math456/linearwave.pdf.
- ↑ V Guruprasad (2015), "Observational evidence for travelling wave modes bearing distance proportional shifts",
*EPL*,**110**(5): 54001, arXiv: 1507.08222 , Bibcode:2015EL....11054001G, doi:10.1209/0295-5075/110/54001 - ↑ John David Jackson, Classical Electrodynamics, 3rd Edition, Wiley, page 425. ISBN 978-0-471-30932-1
- ↑ The initial state for "Investigation by numerical methods" is set with quadratic splines as follows:
- for
- for
- for

**Acoustic theory** is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

The **primitive equations** are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:

- A
*continuity equation*: Representing the conservation of mass. *Conservation of momentum*: Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere- A
*thermal energy equation*: Relating the overall temperature of the system to heat sources and sinks

In physics, a **wave packet** is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere. Each component wave function, and hence the wave packet, are solutions of a wave equation. Depending on the wave equation, the wave packet's profile may remain constant or it may change (dispersion) while propagating.

In mathematics, the **method of characteristics** is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.

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In mathematics and physics, the **Helmholtz equation**, named for Hermann von Helmholtz, is the linear partial differential equation

In mathematics, a **hyperbolic partial differential equation** of order *n* is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first *n* − 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In physics, the **acoustic wave equation** governs the propagation of acoustic waves through a material medium. The form of the equation is a second order partial differential equation. The equation describes the evolution of acoustic pressure or particle velocity * u* as a function of position

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

**Weak formulations** are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, an equation is no longer required to hold absolutely and has instead **weak solutions** only with respect to certain "test vectors" or "test functions". This is equivalent to formulating the problem to require a solution in the sense of a distribution.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

In fluid dynamics, the **mild-slope equation** describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. 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 often used in coastal engineering to compute the wave-field changes near harbours and coasts.

In the finite element method for the numerical solution of elliptic partial differential equations, the **stiffness matrix** represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

**Multipole radiation** is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

**Q-vectors** are used in atmospheric dynamics to understand physical processes such as vertical motion and frontogenesis. Q-vectors are not physical quantities that can be measured in the atmosphere but are derived from the quasi-geostrophic equations and can be used in the previous diagnostic situations. On meteorological charts, Q-vectors point toward upward motion and away from downward motion. Q-vectors are an alternative to the omega equation for diagnosing vertical motion in the quasi-geostrophic equations.

In optics, the **Ewald–Oseen extinction theorem**, sometimes referred to as just "extinction theorem", is a theorem that underlies the common understanding of scattering. It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline and isotropic media, respectively, in 1916 and 1915. Originally, the theorem applied to scattering by an isotropic dielectric objects in free space. The scope of the theorem was greatly extended to encompass a wide variety of bianisotropic media.

In fluid dynamics, **Beltrami flows** are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow where Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

- M. F. Atiyah, R. Bott, L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients I",
*Acta Math.*,**124**(1970), 109–189. - M.F. Atiyah, R. Bott, and L. Garding, "Lacunas for hyperbolic differential operators with constant coefficients II",
*Acta Math.*,**131**(1973), 145–206. - R. Courant, D. Hilbert,
*Methods of Mathematical Physics, vol II*. Interscience (Wiley) New York, 1962. - L. Evans, "Partial Differential Equations". American Mathematical Society Providence, 1998.
- "Linear Wave Equations",
*EqWorld: The World of Mathematical Equations.* - "Nonlinear Wave Equations",
*EqWorld: The World of Mathematical Equations.* - William C. Lane, "MISN-0-201 The Wave Equation and Its Solutions",
*Project PHYSNET*.

- Nonlinear Wave Equations by Stephen Wolfram and Rob Knapp, Nonlinear Wave Equation Explorer by Wolfram Demonstrations Project.
- Mathematical aspects of wave equations are discussed on the Dispersive PDE Wiki.
- Graham W Griffiths and William E. Schiesser (2009). Linear and nonlinear waves. Scholarpedia, 4(7):4308. doi:10.4249/scholarpedia.4308

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