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

**Hermann Ludwig Ferdinand von Helmholtz** was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of research institutions, the Helmholtz Association, is named after him.

In mathematics, a **partial differential equation** (**PDE**) is a differential equation that contains beforehand 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.

- Motivation and uses
- Solving the Helmholtz equation using separation of variables
- Vibrating membrane
- Three-dimensional solutions
- Paraxial approximation
- Inhomogeneous Helmholtz equation
- See also
- Notes
- References
- External links

where is the Laplacian, is the wave number, and is the amplitude. This is also an eigenvalue equation.

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}, or Δ. The Laplacian Δ*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 grows. 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.

In the physical sciences, the **wavenumber** is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. Whereas temporal frequency can be thought of as the number of waves per unit time, wavenumber is the number of waves per unit distance.

The **amplitude** of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a **time-independent** form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis.

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 or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

In mathematics, **separation of variables** is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

For example, consider the wave equation

Separation of variables begins by assuming that the wave function is in fact separable:

Substituting this form into the wave equation and then simplifying, we obtain the following equation:

Notice that the expression on the left side depends only on , whereas the right expression depends only on . As a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. This argument is key in the technique of solving linear partial differential equations by separation of variables. From this observation, we obtain two equations, one for , the other for :

and

where we have chosen, without loss of generality, the expression for the value of the constant. (It is equally valid to use any constant as the separation constant; is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

Likewise, after making the substitution , where is the wave number, and is the angular frequency, the second equation becomes

In physics, **angular frequency***ω* is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

We now have Helmholtz's equation for the spatial variable and a second-order ordinary differential equation in time. The solution in time will be a linear combination of sine and cosine functions, whose exact form is determined by initial conditions, while the form of the solution in space will depend on the boundary conditions. Alternatively, integral transforms, such as the Laplace or Fourier transform, are often used to transform a hyperbolic PDE into a form of the Helmholtz equation.

In mathematics, an **ordinary differential equation** (**ODE**) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term *ordinary* is used in contrast with the term partial differential equation which may be with respect to *more than* one independent variable.

In mathematics, a **linear combination** is an expression constructed from a set of terms by multiplying each term by a constant and adding the results. The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

In mathematics, the **sine** is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle.

Because of its relationship to the wave equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology, and acoustics.

**Physics** is the natural science that studies matter and its motion and behavior through space and time and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

In physics, **electromagnetic radiation** refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

**Seismology** is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies. The field also includes studies of earthquake environmental effects such as tsunamis as well as diverse seismic sources such as volcanic, tectonic, oceanic, atmospheric, and artificial processes such as explosions. A related field that uses geology to infer information regarding past earthquakes is paleoseismology. A recording of earth motion as a function of time is called a seismogram. A seismologist is a scientist who does research in seismology.

The solution to the spatial Helmholtz equation

can be obtained for simple geometries using separation of variables.

The two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The Helmholtz equation was solved for many basic shapes in the 19th century: the rectangular membrane by Siméon Denis Poisson in 1829, the equilateral triangle by Gabriel Lamé in 1852, and the circular membrane by Alfred Clebsch in 1862. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieu's differential equation.

If the edges of a shape are straight line segments, then a solution is integrable or knowable in closed-form only if it is expressible as a finite linear combination of plane waves that satisfy the boundary conditions (zero at the boundary, i.e., membrane clamped).

If the domain is a circle of radius *a*, then it is appropriate to introduce polar coordinates *r* and θ. The Helmholtz equation takes the form

We may impose the boundary condition that *A* vanish if *r* = *a*; thus

The method of separation of variables leads to trial solutions of the form

where Θ must be periodic of period 2π. This leads to

and

It follows from the periodicity condition that

and that *n* must be an integer. The radial component *R* has the form

where the Bessel function *J _{n}*(

and *ρ* = *kr*. The radial function *J*_{n} has infinitely many roots for each value of *n*, denoted by *ρ*_{m,n}. The boundary condition that *A* vanishes where *r* = *a* will be satisfied if the corresponding wavenumbers are given by

The general solution *A* then takes the form of a doubly infinite sum of terms involving products of

These solutions are the modes of vibration of a circular drumhead.

In spherical coordinates, the solution is:

This solution arises from the spatial solution of the wave equation and diffusion equation. Here and are the spherical Bessel functions, and

are the spherical harmonics (Abramowitz and Stegun, 1964). Note that these forms are general solutions, and require boundary conditions to be specified to be used in any specific case. For infinite exterior domains, a radiation condition may also be required (Sommerfeld, 1949).

Writing function has asymptotics

where function *f* is called scattering amplitude and is the value of *A* at each boundary point .

In the paraxial approximation of the Helmholtz equation,^{ [1] } the complex amplitude *A* is expressed as

where *u* represents the complex-valued amplitude which modulates the sinusoidal plane wave represented by the exponential factor. Then under a suitable assumption, *u* approximately solves

where is the transverse part of the Laplacian.

This equation has important applications in the science of optics, where it provides solutions that describe the propagation of electromagnetic waves (light) in the form of either paraboloidal waves or Gaussian beams. Most lasers emit beams that take this form.

The assumption under which the paraxial approximation is valid is that the *z* derivative of the amplitude function *u* is a slowly-varying function of *z*:

This condition is equivalent to saying that the angle θ between the wave vector **k** and the optical axis *z* is small: .

The paraxial form of the Helmholtz equation is found by substituting the above-stated expression for the complex amplitude into the general form of the Helmholtz equation as follows:

Expansion and cancellation yields the following:

Because of the paraxial inequalitiy stated above, the ∂^{2}u/∂z^{2} term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting then gives the paraxial equation for the original complex amplitude *A*:

The Fresnel diffraction integral is an exact solution to the paraxial Helmholtz equation.^{ [2] }

There is even a subject named "Helmholtz optics" based on the equation, named in honour of Helmholtz. ^{ [3] }^{ [4] }^{ [5] }

The **inhomogeneous Helmholtz equation** is the equation

where *ƒ* : **R**^{n} → **C** is a function with compact support, and *n* = 1, 2, 3. This equation is very similar to the screened Poisson equation, and would be identical if the plus sign (in front of the *k* term) is switched to a minus sign.

In order to solve this equation uniquely, one needs to specify a boundary condition at infinity, which is typically the Sommerfeld radiation condition

uniformly in with , where the vertical bars denote the Euclidean norm.

With this condition, the solution to the inhomogeneous Helmholtz equation is the convolution

(notice this integral is actually over a finite region, since has compact support). Here, is the Green's function of this equation, that is, the solution to the inhomogeneous Helmholtz equation with ƒ equaling the Dirac delta function, so *G* satisfies

The expression for the Green's function depends on the dimension of the space. One has

for *n* = 1,

for *n* = 2,^{ [6] } where is a Hankel function, and

for *n* = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for .

- Laplace's equation (a particular case of the Helmholtz equation)

- ↑ J. W. Goodman.
*Introduction to Fourier Optics*(2nd ed.). pp. 61–62. - ↑ Grella, R. (1982). "Fresnel propagation and diffraction and paraxial wave equation".
*Journal of Optics*.**13**(6): 367–374. doi:10.1088/0150-536X/13/6/006. - ↑ Kurt Bernardo Wolf and Evgenii V. Kurmyshev, Squeezed states in Helmholtz optics, Physical Review A 47, 3365–3370 (1993).
- ↑ Sameen Ahmed Khan, Wavelength-dependent modifications in Helmholtz Optics, International Journal of Theoretical Physics, 44(1), 95http://www.maa.org/programs/maa-awards/writing-awards/can-one-hear-the-shape-of-a-drum125 (January 2005).
- ↑ Sameen Ahmed Khan, A Profile of Hermann von Helmholtz, Optics & Photonics News, Vol. 21, No. 7, pp. 7 (July/August 2010).
- ↑ ftp://ftp.math.ucla.edu/pub/camreport/cam14-71.pdf

**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 mathematics, **Laplace's equation** is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

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

In fluid dynamics, **potential flow** describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

In continuum mechanics, the **infinitesimal strain theory** is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

In mathematics and physical science, **spherical harmonics** are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

**Linear elasticity** is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of continuum mechanics. The fundamental "linearizing" assumptions of linear elasticity are: infinitesimal strains or "small" deformations and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis.

**Fourier optics** is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or *superposition*, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.

**Geometrical optics**, or **ray optics**, describes light propagation in terms of rays. The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.

This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In quantum physics, the **scattering amplitude** is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.

**Stokes flow**, also named **creeping flow** or **creeping motion**, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

In physics and mathematics, the **solid harmonics** are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the *regular solid harmonics* , which vanish at the origin and the *irregular solid harmonics* , which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

**Relativistic heat conduction** refers to the modelling of heat conduction in a way compatible with special relativity. This article discusses models using a wave equation with a dissipative term.

**Lagrangian mechanics** is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

**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.

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

- Abramowitz, Milton; Stegun, Irene, eds. (1964).
*Handbook of Mathematical functions with Formulas, Graphs and Mathematical Tables*. New York: Dover Publications. ISBN 978-0-486-61272-0.

- Riley, K. F.; Hobson, M. P.; Bence, S. J. (2002). "Chapter 19".
*Mathematical methods for physics and engineering*. New York: Cambridge University Press. ISBN 978-0-521-89067-0.

- Riley, K. F. (2002). "Chapter 16".
*Mathematical Methods for Scientists and Engineers*. Sausalito, California: University Science Books. ISBN 978-1-891389-24-5.

- Saleh, Bahaa E. A.; Teich, Malvin Carl (1991). "Chapter 3".
*Fundamentals of Photonics*. Wiley Series in Pure and Applied Optics. New York: John Wiley & Sons. pp. 80–107. ISBN 978-0-471-83965-1.

- Sommerfeld, Arnold (1949). "Chapter 16".
*Partial Differential Equations in Physics*. New York: Academic Press. ISBN 978-0126546569.

- Howe, M. S. (1998).
*Acoustics of fluid-structure interactions*. New York: Cambridge University Press. ISBN 978-0-521-63320-8.

- Helmholtz Equation at EqWorld: The World of Mathematical Equations.
- Hazewinkel, Michiel, ed. (2001) [1994], "Helmholtz equation",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Vibrating Circular Membrane by Sam Blake, The Wolfram Demonstrations Project.
- Green's functions for the wave, Helmholtz and Poisson equations in a two-dimensional boundless domain

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