# Helmholtz equation

Last updated

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.

## Contents

${\displaystyle (\nabla ^{2}+k^{2})A=0,}$

where ${\displaystyle \nabla ^{2}}$ is the Laplacian, ${\displaystyle k}$ is the wave number, and ${\displaystyle A}$ 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 Δf(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.

## Motivation and uses

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

${\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right)u(\mathbf {r} ,t)=0.}$

Separation of variables begins by assuming that the wave function ${\displaystyle u(\mathbf {r} ,t)}$is in fact separable:

${\displaystyle u(\mathbf {r} ,t)=A(\mathbf {r} )T(t).}$

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

${\displaystyle {\frac {\nabla ^{2}A}{A}}={\frac {1}{c^{2}T}}{\frac {d^{2}T}{dt^{2}}}.}$

Notice that the expression on the left side depends only on ${\displaystyle \mathbf {r} }$, whereas the right expression depends only on ${\displaystyle t}$. 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 ${\displaystyle A(\mathbf {r} )}$, the other for ${\displaystyle T(t)}$:

${\displaystyle {\frac {\nabla ^{2}A}{A}}=-k^{2}}$

and

${\displaystyle {\frac {1}{c^{2}T}}{\frac {d^{2}T}{dt^{2}}}=-k^{2},}$

where we have chosen, without loss of generality, the expression ${\displaystyle -k^{2}}$ for the value of the constant. (It is equally valid to use any constant ${\displaystyle k}$ as the separation constant; ${\displaystyle -k^{2}}$ is chosen only for convenience in the resulting solutions.)

Rearranging the first equation, we obtain the Helmholtz equation:

${\displaystyle \nabla ^{2}A+k^{2}A=(\nabla ^{2}+k^{2})A=0.}$

Likewise, after making the substitution ${\displaystyle \omega =kc}$, where ${\displaystyle k}$ is the wave number, and ${\displaystyle \omega }$ is the angular frequency, the second equation becomes

In physics, angular frequencyω 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.

${\displaystyle {\frac {d^{2}T}{dt^{2}}}+\omega ^{2}T=\left({\frac {d^{2}}{dt^{2}}}+\omega ^{2}\right)T=0,}$

We now have Helmholtz's equation for the spatial variable ${\displaystyle \mathbf {r} }$ 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.

## Solving the Helmholtz equation using separation of variables

The solution to the spatial Helmholtz equation

${\displaystyle (\nabla ^{2}+k^{2})A=0}$

can be obtained for simple geometries using separation of variables.

### Vibrating membrane

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

${\displaystyle A_{rr}+{\frac {1}{r}}A_{r}+{\frac {1}{r^{2}}}A_{\theta \theta }+k^{2}A=0.}$

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

${\displaystyle A(a,\theta )=0.\,}$

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

${\displaystyle A(r,\theta )=R(r)\Theta (\theta ),\,}$

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

${\displaystyle \Theta ''+n^{2}\Theta =0,\,}$

and

${\displaystyle r^{2}R''+rR'+r^{2}k^{2}R-n^{2}R=0.\,}$

It follows from the periodicity condition that

${\displaystyle \Theta =\alpha \cos n\theta +\beta \sin n\theta ,\,}$

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

${\displaystyle R(r)=\gamma J_{n}(\rho ),\,}$

where the Bessel function Jn(ρ) satisfies Bessel's equation

${\displaystyle \rho ^{2}J_{n}''+\rho J_{n}'+(\rho ^{2}-n^{2})J_{n}=0,}$

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

${\displaystyle k_{m,n}={\frac {1}{a}}\rho _{m,n}.\,}$

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

${\displaystyle \sin(n\theta ){\text{ or }}\cos(n\theta ),{\text{ and }}J_{n}(k_{m,n}r).}$

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

### Three-dimensional solutions

In spherical coordinates, the solution is:

${\displaystyle A(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(a_{\ell m}j_{\ell }(kr)+b_{\ell m}y_{\ell }(kr)\right)Y_{\ell }^{m}(\theta ,\varphi ).}$

This solution arises from the spatial solution of the wave equation and diffusion equation. Here ${\displaystyle j_{\ell }(kr)}$ and ${\displaystyle y_{\ell }(kr)}$ are the spherical Bessel functions, and

${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}$

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 ${\displaystyle \mathbf {r_{0}} =(x,y,z)}$ function ${\displaystyle A(r_{0})}$ has asymptotics

${\displaystyle A(r_{0})={\frac {e^{ikr_{0}}}{r_{0}}}f\left({\frac {\mathbf {r} _{0}}{r_{0}}},k,u_{0}\right)+o\left({\frac {1}{r_{0}}}\right){\text{ as }}r_{0}\to \infty }$

where function f is called scattering amplitude and ${\displaystyle u_{0}(r_{0})}$ is the value of A at each boundary point ${\displaystyle r_{0}}$.

## Paraxial approximation

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

${\displaystyle A(\mathbf {r} )=u(\mathbf {r} )e^{ikz}}$

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

${\displaystyle \nabla _{\perp }^{2}u+2ik{\frac {\partial u}{\partial z}}=0,}$

where ${\displaystyle \textstyle \nabla _{\perp }^{2}{\stackrel {\mathrm {def} }{=}}{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}}$ 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:

${\displaystyle {\bigg |}{\partial ^{2}u \over \partial z^{2}}{\bigg |}\ll {\bigg |}{k{\partial u \over \partial z}}{\bigg |}.}$

This condition is equivalent to saying that the angle θ between the wave vector k and the optical axis z is small: ${\displaystyle \theta \ll 1}$.

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:

${\displaystyle \nabla ^{2}(u\left(x,y,z\right)e^{ikz})+k^{2}u\left(x,y,z\right)e^{ikz}=0.}$

Expansion and cancellation yields the following:

${\displaystyle \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)u(x,y,z)e^{ikz}+\left({\frac {\partial ^{2}}{\partial z^{2}}}u(x,y,z)\right)e^{ikz}+2\left({\frac {\partial }{\partial z}}u(x,y,z)\right)ik{e^{ikz}}=0.}$

Because of the paraxial inequalitiy stated above, the ∂2u/∂z2 term is neglected in comparison with the k·∂u/∂z term. This yields the paraxial Helmholtz equation. Substituting ${\displaystyle u(\mathbf {r} )=A(\mathbf {r} )e^{-ikz}}$ then gives the paraxial equation for the original complex amplitude A:

${\displaystyle \nabla _{\perp }^{2}A+2ik{\frac {\partial A}{\partial z}}+2k^{2}A=0.}$

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]

## Inhomogeneous Helmholtz equation

The inhomogeneous Helmholtz equation is the equation

${\displaystyle \nabla ^{2}A(x)+k^{2}A(x)=-f(x)\ {\text{ in }}\mathbb {R} ^{n},}$

where ƒ : Rn  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

${\displaystyle \lim _{r\to \infty }r^{\frac {n-1}{2}}\left({\frac {\partial }{\partial r}}-ik\right)A(r{\hat {x}})=0}$

uniformly in ${\displaystyle {\hat {x}}}$ with ${\displaystyle |{\hat {x}}|=1}$, where the vertical bars denote the Euclidean norm.

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

${\displaystyle A(x)=(G*f)(x)=\int \limits _{\mathbb {R} ^{n}}\!G(x-y)f(y)\,dy}$

(notice this integral is actually over a finite region, since ${\displaystyle f}$ has compact support). Here, ${\displaystyle G}$ 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

${\displaystyle \nabla ^{2}G(x)+k^{2}G(x)=-\delta (x){\text{ in }}\mathbb {R} ^{n}.\,}$

The expression for the Green's function depends on the dimension ${\displaystyle n}$ of the space. One has

${\displaystyle G(x)={\frac {ie^{ik|x|}}{2k}}}$

for n = 1,

${\displaystyle G(x)={\frac {i}{4}}H_{0}^{(1)}(k|x|)}$

for n = 2, [6] where ${\displaystyle H_{0}^{(1)}}$ is a Hankel function, and

${\displaystyle G(x)={\frac {e^{ik|x|}}{4\pi |x|}}}$

for n = 3. Note that we have chosen the boundary condition that the Green's function is an outgoing wave for ${\displaystyle |x|\to \infty }$.

## Notes

1. J. W. Goodman. Introduction to Fourier Optics (2nd ed.). pp. 61–62.
2. 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.
3. Kurt Bernardo Wolf and Evgenii V. Kurmyshev, Squeezed states in Helmholtz optics, Physical Review A 47, 3365–3370 (1993).
4. 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).
5. Sameen Ahmed Khan, A Profile of Hermann von Helmholtz, Optics & Photonics News, Vol. 21, No. 7, pp. 7 (July/August 2010).

## Related Research Articles

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.